Recent zbMATH articles in MSC 34https://www.zbmath.org/atom/cc/342022-05-16T20:40:13.078697ZUnknown authorWerkzeugPrefacehttps://www.zbmath.org/1483.000382022-05-16T20:40:13.078697ZFrom the text: The present volume is a Special Issue on Optimization and Differential Equations. The idea for this publication emerged from discussions at the conference IMAME 2019, International Meeting on Applied Mathematics \& Evolution, held in La Rochelle in April 2019.Extremal quasimodular forms of lower depth with integral Fourier coefficientshttps://www.zbmath.org/1483.110842022-05-16T20:40:13.078697Z"Kaminaka, Tsudoi"https://www.zbmath.org/authors/?q=ai:kaminaka.tsudoi"Kato, Fumiharu"https://www.zbmath.org/authors/?q=ai:kato.fumiharuSummary: We show that, based on Grabner's recent results on modular differential equations satisfied by quasimodular forms, there exist only finitely many normalized extremal quasimodular forms of depth \(r\) that have all Fourier coefficients integral for each of \(r = 1, 2, 3\) and \(4\), and partly classifies them, where the classification is complete for \(r = 2, 3\) and \(4\). In fact, we show that there exists no normalized extremal quasimodular forms of depth four with all Fourier coefficients integral. Our result disproves a conjecture by \textit{F. Pellarin} [Kyushu J. Math. 74, No. 2, 401--413 (2020; Zbl 1470.11075)].Generalized weakly singular Gronwall-type inequalities and their applications to fractional differential equationshttps://www.zbmath.org/1483.260102022-05-16T20:40:13.078697Z"Nguyen, Minh Dien"https://www.zbmath.org/authors/?q=ai:nguyen.minh-dienIn this paper, the author obtains some new generalized Gronwall-type inequalities with respect to an appropriate function. A weak-type singular Gronwall-type inequality involving an appropriate function was given. Moreover, a Gronwall-type inequality in which the right-hand side includes two integrals with doubly singular kernels was established. Finally, some applications of the above results were presented to investigate the existence and stability of solutions of fractional differential equations involving \(\psi\)-Caputo and \(\psi\)-Hilfer fractional derivatives.
Reviewer: Chuanzhi Bai (Huaian)On the spatial Julia set generated by fractional Lotka-Volterra system with noisehttps://www.zbmath.org/1483.280112022-05-16T20:40:13.078697Z"Wang, Yupin"https://www.zbmath.org/authors/?q=ai:wang.yupin"Liu, Shutang"https://www.zbmath.org/authors/?q=ai:liu.shutang"Li, Hui"https://www.zbmath.org/authors/?q=ai:li.hui.3|li.hui.1|li.hui.4|li.hui.5|li.hui|li.hui.2"Wang, Da"https://www.zbmath.org/authors/?q=ai:wang.daSummary: This paper investigates the structures and properties of the spatial Julia set generated by a fractional complex Lotka-Volterra system with noise. The influence of several types of dynamic noise upon the system's Julia set is quantitatively analyzed through the Julia deviation index. Then, the symmetry of the Julia set is discussed and the symmetrical structure destruction caused by noise is studied. Numerical simulations are presented to further verify the correctness and effectiveness of the main theoretical results.Properties of analytic solutions of three similar differential equations of the second orderhttps://www.zbmath.org/1483.300482022-05-16T20:40:13.078697Z"Sheremeta, M. M."https://www.zbmath.org/authors/?q=ai:sheremeta.myroslav-m"Trukhan, Yu. S."https://www.zbmath.org/authors/?q=ai:trukhan.yu-sSummary: An analytic univalent in \(\mathbb{D}=\{z:\;|z|<1\}\) function \(f(z)\) is said to be convex if \(f(\mathbb{D})\) is a convex domain. It is well known that the condition \(\operatorname{Re}\{1+zf''(z)/f'(z)\}>0\), \(z\in\mathbb{D} \), is necessary and sufficient for the convexity of \(f\). The function \(f\) is said to be close-to-convex in \(\mathbb{D}\) if there exists a convex in \(\mathbb{D}\) function \(\Phi\) such that \(\operatorname{Re}(f'(z)/\Phi'(z))>0\), \(z\in\mathbb{D} \).
S.M. Shah indicated conditions on real parameters \(\beta_0\), \(\beta_1\), \(\gamma_0\), \(\gamma_1\), \(\gamma_2\) of the differential equation \(z^2w''+(\beta_0 z^2+\beta_1 z)w'+(\gamma_0z^2+\gamma_1 z+\gamma_2) w=0\), under which there exists an entire transcendental solution \(f\) such that \(f\) and all its derivatives are close-to-convex in \(\mathbb{D} \).
Let \(0<R\le+\infty\), \(\mathbb{D}_R=\{z:\;|z|<R\}\) and \(l\) be a positive continuous function on \([0,R)\), which satisfies \((R-r)l(r)>C\), \(C=\operatorname{const}>1\). An analytic in \(\mathbb{D}_R\) function \(f\) is said to be of bounded \(l\)-index if there exists \(N\in\mathbb{Z}_+\) such that for all \(n\in\mathbb{Z}_+\) and \(z\in\mathbb{D}_R\)
\[\frac{|f^{(n)}(z)|}{n!l^n(|z|)}\le \max\left\{\frac{|f^{(k)}(z)|}{k!l^k(|z|)}:\;0\le k\le N\right\}.\]
Here we investigate close-to-convexity and the boundedness of the \(l\)-index for analytic in \(\mathbb{D}\) solutions of three analogues of Shah differential equation: \(z(z-1) w''+\beta z w'+\gamma w=0\), \((z-1)^2 w''+\beta z w'+\gamma w=0\) and \((1-z)^3 w''+\beta(1- z) w'+\gamma w=0\). Despite the similarity of these equations, their solutions have different properties.On meromorphic solutions of nonlinear delay-differential equationshttps://www.zbmath.org/1483.300632022-05-16T20:40:13.078697Z"Mao, Zhiqiang"https://www.zbmath.org/authors/?q=ai:mao.zhiqiang"Liu, Huifang"https://www.zbmath.org/authors/?q=ai:liu.huifangSummary: Using Cartan's second main theorem and Nevanlinna's theorem concerning a group of meromorphic functions, we obtain the growth and zero distribution of meromorphic solutions of the nonlinear delay-differential equation \(f^n(z) + P(z) f^{( k )}(z + \eta) = H_0(z) + H_1(z) e^{\omega_1 z^q} + \cdots + H_m(z) e^{\omega_m z^q}\), where \(n, k, q, m\) are positive integers, \( \eta, \omega_1, \cdots, \omega_m\) are complex numbers with \(\omega_1 \cdots \omega_m \neq 0\), and \(P, H_0, H_1, \cdots, H_m\) are entire functions of order less than \(q\) with \(P H_1 \cdots H_m \not\equiv 0\). Especially for \(\eta = 0\), some sufficient conditions are given to guarantee the above equation has no meromorphic solutions of few poles.Entire solutions of differential-difference equations of Fermat typehttps://www.zbmath.org/1483.300652022-05-16T20:40:13.078697Z"Hu, Peichu"https://www.zbmath.org/authors/?q=ai:hu.peichu"Wang, Wenbo"https://www.zbmath.org/authors/?q=ai:wang.wenbo"Wu, Linlin"https://www.zbmath.org/authors/?q=ai:wu.linlinSummary: In this paper, we extend some previous works by Liu et al. on the existence of transcendental entire solutions of differential-difference equations of Fermat type. In addition, we also present a precise description of the associated entire solutions.Voros coefficients of the Gauss hypergeometric differential equation with a large parameterhttps://www.zbmath.org/1483.330032022-05-16T20:40:13.078697Z"Aoki, T."https://www.zbmath.org/authors/?q=ai:aoki.tosizumi|aoki.toshihiro|aoki.toru|aoki.takayuki|aoki.takuya|aoki.takafumi|aoki.toshihiko|aoki.toshiro|aoki.toshiaki|aoki.takashi|aoki.toshiki|aoki.takaaki|aoki.takayoshi|aoki.takeshi|aoki.takahiro|aoki.toshizumi|aoki.toshitaka|aoki.terumasa|aoki.takanori|aoki.takahira|aoki.toshiyuki"Takahashi, T."https://www.zbmath.org/authors/?q=ai:takahashi.tsuguo|takahashi.tomohiro|takahashi.takashi|takahashi.takenori|takahashi.tomoya|takahashi.toshinori|takahashi.toshitake|takahashi.tomoichi|takahashi.tsutomu|takahashi.tomokuni|takahashi.toshiaki|takahashi.tokiichiro|takahashi.takuya|takahashi.takayuki|takahashi.tomihiko|takahashi.tadashi|takahashi.tetsuya|takahashi.tadataka|takahashi.tadayasu|takahashi.takehito|takahashi.tohru|takahashi.toshio|takahashi.toru-t|takahashi.takao|takahashi.tsunero|takahashi.toyofumi|takahashi.toshimi|takahashi.tatsuji|takahashi.timothy-t|takahashi.teruo|takahashi.takuhiro|takahashi.taiki|takahashi.tomonori|takahashi.tsuyoshi|takahashi.tomokazu|takahashi.toshihiko|takahashi.toru|takahashi.takaaki|takahashi.takeshi.1|takahashi.tomoyuki|takahashi.tomohiko|takahashi.takeo|takahashi.toshimitsu|takahashi.tomoki|takahashi.tomo|takahashi.toshie"Tanda, M."https://www.zbmath.org/authors/?q=ai:tanda.mario|tanda.mikaSummary: The Voros coefficient of the Gauss hypergeometric differential equation with a large parameter is defined for the origin and its explicit form and the details of derivation are given.A generalized sextic Freud weighthttps://www.zbmath.org/1483.330052022-05-16T20:40:13.078697Z"Clarkson, Peter A."https://www.zbmath.org/authors/?q=ai:clarkson.peter-a"Jordaan, Kerstin"https://www.zbmath.org/authors/?q=ai:jordaan.kerstinSummary: We discuss the recurrence coefficients of orthogonal polynomials with respect to a generalized sextic Freud weight
\[
\omega(x;t,\lambda)=|x|^{2\lambda+1} \operatorname{exp}(-x^6 + tx^2), \quad x \in \mathbb{R},
\]
with parameters \(\lambda > -1\) and \(t \in \mathbb{R}\). We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of generalized hypergeometric functions \(_1F_2(a_1;b_1;b_2;z)\). We derive a nonlinear discrete as well as a system of differential equations satisfied by the recurrence coefficients and use these to investigate their asymptotic behaviour. We conclude by highlighting a fascinating connection between generalized quartic, sextic, octic and decic Freud weights when expressing their first moments in terms of generalized hypergeometric functions.Introduction to differential and difference equationshttps://www.zbmath.org/1483.340012022-05-16T20:40:13.078697Z"Lewintan, Alexander"https://www.zbmath.org/authors/?q=ai:lewintan.alexander"Lewintan, Peter"https://www.zbmath.org/authors/?q=ai:lewintan.peterPublisher's description: In einer stärker computerisierten Welt finden Differential- und Differenzengleichungen immer mehr Anwendung. Das vorliegende Lehrbuch ist insbesondere für Studierende der ingenieurwissenschaftlichen, der informatikorientierten und der ökonomischen Studiengänge geeignet. Ausgewählte Kapitel sind auch für Schülerinnen und Schüler aus der Oberstufe mit den Leistungskursen Mathematik/Physik/Informatik interessant.
Der präsentierte Stoff entspricht einer zweistündigen Vorlesung im Grundlagenbereich, wobei Basis-Kenntnisse aus der Analysis und der Linearen Algebra vorausgesetzt sind. Die Autoren zeigen Parallelen bei den Untersuchungen von linearen Differential- und linearen Differenzengleichungen auf, wobei die Vorgehensweisen anhand von vielen Beispielen ausführlich illustriert werden. Es werden lineare Differential- und lineare Differenzengleichungen erster und zweiter Ordnung betrachtet, sowie den Leserinnen und Leser alle Werkzeuge für die Betrachtungen von Gleichungen höherer Ordnung zur Verfügung gestellt.Applied differential equations compact. For engineers and physicistshttps://www.zbmath.org/1483.340022022-05-16T20:40:13.078697Z"Oprandi, Adriano"https://www.zbmath.org/authors/?q=ai:oprandi.adrianoPublisher's description: Dieser Gesamtband stellt mehr noch als die Einzelbände mit gleichnamigem Titel die Differenzialgleichung als Bilanzgleichung einer physikalischen Grösse ins Zentrum der Betrachtung. Die Lesenden lernen Schritt für Schritt, wie ein konkret gestelltes Problem mit Hilfe sinnvoller Voraussetzungen und Idealisierungen modelliert, als Bilanz formuliert und formalisiert und die entstandene Differenzialgleichung exakt oder numerisch gelöst wird. Dieses didaktische Konzept wird durchgehend, angefangen von der Festkörperphysik über die Wärmelehre bis hin zur Strömungsmechanik, sorgfältig und konsequent für jedes Teilgebiet angewandt und ermöglicht auf diese Weise den Studierenden die Bilanz als etwas Grundlegendes zur Beschreibung eines physikalischen Sachverhalts zu begreifen. Jedes Kapitel enthält viele, praxisorientierte und vollständig gelöste Beispiele.
\begin{itemize}
\item Klar und bündig -- auch für Anwender in den Ingenieurwissenschaften und der Physik.
\item Didaktische Muster zur Modellierung praktischer Probleme.
\item Mit Beispielen und Übungen nach jedem Hauptkapitel.
\end{itemize}Hyperbolicity in delay equationshttps://www.zbmath.org/1483.340032022-05-16T20:40:13.078697Z"Barreira, Luis"https://www.zbmath.org/authors/?q=ai:barreira.luis-m"Valls, Claudia"https://www.zbmath.org/authors/?q=ai:valls.claudiaThis is a very dense book, exclusively theoretical. It covers a gap (at least up to reviewer's knowledge) in the theory of time varying linear systems (in general). It starts from the basics given by Perron condition and, more general, admissibility as viewed by Massera and Schaffer. The time delay equations are viewed within the most general framework due to J. K. Hale, having the space \(C\) as basic (another proof that, regretfully, the Hilbert space approach summarized in the book of Bensoussan, DaPrato, Delfour and Mitter is carefully avoided).
The book is based on authors' contributions and results and is organized in four parts: I. Prelude (with three chapters: 1. Introduction. 2. Basic Notions. 3. Hyperbolicity) II. Linear Stability (with three chapters: 4. Two-sided robustness. 5. Admissibility. 6. Robustness and Parameters) III. Nonlinear Stability (with two chapters: 7. Lipschitz Perturbations. 8. Smooth Invariant Manifolds) IV. Further Topics (with two chapters: 9. Center Manifolds. 10. Spectral Theory). The book is endowed with a Reference list of 142 positions.
The somehow abstract framework makes the book not very attractive to people dedicated to various applications. Nevertheless, under its excellent graphical presentation, it is highly recommendable for researchers in Functional Differential Equations. And, a short question: Neutral Equations?
Reviewer: Vladimir Răsvan (Craiova)Dynamics and bifurcations in networks designed for frequency conversionhttps://www.zbmath.org/1483.340042022-05-16T20:40:13.078697Z"In, Visarath"https://www.zbmath.org/authors/?q=ai:in.visarath"Palacios, Antonio"https://www.zbmath.org/authors/?q=ai:palacios.antonioThis is a long review article covering the concept that certain frequency up- and down-conversion patterns can be induced by the topology of connections of cascaded arrays of oscillators. The fundamental idea is to exploit the inherent symmetry of networks to produce collective behavior in which certain oscillators oscillate at different frequencies. These ideas and methods are model-independent, so similar results can be obtained with a wide range of oscillator systems so long as certain conditions, dictated by symmetry, are satisfied. Theoretical concepts using equivariant bifurcation theory are given, along with numerical and experimental results from the implementation of electronic circuits.
Reviewer: Carlo Laing (Auckland)Approaches toward understanding delay-induced stability and instabilityhttps://www.zbmath.org/1483.340052022-05-16T20:40:13.078697Z"Menon, Shreya"https://www.zbmath.org/authors/?q=ai:menon.shreya"Cain, John W."https://www.zbmath.org/authors/?q=ai:cain.john-wThis is an interesting survey on stability methods for delay differential equations. Instead of a comprehensive search, summary and comparisons on the available results on this topic, the authors aimed at inspiring the interest of students in exploring this area, and thus focused on various approaches to understanding these typical methods and results for stability and instability caused by delays. A good piece of work that deserves a recommendation to new researchers.
Reviewer: Zhanyuan Hou (London)Analytic integrability around a nilpotent singularity: the non-generic casehttps://www.zbmath.org/1483.340062022-05-16T20:40:13.078697Z"Algaba, Antonio"https://www.zbmath.org/authors/?q=ai:algaba.antonio"Díaz, María"https://www.zbmath.org/authors/?q=ai:diaz.maria-elena"García, Cristóbal"https://www.zbmath.org/authors/?q=ai:garcia.cristobal"Giné, Jaume"https://www.zbmath.org/authors/?q=ai:gine.jaumeLocal integrability of the analytic system \[ \dot x=y+\dots, \qquad \dot y= b x^n y+\dots. \tag{1} \] where the dots stand for the quasi-homogeneous components of higher order, is investigated. The following statements are proved:
1) system (1) is analytically integrable if, and only if, it is formally orbitally equivalent to the system \[ \dot x=y, \qquad \dot y= (n+1) x^n y; \]
2) system (1) is analytically integrable if, and only if, there exists a formal inverse integrating factor of system (1) of the form \(V = h +\dots\), where \(h = y(y -x^{n+1});\)
3) system (1) is analytically integrable if, and only if, there exists a vectorfield \( G = D_0 + \dots\) (where \(D_0= (x, (n + 1)y) \)) and a scalar function \(\mu\), with \(\mu(0) = n + 1 \), such that \([F,G] = \mu F.\)
Reviewer: Valery Romanovski (Maribor)Formal Weierstrass integrability for a Liénard differential systemhttps://www.zbmath.org/1483.340072022-05-16T20:40:13.078697Z"Ferčec, Brigita"https://www.zbmath.org/authors/?q=ai:fercec.brigita"Giné, Jaume"https://www.zbmath.org/authors/?q=ai:gine.jaumeIn the present paper, the authors study the following real Liénard differential system
\[
x'=y,\quad y'=-(\zeta x^{3}+\alpha x+\beta)y-(\epsilon x^{3}+\sigma x). \tag{\(*\)}
\]
The cases \(\zeta \neq 0\) and \(\zeta =0\) with \(\alpha\neq 0\) are studied here, since the case \(\zeta = \alpha =0\) is studied earlier in [\textit{J. Giné} and \textit{J. Llibre}, Electron. J. Qual. Theory Differ. Equ. 2020, Paper No. 1, 16 p. (2020; Zbl 1449.34001)]. The authors apply the techniques developed in [\textit{J. Giné} and \textit{J. Llibre}, Int. J. Bifurcation Chaos Appl. Sci. Eng. 30, No. 4, Article ID 2050064, 7 p. (2020; Zbl 1446.34001)] and [loc. cit.] to equation (\(*\)) for finding non-Liouvillian integrable systems.
Reviewer: Narahari Parhi (Bhubaneswar)On a conjecture on the integrability of Liénard systemshttps://www.zbmath.org/1483.340082022-05-16T20:40:13.078697Z"Llibre, Jaume"https://www.zbmath.org/authors/?q=ai:llibre.jaume"Murza, Adrian C."https://www.zbmath.org/authors/?q=ai:murza.adrian-calin"Valls, Claudia"https://www.zbmath.org/authors/?q=ai:valls.claudiaLocal analytical integrability of the Liénard system \[ \dot x=y+F(x), \qquad \dot y= x, \qquad (1) \] where \(F(x)\) is an analytic function satisfying \(F(0) = 0\) and \(F'(0) \ne 0\), is investigated. First, the authors prove that if system (1) has a local analytic first integral defined in a neighborhood of the origin, then \(a = F'(0) = \pm (k_1 - k_2)/\sqrt{k_1 k_2},\) where \(a \ne 0\) and \(k_1\) and \(k_2\) are coprime positive integers. Then, it is proved that if system (1), with \(a\) defined above, has an analytic first integral in a neighborhood of the origin, then \(F(x) = ax\) and the system has a polynomial first integral of a certain form.
Reviewer: Valery Romanovski (Maribor)Nonlinear Hadamard fractional boundary value problems with different ordershttps://www.zbmath.org/1483.340092022-05-16T20:40:13.078697Z"Abuasbeh, Kinda"https://www.zbmath.org/authors/?q=ai:abuasbeh.kinda"Awadalla, Muath"https://www.zbmath.org/authors/?q=ai:awadalla.muath-m"Jneid, Maher"https://www.zbmath.org/authors/?q=ai:jneid.maherSummary: In this paper, we study the existence (uniqueness) of solutions for nonlinear fractional differential equations with different orders of Hadamard fractional derivatives involving associated with nonlocal boundary conditions. Several fixed point theorems are used for sufficient conditions of the existence (uniqueness) of solutions to nonlinear differential equations such as Banach's contraction principle, the Leray-Schauder nonlinear alternative, and Krasnoselskii's fixed point theorem. Applications of the main results are also presented.Mittag-Leffler stability for impulsive Caputo fractional differential equationshttps://www.zbmath.org/1483.340102022-05-16T20:40:13.078697Z"Agarwal, R."https://www.zbmath.org/authors/?q=ai:agarwal.ravi-p|agarwal.r-b"Hristova, S."https://www.zbmath.org/authors/?q=ai:khristova.snezhana-g|hristova.snehana"O'Regan, D."https://www.zbmath.org/authors/?q=ai:oregan.donalThe authors of this paper study stability properties of Caputo fractional differential equations with impulses. They consider various types of impulses, such as: non-instantaneous impulses as well as instantaneous impulses. They present and discuss two approaches for the interpretation of solutions of impulsive Caputo fractional differential equations. They generalize Mittag-Leffler stability with respect to both types of impulses.
Reviewer: Samir Bashir Hadid (Ajman)On some extended Routh-Hurwitz conditions for fractional-order autonomous systems of order \(\alpha\in (0, 2)\) and their applications to some population dynamic modelshttps://www.zbmath.org/1483.340112022-05-16T20:40:13.078697Z"Bourafa, S."https://www.zbmath.org/authors/?q=ai:bourafa.s"Abdelouahab, M.-S."https://www.zbmath.org/authors/?q=ai:abdelouahab.mohammed-salah"Moussaoui, A."https://www.zbmath.org/authors/?q=ai:moussaoui.ahmed|moussaoui.abdelkrim|moussaoui.abdelhamid|moussaoui.abdelouahab|moussaoui.aliSummary: The Routh-Hurwitz stability criterion is a useful tool for investigating the stability property of linear and nonlinear dynamical systems by analyzing the coefficients of the corresponding characteristic polynomial without calculating the eigenvalues of its Jacobian matrix. Recently some of these conditions have been generalized to fractional systems of order \(\alpha\in[0,1)\). In this paper we extend these results to fractional systems of order \(\alpha\in[0,2)\). Stability diagram and phase portraits classification in the \(\tau,\Delta)\)-plane for planer fractional-order system are reported. Finally some numerical examples from population dynamics are employed to illustrate our theoretical results.Existence and uniqueness of positive mild solutions for a class of fractional evolution equations on infinite intervalhttps://www.zbmath.org/1483.340122022-05-16T20:40:13.078697Z"Chen, Yi"https://www.zbmath.org/authors/?q=ai:chen.yi"Lv, Zhanmei"https://www.zbmath.org/authors/?q=ai:lv.zhanmei"Zhang, Liang"https://www.zbmath.org/authors/?q=ai:zhang.liang.3Summary: Based on an equivalent integral equation of a new type for a class of fractional evolution equations, which is different from those obtained in the existing literature, the paper investigates a class of fractional evolution equations with nonlocal conditions on infinite interval. Without the assumption of lower and upper solutions, we present a new result on the existence and uniqueness of positive mild solutions for the abstract fractional evolution equations by using the monotone iterative method.Some new results for \(\psi\)-Hilfer fractional pantograph-type differential equation depending on \(\psi\)-Riemann-Liouville integralhttps://www.zbmath.org/1483.340132022-05-16T20:40:13.078697Z"Foukrach, Djamal"https://www.zbmath.org/authors/?q=ai:foukrach.djamal"Bouriah, Soufyane"https://www.zbmath.org/authors/?q=ai:bouriah.soufyane"Benchohra, Mouffak"https://www.zbmath.org/authors/?q=ai:benchohra.mouffak"Karapinar, Erdal"https://www.zbmath.org/authors/?q=ai:karapinar.erdalSummary: The aim of the present work is to study a large class of \(\psi\)-Hilfer fractional differential equation of Pantograph-type depending on \(\psi\)-Riemann-Liouville fractional integral operator associated with periodic-type fractional integral boundary conditions in a weighted space of continuous functions. We shall prove the existence and uniqueness results by means of Mawhin's coincidence degree theory. At the end, an illustrative example will be constructed to approve our findings.Existence of the solution and stability for a class of variable fractional order differential systemshttps://www.zbmath.org/1483.340142022-05-16T20:40:13.078697Z"Jiang, Jingfei"https://www.zbmath.org/authors/?q=ai:jiang.jingfei"Chen, Huatao"https://www.zbmath.org/authors/?q=ai:chen.huatao"Guirao, Juan L. G."https://www.zbmath.org/authors/?q=ai:garcia-guirao.juan-luis"Cao, Dengqing"https://www.zbmath.org/authors/?q=ai:cao.dengqingSummary: In this paper, the existence results of the solution and stability are focused for the variable fractional order differential equation. In view of the definitions of three kinds of Caputo variable fractional order operator, the sufficient condition of the solution existence for the variable fractional order differential system is obtained by use of Arzela-Ascoli theorem. Moreover, some criterions of the Mittag-Leffler stability and asymptotical stability are proposed for the variable fractional order differential system according to the Fractional Comparison Principle.On impulsive boundary value problem with Riemann-Liouville fractional order derivativehttps://www.zbmath.org/1483.340152022-05-16T20:40:13.078697Z"Khan, Zareen A."https://www.zbmath.org/authors/?q=ai:khan.zareen-a-a|khan.zareen-abdulhameed"Gul, Rozi"https://www.zbmath.org/authors/?q=ai:gul.rozi"Shah, Kamal"https://www.zbmath.org/authors/?q=ai:shah.kamalA class of impulsive boundary value problems for Riemann-Liouville fractional differential equations is studied. Unfortunately, it is not taken into account that the lower limit of the Riemann-Liouville fractional order derivative is very important. According to the equation~(4), the Riemann-Liouville fractional order derivative has a lower limit at zero. Therefore, Lemma~5 and equation~(5) are true if both the fractional integral and the fractional derivative have one and the same lower limit. At the same time this Lemma is applied in the proof of the main Lemma~6 in order to obtain equality~(12) for the lower limit of the integral \(z_1\) and for the fractional derivative~$0$, which does not lead to~(12). It has a huge influence on the other results in this paper.
Reviewer: Snezhana Hristova (Plovdiv)Differential equations with tempered \(\Psi\)-Caputo fractional derivativehttps://www.zbmath.org/1483.340162022-05-16T20:40:13.078697Z"Medveď, Milan"https://www.zbmath.org/authors/?q=ai:medved.milan"Brestovanská, Eva"https://www.zbmath.org/authors/?q=ai:brestovanska.evaSummary: In this paper we define a new type of the fractional derivative, which we call tempered \(\Psi\)-Caputo fractional derivative. It is a generalization of the tempered Caputo fractional derivative and of the \(\Psi\)-Caputo fractional derivative. The Cauchy problem for fractional differential equations with this type of derivative is discussed and some existence and uniqueness results are proved. We present a Henry-Gronwall type inequality for an integral inequality with the tempered \(\Psi\)-fractional integral. This inequality is applied in the proof of an existence theorem. A result on a representation of solutions of linear systems of \(\Psi\)-Caputo fractional differential equations is proved and in the last section an example is presented.Transmission dynamics of fractional order Typhoid fever model using Caputo-Fabrizio operatorhttps://www.zbmath.org/1483.340172022-05-16T20:40:13.078697Z"Shaikh, Amjad S."https://www.zbmath.org/authors/?q=ai:shaikh.amjad-salim"Sooppy Nisar, Kottakkaran"https://www.zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaranSummary: In this manuscript, we develop existence, uniqueness and stability criteria for fractional order Typhoid fever model having Caputo-Fabrizio operator by using fixed point theory. This approach of the fractional derivative is relatively new for such kind of biological models. We have also obtained the first accessible approximate solutions for a proposed model by utilizing iterative Laplace transform method. This technique is a combination of one of the reliable method known as new iterative method and Laplace transform method. Finally, we have evaluated parameters that portray the conduct of illness and present the numerical simulations using plots.Solutions of fractional differential equations with \(p\)-Laplacian operator in Banach spaceshttps://www.zbmath.org/1483.340182022-05-16T20:40:13.078697Z"Tan, Jingjing"https://www.zbmath.org/authors/?q=ai:tan.jingjing"Li, Meixia"https://www.zbmath.org/authors/?q=ai:li.meixiaSummary: In this paper, we study the solutions for nonlinear fractional differential equations with \(p\)-Laplacian operator nonlocal boundary value problem in a Banach space. By means of the technique of the properties of the Kuratowski noncompactness measure and the Sadovskii fixed point theorem, we establish some new existence criteria for the boundary value problem. As application, an interesting example is provided to illustrate the main results.Linear inverse problems for multi-term equations with Riemann-Liouville derivativeshttps://www.zbmath.org/1483.340192022-05-16T20:40:13.078697Z"Turov, Mikhail Mikhailovich"https://www.zbmath.org/authors/?q=ai:turov.mikhail-mikhailovich"Fëdorov, Vladimir Evgen'evich"https://www.zbmath.org/authors/?q=ai:fedorov.v-e"Kien, Bui Trong"https://www.zbmath.org/authors/?q=ai:kien.bui-trongSummary: The issues of well-posedness of linear inverse coefficient problems for multi-term equations in Banach spaces with fractional Riemann-Liouville derivatives and with bounded operators at them are considered. Well-posedness criteria are obtained both for the equation resolved with respect to the highest fractional derivative, and in the case of a degenerate operator at the highest derivative in the equation. Two essentially different cases are investigated in the degenerate problem: when the fractional part of the order of the second-oldest derivative is equal to or different from the fractional part of the order of the highest fractional derivative. Abstract results are applied in the study of inverse problems for partial differential equations with polynomials from a self-adjoint elliptic differential operator with respect to spatial variables and with Riemann-Liouville derivatives in time.Multiplicity results for impulsive fractional differential equations with \(p\)-Laplacian via variational methodshttps://www.zbmath.org/1483.340202022-05-16T20:40:13.078697Z"Zhao, Yulin"https://www.zbmath.org/authors/?q=ai:zhao.yulin"Tang, Liang"https://www.zbmath.org/authors/?q=ai:tang.liangSummary: In this paper, we apply critical point theory and variational methods to study the multiple solutions of boundary value problems for an impulsive fractional differential equation with \(p\)-Laplacian. Some new criteria guaranteeing the existence of multiple solutions are established for the considered problem.On the stability of linear quaternion-valued differential equationshttps://www.zbmath.org/1483.340212022-05-16T20:40:13.078697Z"Chen, Dan"https://www.zbmath.org/authors/?q=ai:chen.dan"Fečkan, Michal"https://www.zbmath.org/authors/?q=ai:feckan.michal"Wang, JinRong"https://www.zbmath.org/authors/?q=ai:wang.jinrongSummary: This paper deals with the stability of linear quaternion-valued differential equations. First, we derive an explicit norm estimation like the matrix exponential function in the sense of quaternion-valued. Second, we use this norm to show that the first-order linear equations are asymptotically stable and Hyers-Ulam's type stable. Further, we show that \(n\)th-order equations are also generalized Hyers-Ulam stability. Some examples which can effectively illustrate the theoretical results are presented.Dynamics of nonlocal and local discrete Ginzburg-Landau equations: global attractors and their congruencehttps://www.zbmath.org/1483.340222022-05-16T20:40:13.078697Z"Hennig, Dirk"https://www.zbmath.org/authors/?q=ai:hennig.dirk"Karachalios, Nikos I."https://www.zbmath.org/authors/?q=ai:karachalios.nikos-iSummary: Discrete Ginzburg-Landau (DGL) equations with non-local nonlinearities have been established as significant inherently discrete models in numerous physical contexts, similar to their counterparts with local nonlinear terms. We study two prototypical examples of non-local and local DGLs on the one-dimensional infinite lattice. For the non-local DGL, we identify distinct scenarios for the asymptotic behavior of the globally existing in time solutions depending on certain parametric regimes. One of these scenarios is associated with a restricted compact attractor according to J. K. Hale's definition. We also prove the closeness of the solutions of the two models in the sense of a ``continuous dependence on their initial data'' in the \(l^2\) metric under general conditions on the intrinsic linear gain or loss incorporated in the model. As a consequence of the closeness results, in the dissipative regime we establish the congruence of the attractors possessed by the semiflows of the non-local and of the local model respectively, for initial conditions in a suitable domain of attraction defined by the non-local system.On coupled systems of Lidstone-type boundary value problemshttps://www.zbmath.org/1483.340232022-05-16T20:40:13.078697Z"de Sousa, Robert"https://www.zbmath.org/authors/?q=ai:de-sousa.robert"Minhós, Feliz"https://www.zbmath.org/authors/?q=ai:minhos.feliz-manuel"Fialho, João"https://www.zbmath.org/authors/?q=ai:fialho.joao-fSummary: This research concerns the existence and location of solutions for coupled system of differential equations with Lidstone-type boundary conditions. Methodology used utilizes three fundamental aspects: upper and lower solutions method, degree theory and nonlinearities with monotone conditions. In the last section an application to a coupled system composed by two fourth order equations, which models the bending of coupled suspension bridges or simply supported coupled beams, is presented.The influence function properties for a problem with discontinuous solutionshttps://www.zbmath.org/1483.340242022-05-16T20:40:13.078697Z"Kamenskii, Mikhail"https://www.zbmath.org/authors/?q=ai:kamenskii.mikhail-igorevich"Wen, Ching-Feng"https://www.zbmath.org/authors/?q=ai:wen.chingfeng"Zalukaev, Zhanna"https://www.zbmath.org/authors/?q=ai:zalukaev.zhanna"Zvereva, Margarita"https://www.zbmath.org/authors/?q=ai:zvereva.margarita-borisovnaSummary: We consider the boundary value problem describing deformations of a discontinuous Stieltjes string. Properties of the influence function (Green function) are investigated. The analysis is based on a refined Stieltjes integral.Convergence analysis for iterative learning control of conformable impulsive differential equationshttps://www.zbmath.org/1483.340252022-05-16T20:40:13.078697Z"Qiu, Wanzheng"https://www.zbmath.org/authors/?q=ai:qiu.wanzheng"Fečkan, Michal"https://www.zbmath.org/authors/?q=ai:feckan.michal"O'Regan, Donal"https://www.zbmath.org/authors/?q=ai:oregan.donal"Wang, JinRong"https://www.zbmath.org/authors/?q=ai:wang.jinrongSummary: This paper deals with iterative learning control for conformable impulsive differential equations. For nonlinear and linear problems varying with the initial state, we design standard \(P\)-type, \(D_{\gamma}\)-type, and conformable \(PI_{\gamma} D_{\gamma}\)-type learning update laws. Next, we establish sufficient conditions for tracking error convergence and use impulsive Gronwall inequality and mathematical analysis tools to prove the main results. Finally, three numerical examples are given to illustrate our theoretical results.Simple numerical methods of second- and third-order convergence for solving a fully third-order nonlinear boundary value problemhttps://www.zbmath.org/1483.340262022-05-16T20:40:13.078697Z"Dang, Quang A"https://www.zbmath.org/authors/?q=ai:dang-quang-a."Dang, Quang Long"https://www.zbmath.org/authors/?q=ai:dang.quang-longThis paper is concerned with the following fully third-order nonlinear boundary value problem that is of great interest of many researchers
\begin{align*}
&u^{(3)}(t)=f\left(t, u(t), u^{\prime}(t), u^{\prime \prime}(t)\right), \quad 0<t<1, \\
&u(0)=c_{1}, u^{\prime}(0)=c_{2}, u^{\prime}(1)=c_{3}.
\end{align*}
First, the existence and uniqueness of solution are discussed. Next, the simple iterative methods on both continuous and discrete levels are proposed. The discrete methods are of second-order and third-order of accuracy due to the use of appropriate formulas for numerical integration. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the iterative methods. A completely different method, specifically, an iterative method on both continuous and discrete levels for the considered fully third-order differential equations is developed. These methods are based on the popular trapezoidal rule and a modified Simpson rule for numerical integration. Further, an analysis of the total error of the solution is obtained. The obtained error includes the error of the iterative method on continuous level and the error arising in the numerical realization of this iterative method. The obtained total error estimate suggests how to choose a suitable grid size for discretization to get an approximate solution with a given accuracy. In order to justify the total error estimate, some results on existence and uniqueness of solution are established. Also, the applicability shows that, these methods can easily be extend to higher order nonlinear boundary value problems. Numerical results are also given.
Reviewer: Saurabh Tomar (Kharagpur)Simplified Liénard equation by homotopy analysis methodhttps://www.zbmath.org/1483.340272022-05-16T20:40:13.078697Z"Mitchell, Jonathan"https://www.zbmath.org/authors/?q=ai:mitchell.jonathanIn this article the author used Homotopy analysis method (HAM) to solve a simplified Lienard's equation. (HAM) method is one of the easiest way to assure the convergence of solution to a series so that it is valid even if nonlinearity becomes quite strong. The result obtained by HAM is compared with the traditional perturbation technique Poincaré-Lindsted method for the weakly nonlinear case. The approximations obtained by HAM are in good agreement with the traditional P-L method. The authors also discuss the limitations of the method using conditional convergence.
Reviewer: J. Peter Praveen (Guntur)The uniqueness theorems in the inverse problems for Dirac operatorshttps://www.zbmath.org/1483.340282022-05-16T20:40:13.078697Z"Harutyunyan, T. N."https://www.zbmath.org/authors/?q=ai:harutyunyan.tigran-nSummary: We introduce new supplementary data to the set of the eigenvalues, to determine uniquely potential matrix in the inverse problem for Dirac canonical operator. Besides, we obtain others uniqueness theorems in inverse problems, which are the analogues of well-known Borg, Marchenko and McLaughlin-Rundell theorems in inverse Sturm-Liouville problems.Inverse spectral problem of an anharmonic oscillator on a half-axis with the Neumann boundary conditionhttps://www.zbmath.org/1483.340292022-05-16T20:40:13.078697Z"Khanmamedov, Agil K."https://www.zbmath.org/authors/?q=ai:khanmamedov.agil-kh"Gafarova, Nigar F."https://www.zbmath.org/authors/?q=ai:gafarova.nigar-fThe paper under review deals with the half-line Neumann problem for the equation \[-y''+ x^2 y + q(x) y = \lambda y\] under some smoothness and integrability conditions on \(q\). The authors consider the inverse problem of recovering this boundary value problem by its spectrum and norming constants. They obtain a Gelfand-Levitan-Marchenko-type integral equation, prove its unique solvability, and indicate a constructive algorithm for recovering \(q\).
Reviewer: Namig Guliyev (Baku)One remark on the transformation operator for perturbed Hill operatorshttps://www.zbmath.org/1483.340302022-05-16T20:40:13.078697Z"Khanmamedov, Agil Kh."https://www.zbmath.org/authors/?q=ai:khanmamedov.agil-kh"Mamedova, A. F."https://www.zbmath.org/authors/?q=ai:mamedova.afag-fSummary: The transformation operator for the perturbed Hill operators is considered. An error is indicated in the proof of the previously proved theorem on the triangular representation of the solution of the perturbed Hill equation. A method for eliminating this disadvantage for a special periodic potential is proposed.Real-time reconstruction of external impact on fractional order system under measuring a part of coordinateshttps://www.zbmath.org/1483.340312022-05-16T20:40:13.078697Z"Surkov, P. G."https://www.zbmath.org/authors/?q=ai:surkov.platon-gennadevichSummary: For a system of nonlinear fractional differential equations, the problem of reconstructing an unknown external impact is considered. It is complicated by the fact that only a part of system's parameters is available for measuring. An algorithm for solving this problem is proposed, which is resistant to informational noises and computational errors, and is based on regularization methods and constructions of guaranteed control theory. A numerical example illustrating the operation of the algorithm is considered.Controlled differential equations with a parameter and with multivalued impulseshttps://www.zbmath.org/1483.340322022-05-16T20:40:13.078697Z"Filippova, Ol'ga Viktorovna"https://www.zbmath.org/authors/?q=ai:filippova.olga-viktorovnaSummary: We study the Cauchy problem for a controlled differential system with a parameter which is an element of some metric space \(\Xi\) containing phase constraints on the control. It is assumed that at the given time instants \(t_k\), \(k=1,2,\ldots, p\), the solution \(x\) is continuous from the left and suffers a discontinuity, the value of which is \(x(t_k + 0)-x(t_k)\), belongs to some non-empty compact set of the space \(\mathbb{R}^n\). The notions of an admissible pair of this controlled impulsive system are introduced. The questions of continuity of admissible pairs are considered. Definitions of a priori boundedness and a priori collective boundedness on a given set \(S \times K\) (where \(S\subset \mathbb{R}^n\) is a set of initial values, \(K \subset \Xi\) is a set of parameter values) of the set of phase trajectories are considered. It is proved that if at some point \((x_0, \xi) \in \mathbb{R}^n \times \Xi\) the set of phase trajectories is a priori bounded, then it will be a priori bounded in some neighborhood of this point.Accurate approximating solution of the differential inclusion based on the ordinary differential equationhttps://www.zbmath.org/1483.340332022-05-16T20:40:13.078697Z"Nguyen, T. H."https://www.zbmath.org/authors/?q=ai:nguyen.tuong-huy|nguyen.thanh-hieu|nguyen.thu-huong|nguyen.thi-hoa|nguyen.thi-hong-van|nguyen.ta-h-d|nguyen.thi-huyen-thu|nguyen.thanh-hong|nguyen.tuan-ha|nguyen.thanh-hung|nguyen.tam-h|nguyen-thieu-huy.|nguyen-thi-hong-loan.|nguyen.thuy-hong|nguyen.trieu-hai|nguyen.thi-huyen-chau|nguyen.thai-huu|nguyen.thi-hanh|nguyen.thi-hien|nguyen.tuan-h-a|nguyen.thanhvu-h|nguyen-huy-tuan.|nguyen.truong-khanh|nguyen.thi-hoai-thuong|nguyen.thanh-hai|nguyen.thuong-huyen|nguyen.thong-h|nguyen.thanh-ha|nguyen.trong-hieu|nguyen-thua-hop.|nguyen.thanh-huong|nguyen.thanh-huyen-thi|nguyen.the-hung|nguyen.tu-hoang-huy|nguyen.thu-hien-thi|nguyen.tung-h|nguyen.thi-hong-khanhLet \(Q \subset \mathbb{R}^n\) be a closed bounded convex set, \((t_0,x_0) \in \mathbb{R} \times Q\) and \(f \colon [t_0,T] \times Q \to \mathbb{R}^n\) be a continuous function, Lipschitz in the second argument with a constant \(L>0\) and bounded by a constant \(C>0.\) Then the Cauchy problem \[ \left\{\begin{array}{l} \dot{x} \in f(t,x) - N_Qx, \quad t \in [t_0,T], \quad x(t) \in Q, \\
x(t_0)=x_0 \in Q, \end{array} \right. \] where \(N_Qx\) denotes the normal cone to \(Q\) at a point \(x,\) has a unique solution. Moreover, if \(y_K(t)\) is a solution of the problem \[ \left\{\begin{array}{l} \dot{y} = f(t,\bar{y}) - K(y - \bar{y}), \\
y(t_0)=x_0 \in Q \end{array} \right. \] with a fixed \(K>0\), where \(\bar{y} = P_Q(y)\) is the metric projection, then
\[
\parallel x(t) - y_K(t)\parallel \leq \frac{Ce^{L(T-t_0)}}{\sqrt{L}}\frac{1}{\sqrt{K}}, \quad t \in [t_0,T].
\]
Reviewer: Valerii V. Obukhovskij (Voronezh)Solvability of inclusions of Hammerstein typehttps://www.zbmath.org/1483.340342022-05-16T20:40:13.078697Z"Pietkun, Radosław"https://www.zbmath.org/authors/?q=ai:pietkun.radoslawSummary: We establish a universal rule for solving operator inclusions of Hammerstein type in Lebesgue-Bochner spaces with the aid of some recently proven continuation theorem of Leray-Schauder type for the class of so-called admissible multimaps. Examples illustrating the legitimacy of this approach include the initial value problem for perturbation of \(m\)-accretive multivalued differential equation, the nonlocal Cauchy problem for semilinear differential inclusion, abstract integral inclusion of Fredholm and Volterra type and the two-point boundary value problem for nonlinear evolution inclusion.Existence and relaxation for subdifferential inclusions with unbounded perturbationhttps://www.zbmath.org/1483.340352022-05-16T20:40:13.078697Z"Timoshin, Sergey A."https://www.zbmath.org/authors/?q=ai:timoshin.sergey-aSummary: We consider a differential inclusion of subdifferential type with a nonconvex and unbounded valued perturbation. Existence and relaxation results are obtained for this inclusion. By relaxation we mean approximation of a solution of the differential inclusion with convexified perturbation by solutions of the given inclusion. The traditional condition of Lipschitz continuity for such kind of problems is weakened and a somehow more appropriate in the context of unbounded valued multifunctions ``truncated'' version of it is considered instead.Well ordered monotone iterative technique for nonlinear second order four point Dirichlet BVPshttps://www.zbmath.org/1483.340362022-05-16T20:40:13.078697Z"Verma, Amit K."https://www.zbmath.org/authors/?q=ai:verma.amit-kumar"Urus, Nazia"https://www.zbmath.org/authors/?q=ai:urus.naziaSummary: In this article, we develop a monotone iterative technique (MI-technique) with lower and upper (L-U) solutions for a class of four-point Dirichlet nonlinear boundary value problems (NLBVPs), defined as,
\[
-\psi^{\prime \prime}(x) = F (x, \psi, \psi^{\prime}), \quad 0 < x < 1, \text{ BCs}(i) \equiv \psi(i) - c_i\psi(\eta_i) = 0, \quad i = 0, 1,
\]
where \(0 < c_0 < 1\), \(c_1 > 0\), \(0 < \eta_0 \leq \eta_1 < 1\), \(\psi(x) \in C^2[0, 1]\), the non linear term \(F (x, \psi, \psi^{\prime})\) is continuous function in \(x\), one sided Lipschitz in \(\psi\) and Lipschitz in \(\psi^{\prime}\). To show the existence result, we construct Green's function and iterative sequences for the corresponding linear problem. We use quasilinearization to construct these iterative schemes. We prove maximum principle and establish monotonicity of sequences of lower solution \((l_m(x))_m\) and upper solution \((u_m(x))_m\) such that \(l_m(x) \leq u_m(x)\), \(\forall m \in \mathbb{N}\). Then under certain sufficient conditions we prove that these sequences converge uniformly to the solution \(\psi (x)\) in a specific region where \(\frac{\partial F}{\partial\psi} \neq 0\).Eigenvalues of a class of fourth-order boundary value problems with transmission conditions using matrix theoryhttps://www.zbmath.org/1483.340372022-05-16T20:40:13.078697Z"Ao, Ji-jun"https://www.zbmath.org/authors/?q=ai:ao.jijun"Sun, Jiong"https://www.zbmath.org/authors/?q=ai:sun.jiongThe authors study the differential equation \[ (p u'')'' + q u = \lambda w u \] on \(J = (a, c) \cup (c, b)\) for finite \(a < c < b\), together with boundary conditions \[ A U (a) + B U (b) = 0, \quad U = \begin{pmatrix} u \\
u' \\
p u'' \\
(p u'')' \end{pmatrix}, \quad A, B \in M_4 (\mathbb{R}), \] and transmission conditions \[ C U (c-) + D U (c+) = 0. \] Here \(C, D\) are real-valued \(4 \times 4\)-matrices with positive determinants and the coefficient functions \(1/p, q, w\) are integrable on \(J\). Moreover, conditions on the matrices \(A, B\) are imposed which make the problem self-adjoint. As the main result, sufficient conditions are provided under which this eigenvalue problem is equivalent to a matrix eigenvalue problem of the form \((\mathbb P + \mathbb Q) \mathbb Y = \lambda \mathbb W \mathbb Y\), where \(\mathbb {P, Q, W}\) are constructed explicitly.
Reviewer: Jonathan Rohleder (Stockholm)On the spectra of boundary value problems generated by some one-dimensional embedding theoremshttps://www.zbmath.org/1483.340382022-05-16T20:40:13.078697Z"Minarsky, A. M."https://www.zbmath.org/authors/?q=ai:minarsky.a-m"Nazarov, A. I."https://www.zbmath.org/authors/?q=ai:nazarov.alexander-iSummary: The spectra of boundary value problems related to one-dimensional high order embedding theorems are considered. It is proved that for some orders, the eigenvalues corresponding to even eigenfunctions of different problems cannot coincide.Global structure for a fourth-order boundary value problem with sign-changing weighthttps://www.zbmath.org/1483.340392022-05-16T20:40:13.078697Z"Ye, Fumei"https://www.zbmath.org/authors/?q=ai:ye.fumeiSummary: We study the fourth-order boundary value problem with a sign-changing weight function:
\[
\begin{cases}
u''''=\lambda m(t)u+f_1(t,u,u',u'',u''',\lambda)+f_2(t,u,u',u'',u''',\lambda),\quad t\in(0,1),\\
u(0)=u(1)=u''(0)=u''(1)=0,
\end{cases}
\]
where \(\lambda\in\mathbb{R}\) is a parameter, \(f_1,f_2\in C([0, 1] \times\mathbb{R}^5,\mathbb{R}),f_1\) is not differentiable at the origin and infinity. Under some suitable conditions on nonlinear terms, we prove the existence of unbounded continua of positive and negative solutions of this problem which bifurcating from intervals of the line of trivial solutions or from infinity, respectively.Dependence of eigenvalues of \(2m\)th-order spectral problemshttps://www.zbmath.org/1483.340402022-05-16T20:40:13.078697Z"Zheng, Zhaowen"https://www.zbmath.org/authors/?q=ai:zheng.zhaowen"Ma, Yujuan"https://www.zbmath.org/authors/?q=ai:ma.yujuanSummary: A regular \(2m\)th-order spectral problem with self-adjoint boundary conditions is considered in this paper. The continuous dependence of eigenvalues and normalized eigenfunctions on the problem is researched. The derivative formulas of eigenvalues with respect to the given parameters are obtained: endpoints, boundary conditions, coefficients and the weight function. These are of both theoretical and computational importance.Bound sets for a class of \(\varphi \)-Laplacian operatorshttps://www.zbmath.org/1483.340412022-05-16T20:40:13.078697Z"Feltrin, Guglielmo"https://www.zbmath.org/authors/?q=ai:feltrin.guglielmo"Zanolin, Fabio"https://www.zbmath.org/authors/?q=ai:zanolin.fabioThe authors provide an extension of the Hartman-Knobloch theorem for periodic solutions of vector differential systems to a general class of \(\phi\)-Laplacian differential operators. Their main tool is a variant of the Manásevich-Mawhin continuation theorem developed for this class of operator equations, together with the theory of bound sets. They also extend a classical theorem by Reissig for scalar periodically perturbed Liénard equations.
Reviewer: Alessandro Fonda (Trieste)The Dirichlet problem for the fourth order nonlinear ordinary differential equations at resonancehttps://www.zbmath.org/1483.340422022-05-16T20:40:13.078697Z"Mukhigulashvili, S."https://www.zbmath.org/authors/?q=ai:mukhigulashvili.sulkhan"Manjikashvili, M."https://www.zbmath.org/authors/?q=ai:manjikashvili.mariamThis paper discusses the solvability of the following fourth order boundary value problem \[u^{(4)}(t)=p(t)u(t)+f(t,u(t))+h(t),\;t\in I=[a,b],\tag{1}\] \[u^{(i)}(a)=0,\,u^{(i)}(b)=0,\,i=0,1,\tag{2}\] where \(h, p\in L(I, \mathbb R)\) and \(f:I\times \mathbb R\to \mathbb R\) is a Carathéodory function.
The authors suppose that the problem \[w^{(4)}(t)=p(t)w(t),\;t\in I,\tag{3}\] \[w^{(i)}(a)=0,\,w^{(i)}(b)=0,\,i=0,1\tag{4}\] has a nonzero solution \(w,\) introduce the set \({N_p:=\{t\in I: w(t)=0\}}\) and for a finite subset \(A=\{t_1,\dots, t_k\}\) of \(I\) introduce the set \(E(A)\) of all Carathéodory functions \(f : I\times \mathbb R\to \mathbb R\) such that for an arbitrary neighbourhood \(U(A)\) of \(A\) and a positive constant \(r\) there exists \(\alpha_1>0\) with the property \[\int\limits_{U'(A)\setminus U_\alpha}|f(s,x)|ds\, - \, \int\limits_{U_\alpha}|f(s,x)|ds\geq0\;\;\text{for}\;\;|x|\geq r,\;\alpha\leq\alpha_1,\] where \(U'(A)=I\cap U(A),\) and \(U_\alpha=I\cap\Bigl(\cup_{j=1}^k[t_j-\alpha, t_j+\alpha]\Bigr).\) Besides for an arbitrary \(r>0\) \(f^*(t,r)=\sup\{|f(t,x)|: |x|\leq r\}\in L(I, [0,+\infty))\) and \[[x(t)]_+=(|x(t)|+x(t))/2,\;[x(t)]_-=(|x(t)|-x(t))/2\] for a function \(x: I\to \mathbb R.\)
One of the main results guarantees at least one solution of (1), (2), i.e. at least one function \(u\in\widetilde{C}^3(I, \mathbb R)\) wich satisfies (1) almost everywhere on \(I\) and satisfies (2), under the assumptions that \(r>0\) and the functions \(f\in E(N_p), f^+, f^-\in L(I, [0,+\infty))\) are such that for \(j\in\{0,1\}\) \[(-1)^j f(t,x)\leq -f^-(t)\;\;\text{for}\;\;x\leq-r,\;t\in I,\] \[f^+(t)\leq(-1)^j f(t,x)\;\;\text{for}\;\;x\geq r,\;t\in I,\] \[\lim_{\rho\to+\infty} \frac{1}{\rho}\int\limits_a^b f^*(s,\rho)ds=0,\] and there exists \(\varepsilon>0\) such that for an arbitrary nonzero solution \(w\) of (3), (4) the following holds \[-\int\limits_a^b\Bigl( f^+(s)[w(s)]_-+f^-(s)[w(s)]_+\Bigr)ds+\varepsilon\gamma_r||w||_C\] \[\leq(-1)^{j+1}\int\limits_a^b h(s)w(s)ds\leq\int\limits_a^b\Bigl( f^-(s)[w(s)]_-+f^+(s)[w(s)]_+\Bigr)ds - \varepsilon\gamma_r||w||_C,\] where \(\gamma_r=\int\limits_a^b f^*(s,r)ds.\)
Reviewer: Petio S. Kelevedjiev (Sliven)Positive solutions to classes of infinite semipositone \((p,q)\)-Laplace problems with nonlinear boundary conditionshttps://www.zbmath.org/1483.340432022-05-16T20:40:13.078697Z"Sim, Inbo"https://www.zbmath.org/authors/?q=ai:sim.inbo"Son, Byungjae"https://www.zbmath.org/authors/?q=ai:son.byungjaeIn this interesting paper the authors study the existence, multiplicity and nonexistence of positive solutions for the one-dimensional \((p,q)\)-Laplacian problems:
\begin{gather*}
-(\varphi(u'))'=\lambda h(t)f(u),\qquad t\in (0,1),\\
u(0)=0=au'(1)+g(\lambda,u(1))u(1),
\end{gather*}
where \(\lambda>0\), \(a\geq 0\), \(\varphi(s)=|s|^{p-2}s+|s|^{q-2}s\) with \(1<p<q<+\infty\), \(h\in C((0,1),(0,+\infty))\), and \(f\in C((0,+\infty),\mathbb{R})\) may have a singularity at 0 of repulsive type. The proofs are based on a classical Krasnoselskii type fixed point theorem which is fit to overcome a lack of homogeneity.
Reviewer: Manuel Zamora (Concepción)New multiple positive solutions for Hadamard-type fractional differential equations with nonlocal conditions on an infinite intervalhttps://www.zbmath.org/1483.340442022-05-16T20:40:13.078697Z"Zhang, Wei"https://www.zbmath.org/authors/?q=ai:zhang.wei.10|zhang.wei.19|zhang.wei.6|zhang.wei.5|zhang.wei.3|zhang.wei.17|zhang.wei.15|zhang.wei.16|zhang.wei.9|zhang.wei.7|zhang.wei.1|zhang.wei.12|zhang.wei.4|zhang.wei.13|zhang.wei.2|zhang.wei.18"Ni, Jinbo"https://www.zbmath.org/authors/?q=ai:ni.jinboIn this paper, the authors consider nonlinear Hadamard-type fractional differential equations with nonlocal boundary conditions on an infinite interval. The existence of multiple positive solutions of the addressed problem is obtained by applying the generalized Avery-Henderson fixed point theorem. Finally, an example was given to show the effectiveness of the main result. This paper provides a new fixed point theorem to study multiple solutions.
Reviewer: Wengui Yang (Sanmenxia)Oscillation properties for non-classical Sturm-Liouville problems with additional transmission conditionshttps://www.zbmath.org/1483.340452022-05-16T20:40:13.078697Z"Mukhtarov, Oktay Sh."https://www.zbmath.org/authors/?q=ai:mukhtarov.oktay-sh"Aydemir, Kadriye"https://www.zbmath.org/authors/?q=ai:aydemir.kadriyeSummary: This work is aimed at studying some comparison and oscillation properties of boundary value problems (BVP's) of a new type, which differ from classical problems in that they are defined on two disjoint intervals and include additional transfer conditions that describe the interaction between the left and right intervals. This type of problems we call boundary value-transmission problems (BVTP's). The main difficulty arises when studying the distribution of zeros of eigenfunctions, since it is unclear how to apply the classical methods of Sturm's theory to problems of this type. We established new criteria for comparison and oscillation properties and new approaches used to obtain these criteria. The obtained results extend and generalizes the Sturm's classical theorems on comparison and oscillation.On a series representation for integral kernels of transmutation operators for perturbed Bessel equationshttps://www.zbmath.org/1483.340462022-05-16T20:40:13.078697Z"Kravchenko, V. V."https://www.zbmath.org/authors/?q=ai:kravchenko.vladislav-v"Shishkina, E. L."https://www.zbmath.org/authors/?q=ai:shishkina.elina-leonidovna"Torba, S. M."https://www.zbmath.org/authors/?q=ai:torba.sergii-mSummary: A representation for the kernel of the transmutation operator relating a perturbed Bessel equation to the unperturbed one is obtained in the form of a functional series with coefficients calculated by a recurrent integration procedure. New properties of the transmutation kernel are established. A new representation of a regular solution of a perturbed Bessel equation is given, which admits a uniform error bound with respect to the spectral parameter for partial sums of the series. A numerical illustration of the application of the obtained result to solve Dirichlet spectral problems is presented.Third-order generalized discontinuous impulsive problems on the half-linehttps://www.zbmath.org/1483.340472022-05-16T20:40:13.078697Z"Minhós, Feliz"https://www.zbmath.org/authors/?q=ai:minhos.feliz-manuel"Carapinha, Rui"https://www.zbmath.org/authors/?q=ai:carapinha.ruiSummary: In this paper, we improve the existing results in the literature by presenting weaker sufficient conditions for the solvability of a third-order impulsive problem on the half-line, having generalized impulse effects. More precisely, our nonlinearities do not need to be positive nor sublinear and the monotone assumptions are local ones. Our method makes use of some truncation and perturbed techniques and on the equiconvergence at infinity and the impulsive points. The last section contains an application to a boundary layer flow problem over a stretching sheet with and without heat transfer.On Pleijel's nodal domain theorem for quantum graphshttps://www.zbmath.org/1483.340482022-05-16T20:40:13.078697Z"Hofmann, Matthias"https://www.zbmath.org/authors/?q=ai:hofmann.matthias"Kennedy, James B."https://www.zbmath.org/authors/?q=ai:kennedy.james-b"Mugnolo, Delio"https://www.zbmath.org/authors/?q=ai:mugnolo.delio"Plümer, Marvin"https://www.zbmath.org/authors/?q=ai:plumer.marvinSummary: We establish metric graph counterparts of Pleijel's theorem on the asymptotics of the number of nodal domains \(\nu_n\) of the \(n\)th eigenfunction(s) of a broad class of operators on compact metric graphs, including Schrödinger operators with \(L^1\)-potentials and a variety of vertex conditions as well as the \(p\)-Laplacian with natural vertex conditions, and without any assumptions on the lengths of the edges, the topology of the graph, or the behaviour of the eigenfunctions at the vertices. Among other things, these results characterise the accumulation points of the sequence \(\left(\frac{\nu_n}{n}\right)_{n\in \mathbb{N}} \), which are shown always to form a finite subset of \((0, 1]\). This extends the previously known result that \(\nu_n\sim n\) \textit{generically}, for certain realisations of the Laplacian, in several directions. In particular, in the special cases of the Laplacian with natural conditions, we show that for graphs any graph with pairwise commensurable edge lengths and at least one cycle, one can find eigenfunctions thereon for which \({\nu_n}\not \sim{n} \); but in this case even the set of points of accumulation may depend on the choice of eigenbasis.Boundedness of a class of spatially discrete reaction-diffusion systemshttps://www.zbmath.org/1483.340492022-05-16T20:40:13.078697Z"Wentz, Jacqueline M."https://www.zbmath.org/authors/?q=ai:wentz.jacqueline-m"Bortz, David M."https://www.zbmath.org/authors/?q=ai:bortz.david-mIt is considered the \(1D\) reaction diffusion system spatially discretized as a compartmental system online described by \[ \displaystyle{u_t = \gamma f(u,v) + \frac{1}{h^2}Du\ ;\ v_t = \gamma g(u,v) + \frac{1}{h^2}dDu} \] with \(f(u,v)=\mathrm{col}\{f(u_i,v_i),i=\overline{1,n}\}\), \(g(u,v)=\mathrm{col}\{g(u_i,v_i),i=\overline{1,n}\}\), \(\gamma>0\), \(d>0\) -- constants and \(D\) -- a three band matrix.
To this system there is associated a Lyapunov-like function \(W(u,v):\mathbb R^2_{\geq 0}\mapsto \mathbb R_+\) with the properties:
(\(P_1\)) \(\exists K>0\) such that if \(\|(u,v)\|\geq K\) then \[ \displaystyle{\nabla W(u,v)(f(u,v)\;g(u,v))^T\leq 0}; \]
(\(P_2\)) \(W(u,v)=w_1(u) + w_2(v)\);
(\(P_3\)) \(\partial_{uu}W>0\ ,\ \partial_{vv}W>0\);
(\(P_4\)) \(\lim_{\|(u,v)\|\rightarrow\infty} W(u,v)=\infty\)
This function is used for obtaining global boundedness of the solutions of the compartmental systems.
Reviewer: Vladimir Răsvan (Craiova)Backbone curves of coupled cubic oscillators in one-to-one internal resonance: bifurcation scenario, measurements and parameter identificationhttps://www.zbmath.org/1483.340502022-05-16T20:40:13.078697Z"Givois, Arthur"https://www.zbmath.org/authors/?q=ai:givois.arthur"Tan, Jin-Jack"https://www.zbmath.org/authors/?q=ai:tan.jin-jack"Touzé, Cyril"https://www.zbmath.org/authors/?q=ai:touze.cyril"Thomas, Olivier"https://www.zbmath.org/authors/?q=ai:thomas.olivierSummary: A system composed of two cubic nonlinear oscillators with close natural frequencies, and thus displaying a 1:1 internal resonance, is studied both theoretically and experimentally, with a special emphasis on the free oscillations and the backbone curves. The instability regions of uncoupled solutions are derived and the bifurcation scenario as a function of the parameters of the problem is established, showing in an exhaustive manner all possible solutions. The backbone curves are then experimentally measured on a circular plate, where the asymmetric modes are known to display companion configurations with close eigenfrequencies. A control system based on a Phase-Locked Loop (PLL) is used to measure the backbone curves and also the frequency response function in the forced and damped case, including unstable branches. The model is used for a complete identification of the unknown parameters and an excellent comparison is drawn out between theoretical prediction and measurements.Emergence of stripe-core mixed spiral chimera on a spherical surface of nonlocally coupled oscillatorshttps://www.zbmath.org/1483.340512022-05-16T20:40:13.078697Z"Kim, Ryong-Son"https://www.zbmath.org/authors/?q=ai:kim.ryong-son"Tae, Gi-Hun"https://www.zbmath.org/authors/?q=ai:tae.gi-hun"Choe, Chol-Ung"https://www.zbmath.org/authors/?q=ai:choe.chol-ungThe authors consider the model
\[
\frac{\partial\psi(\mathbf{r},t)}{\partial t}=\omega+\frac{1}{4\pi}\int_{\mathbb{S}^2}G(\mathbf{r},\mathbf{r}')\sin{[\psi(\mathbf{r}',t)-\psi(\mathbf{r},t)-\alpha]} d\mathbf{r}'
\]
describing the evolution of the phase \(\psi(\mathbf{r},t)\) of an oscillator at position \(\mathbf{r}\) on the unit sphere at time \(t\). The nonlocal coupling function is
\[
G(\mathbf{r},\mathbf{r}')=\cos{\gamma}+\frac{\kappa}{4}(3\cos{(2\gamma)}+1)
\]
where \(\kappa\) is a parameter and \(\gamma\) is the great circle distance between points \(\mathbf{r}\) and \(\mathbf{r}'\). Their main finding is the existence of a stable stripe-core mixed spiral chimera state, in which two spiral waves separated by a stripe-type region of incoherent oscillators on the equator rotate around phase-randomized cores at the poles; these spirals are in anti-phase to each other. Such a state can exist only when \(\kappa\neq 0\). The authors use the Ott/Antonsen ansatz to describe the evolution of the system and largely analytically determine the existence and stability of this and other states such as the two-core spiral state and the incoherent state, exploring the \((\alpha,\kappa)\) plane. Numerical simulations of a discretisation of the phase model are shown, which agree with the analysis given.
Reviewer: Carlo Laing (Auckland)Dynamics of non-autonomous oscillator with a controlled phase and frequency of external forcinghttps://www.zbmath.org/1483.340522022-05-16T20:40:13.078697Z"Krylosova, D. A."https://www.zbmath.org/authors/?q=ai:krylosova.d-a"Seleznev, E. P."https://www.zbmath.org/authors/?q=ai:seleznev.eugene-p"Stankevich, N. V."https://www.zbmath.org/authors/?q=ai:stankevich.nataliya-vladimirovnaSummary: The dynamics of a non-autonomous oscillator in which the phase and frequency of the external force depend on the dynamical variable is studied. Such a control of the phase and frequency of the external force leads to the appearance of complex chaotic dynamics in the behavior of oscillator. A hierarchy of various periodic and chaotic oscillations is observed. The structure of the space of control parameters is studied. It is shown there are oscillatory modes similar to those of a non-autonomous oscillator with a potential in the form of a periodic function in the system dynamics, but there are also significant differences. Physical experiments of such systems are implemented.Stabilizing Stuart-Landau oscillators via time-varying networkshttps://www.zbmath.org/1483.340532022-05-16T20:40:13.078697Z"Pereti, Claudio"https://www.zbmath.org/authors/?q=ai:pereti.claudio"Fanelli, Duccio"https://www.zbmath.org/authors/?q=ai:fanelli.duccioSummary: A procedure is developed and tested to enforce synchronicity in a family of Stuart-Landau oscillators, coupled through a symmetric network. The proposed method exploits network plasticity, as an inherent non autonomous drive. More specifically, we assume that the system is initially confined on a network which turns the underlying homogeneous synchronous state unstable. A properly engineered network can be always generated, which links the same set of nodes, and allows for synchronicity to be eventually restored, upon performing continuously swappings, at a sufficient rate, between the two aforementioned networks. The result is cast in rigorous terms, as follows an application of the average theorem and the critical swapping rate determined analytically.S-shaped connected component of positive solutions for a Minkowski-curvature Dirichlet problem with indefinite weighthttps://www.zbmath.org/1483.340542022-05-16T20:40:13.078697Z"He, Zhiqian"https://www.zbmath.org/authors/?q=ai:he.zhiqian"Miao, Liangying"https://www.zbmath.org/authors/?q=ai:miao.liangyingSummary: In this paper, we investigate the existence of an S-shaped connected component in the set of positive solutions for a Minkowski-curvature Dirichlet problem with indefinite weight. By figuring the shape of unbounded continua of solutions, we show the existence and multiplicity of positive solutions with respect to the parameter \(\lambda\). In particular, we obtain the existence of at least three positive solutions for \(\lambda\) being in a certain interval.Bifurcation of limit cycles in a piecewise smooth near-integrable systemhttps://www.zbmath.org/1483.340552022-05-16T20:40:13.078697Z"Tian, Yun"https://www.zbmath.org/authors/?q=ai:tian.yun"Shang, Xinyu"https://www.zbmath.org/authors/?q=ai:shang.xinyu"Han, Maoan"https://www.zbmath.org/authors/?q=ai:han.maoanAuthors' abstract: In this paper, we study the bifurcation of limit cycles in a class of piecewise smooth quadratic integrable systems under small polynomial perturbations of degree \(n\). By using the first order Melnikov function, we derive a lower bound for the number of limit cycles which bifurcate from the period annulus.
Reviewer: Majid Gazor (Isfahan)On the maximum number of period annuli for second order conservative equationshttps://www.zbmath.org/1483.340562022-05-16T20:40:13.078697Z"Gritsans, Armands"https://www.zbmath.org/authors/?q=ai:gritsans.armands"Yermachenko, Inara"https://www.zbmath.org/authors/?q=ai:yermachenko.inaraSummary: We consider a second order scalar conservative differential equation whose potential function is a Morse function with a finite number of critical points and is unbounded at infinity. We give an upper bound for the number of nonglobal nontrivial period annuli of the equation and prove that the upper bound obtained is sharp. We use tree theory in our considerations.Periodic solutions for a class of \(n\)-dimensional prescribed mean curvature equationshttps://www.zbmath.org/1483.340572022-05-16T20:40:13.078697Z"Liang, Zai-tao"https://www.zbmath.org/authors/?q=ai:liang.zaitao"Lu, Shi-ping"https://www.zbmath.org/authors/?q=ai:lu.shipingIn this paper, there is investigated the periodic problem
\begin{align*}
\frac{d}{dt}\phi(x^{\prime})+\nabla W(x)&= p(t),\\ x(0)=x(T),\quad x^{\prime}(0)&= x^{\prime}(T),
\end{align*}
where \(x^\top=(x_1, x_2, \dots, x_n)\), \(W\in \mathcal{C}^1(\mathbb{R}^n)\), \(p\in \mathcal{C}(\mathbb{R})\), \(p(t+T)=p(t)\), \(\phi(x)= \frac{x}{\sqrt{1+|x|}}\). The authors provide conditions for the functions \(p\) and \(W\) so that the considered periodic problem has at least one periodic solution. The proof of the main result is based upon an extension of the Mawhin continuation theorem. The authors provide the main result with a suitable example
Reviewer: Svetlin Georgiev (Sofia)Investigation of dynamical properties in hysteresis-based a simple chaotic waveform generator with two stable equilibriumhttps://www.zbmath.org/1483.340582022-05-16T20:40:13.078697Z"Joshi, Manoj"https://www.zbmath.org/authors/?q=ai:joshi.manoj"Ranjan, Ashish"https://www.zbmath.org/authors/?q=ai:ranjan.ashishSummary: This research article describes a novel simple chaotic oscillator using bistable operation to generate chaotic waveform. In this design, chaos generation uses differential hysteresis phenomena of an Operational Amplifier (Op-Amp) with tank circuit. The behavior of the proposed chaotic system is investigated in terms of basic dynamical characteristics viz. equilibrium point stability, divergence, Lyapunov exponents, influence of initial condition, routes of chaos, basin of attraction and phase portraits by using theoretical analysis in MATLAB. We observed that proposed chaotic system belongs to the class of hidden attractor with two stable equilibrium points without quadratic or multiplying term that reduced the circuit complexity. Finally, an experimental investigation of the proposed design is performed that validates the theoretical and PSPICE results.Bursting oscillations and bifurcation mechanism in memristor-based Shimizu-Morioka system with two time scaleshttps://www.zbmath.org/1483.340592022-05-16T20:40:13.078697Z"Wen, Zihao"https://www.zbmath.org/authors/?q=ai:wen.zihao"Li, Zhijun"https://www.zbmath.org/authors/?q=ai:li.zhijun"Li, Xiang"https://www.zbmath.org/authors/?q=ai:li.xiang.4|li.xiang.3|li.xiang.2Summary: Bursting oscillators have received great attention in recent years, however, the research on this issue associated with memristive systems has been rarely reported. In this paper, bursting oscillations and bifurcation mechanism in a memristor-based Shimizu-Morioka system are investigated when an order gap exists between the excitation frequency and the natural frequency. Firstly, the bifurcation properties of the fast system are exploited by considering the periodic excitation as a slow-varying parameter. And the stability of different attractors and the critical values of different bifurcations are obtained. Secondly, Complex bursting oscillators are revealed when the slow-varying parameter passes through these critical values. The corresponding bifurcation mechanism, namely, symmetric Fold/Fold, symmetric compound Fold/Fold-delayed supHopf/supHopf, symmetric compound subHopf/subHopf-supHopf/supHopf, symmetric subHopf/subHopf, supHopf/saddle on limit cycle, symmetric delayed supHopf/delayed supHopf, symmetric delay supHopf-supHopf/supHopf are analyzed by the transformed phase portraits, the time series, and the phase portraits. Furthermore, the effect of the excitation frequency on the symmetric Fold/Fold bursting is also revealed. Finally, some numerical and circuit simulation results are provided to verify the validity of the study.An example of Silnikov focus-focus homoclinic orbitshttps://www.zbmath.org/1483.340602022-05-16T20:40:13.078697Z"Battelli, Flaviano"https://www.zbmath.org/authors/?q=ai:battelli.flaviano"Palmer, Kenneth J."https://www.zbmath.org/authors/?q=ai:palmer.kenneth-jamesIn the present manuscript, the authors provide a concrete example of a \textit{four-dimensional} autonomous system of ODEs possessing a \textit{Silnikov homoclinic orbit}.
We remind that, given an autonomous system of ODEs
\[
\dot x = F(x)\qquad\text{(where $F\in C^1(\mathbb{R}^n)$)}\tag{S}
\]
with a hyperbolic equilibrium $q$, a solution $x = x(t)$ of (S) is a \textit{Silnikov homoclinic orbit} if it satisfies the following properties:
\begin{itemize}
\item[(D0)] $x(t) \neq q$ and $|x(t)-q|\to 0$ as $|t|\to\infty$;
\item[(D1)] the eigenvalues of $F'(q)$ having the smallest positive real part are of the form $\sigma+i\omega$, with $\omega > 0$, each having algebraic multiplicity one, and such that $0 < \sigma < -\mathrm{Re}(\lambda)$ for all eigenvalues $\lambda$ with $\mathrm{Re}(\lambda) < 0$;
\item[(D2)] up to a scalar multiple, $x'(t)$ is the unique nontrivial bounded solution of
\[
\dot y = F'(x(t))y;
\]
\item[(D3)] $x(t)e^{-\nu t}$ is unbounded on $t\leq 0$, where $\nu > 0$ is such that
\[
\text{$\sigma < \nu < \lambda$ for all eigenvalues $\lambda$ of $F'(q)$ with $\mathrm{Re}(\lambda) > \sigma$};
\]
\item[(D4)] there does not exist $\xi\neq 0$ such that the solution $y(t)$ of
\[
\begin{cases} \dot y = -F'(x(t))^*y, \\ y(0) = \xi \end{cases}
\]
is bounded on $\mathbb{R}_-$, and the solution of
\[
\begin{cases} \dot y = -\big(F'(x(t))^*-\nu\big)y, \\ y(0) = \xi \end{cases}
\]
is bounded on $\mathbb{R}_+$.
\end{itemize}
As pointed out by the authors in the Introduction, the importance of these orbits is that there exists \textit{chaotic behavior in their neighborhood}.
In order to construct the mentioned example, the authors exploit essentially a \textit{perturbation approach}, which is carefully explained in the Introduction. \begin{itemize}
\item[(a)] First of all, they construct a four-dimensional system having a homoclinic solution to a hyperbolic fixed point at the origin, and such that the linearization at 0 has two real eigenvalues with both algebraic and geometrical multiplicity 2.
\item[(b)] Then, they add a two-parameter dependent perturbation of the form
\[
A(\gamma)x+\mu h(x)
\]
where $A(\gamma)$ is a matrix such that $A(0) = 0$ and $h(x)$ is a nonlinear function such that $h(0) = 0$ and $h'(0) = 0$.
\end{itemize}
Under suitable `technical' assumptions on the function $h$, the authors are able to deduce from a general result (Theorem 1 of the paper) that the resulting system has a Silnikov homoclinic orbit.
Reviewer: Stefano Biagi (Milano)Existence of two-point oscillatory solutions of a relay nonautonomous system with multiple eigenvalue of a real symmetric matrixhttps://www.zbmath.org/1483.340612022-05-16T20:40:13.078697Z"Yevstafyeva, V. V."https://www.zbmath.org/authors/?q=ai:yevstafyeva.victoria-v|yevstafyeva.vistoria-vSummary: We study an \(n\)-dimensional system of ordinary differential equations with hysteresis type relay nonlinearity and a periodic perturbation function on the right-hand side. It is supposed that the matrix of the system is real and symmetric and, moreover, it has an eigenvalue of multiplicity two. In the phase space of the system, we consider continuous bounded oscillatory solutions with two fixed points and the same time of return to each of these points. For these solutions, we prove the existence and nonexistence theorems. For a three-dimensional system, these results are illustrated by a numerical example.The influence of vaccination on the control of JE with a standard incidence rate of mosquitoes, pigs and humanshttps://www.zbmath.org/1483.340622022-05-16T20:40:13.078697Z"Baniya, Vinod"https://www.zbmath.org/authors/?q=ai:baniya.vinod"Keval, Ram"https://www.zbmath.org/authors/?q=ai:keval.ramSummary: In this article, a nonlinear mathematical model used for the impact of vaccination on the control of infectious disease, Japanese encephalitis with a standard incidence rate of mosquitoes, pigs and humans has been planned and analyzed. During the modeling process, it is expected that the disease spreads only due to get in touch with the susceptible and infected class only. It is also assumed that due to the effect of vaccination, the total human population forms a separate class and avoids contact with the infection. The dynamical behaviors of the system have been explored by using the stability theory of differential equations and numerical simulations. The local and global stability of the system for both equilibrium states under certain conditions has been studied. We have set up a threshold condition in the language of the vaccine-induced reproduction number \(R(\alpha_1)\), which is fewer than unity, the disease dies in the absence of the infected population, otherwise, the infection remains in the population. Furthermore, it is found that vaccine coverage has a substantial effect on the basic reproduction number. Also, by continuous efforts and effectiveness of vaccine coverage, the disease can be eradicated. It is also found a more sensitive parameter for the transmission of Japanese encephalitis virus by using sensitivity analysis. In addition, numerical results are used to investigate the effect of some parameters happening the control of JE infection, for justification of analytical results.Harvesting and refugia control chaos-conclusion drawn from a tri-trophic food chainhttps://www.zbmath.org/1483.340632022-05-16T20:40:13.078697Z"Das, Krishna Pada"https://www.zbmath.org/authors/?q=ai:das.krishnapada"Agnihotri, Kulbhushan"https://www.zbmath.org/authors/?q=ai:agnihotri.kulbhushan"Kaur, Harpreet"https://www.zbmath.org/authors/?q=ai:kaur.harpreetSummary: In the present work, our thought process is to investigate the effect of harvesting and refugia on the dynamics of a continuous-time tri-trophic food chain model. To peruse these features we have explored the local stability behavior of various equilibrium points. Conditions for Hopf-bifurcation and persistence have been inferred. Extensive numerical simulation work has been performed to reveal the dynamics of the system. Simulation results exhibit the chaotic dynamics of the system when the value of the half-saturation constant is increased. Further, it is established that the chaotic behavior is controlled by increasing the harvesting parameter value. Again, the chaotic behavior is observed to be controlled by increasing the value of the refugia parameter. Thus we infer that harvesting and refugia parameters can be used to restrain the chaotic dynamics of the model system.A mathematical model of anaerobic digestion with syntrophic relationship, substrate inhibition, and distinct removal rateshttps://www.zbmath.org/1483.340642022-05-16T20:40:13.078697Z"Fekih-Salem, Radhouane"https://www.zbmath.org/authors/?q=ai:fekih-salem.radhouane"Daoud, Yessmine"https://www.zbmath.org/authors/?q=ai:daoud.yessmine"Abdellatif, Nahla"https://www.zbmath.org/authors/?q=ai:abdellatif.nahla"Sari, Tewfik"https://www.zbmath.org/authors/?q=ai:sari.tewfikA control-based mathematical study on psoriasis dynamics with special emphasis on \(\text{IL}-21\) and \(\text{IFN} - \gamma\) interaction networkhttps://www.zbmath.org/1483.340652022-05-16T20:40:13.078697Z"Roy, Amit Kumar"https://www.zbmath.org/authors/?q=ai:roy.amit-kumar"Nelson, Mark"https://www.zbmath.org/authors/?q=ai:nelson.mark-p|nelson.mark-e|nelson.mark-ian"Roy, Priti Kumar"https://www.zbmath.org/authors/?q=ai:kumar-roy.pritiSummary: Psoriasis is characterized by the excessive growth of keratinocytes (skin cells), which is initiated by chaotic signaling within the immune system and irregular release of cytokines. Pro-inflammatory cytokines: Interleukin \(21 (\text{IL} - 21)\) and Interferon gamma \(( \text{IFN} - \gamma )\), released by \(\text{Th}_1\) cell and activated natural killer cells (NK cells) respectively, play central role in the disease pathogenesis. In this work, we have constructed two sets of nonlinear differential equations. One is representing the growth of three vital immune cells (T helper cells (type I and II) and activated NK cells) along with keratinocyte and the other set is for cytokines' \((\text{IL} - 21\) and \(\text{IFN} - \gamma )\) dynamics. The hazardous effects of these cytokines, preconditions for disease persistence and validation of the stability criteria of endemic equilibrium have been studied analytically. We have also observed the effect of the combined biologic therapy (anti \(\text{IFN} - \gamma\) and \(\text{IL} - 21\) inhibitor) by considering an optimal control problem. Analytical and numerical results reveal that the impact of activated NK cells on excessive formation of keratinocytes is mostly regulated by the effects of \(\text{IL} - 21\) and \(\text{IFN} - \gamma \).External localized harmonic influence on an incoherence cluster of chimera stateshttps://www.zbmath.org/1483.340662022-05-16T20:40:13.078697Z"Shepelev, I. A."https://www.zbmath.org/authors/?q=ai:shepelev.igor-aleksandrovich"Vadivasova, T. E."https://www.zbmath.org/authors/?q=ai:vadivasova.tatyana-evgenevna|vadivasova.tatiana-eSummary: We study impacts of external harmonic forces on chimera states in an ensemble of chaotic Rössler oscillators with nonlocal interaction. The main attention is paid to control the spatial structure by applying a targeted localized excitation on an incoherence cluster. This influence on a phase chimera enables us to eliminate the incoherence cluster and to realize the regime with a piecewise smooth spatial profile. The mechanism of elimination of the incoherence cluster of the phase chimera consists in-phase synchronization of all oscillators within the region of influence of the external force. This phenomenon is observed for a sufficiently wide range of the external force frequency, especially when its value is less than the natural frequency. Increasing the external force amplitude can lead to two scenarios depending on the dynamics of individual oscillators. In the case of regular dynamics, a strong force induces another type of the incoherence cluster within the region of the external force influence. The oscillator dynamics within this region becomes chaotic. Thus, the features of this cluster are similar to those for the incoherence cluster of an amplitude chimera. When the dynamics is chaotic, the force can cause the system to switch to the regime of a metastable spatial distribution with a qualitatively different character at different time intervals. It is impossible to eliminate the incoherence cluster of the amplitude chimera by means of the localized harmonic influence for any values of its parameters. The destruction of the amplitude chimera structure under the influence of the external force leads either to the intermittent regime or to inducing the stable incoherence cluster.Geometric analysis of oscillations in the Frzilator modelhttps://www.zbmath.org/1483.340672022-05-16T20:40:13.078697Z"Taghvafard, Hadi"https://www.zbmath.org/authors/?q=ai:taghvafard.hadi"Jardón-Kojakhmetov, Hildeberto"https://www.zbmath.org/authors/?q=ai:jardon-kojakhmetov.hildeberto"Szmolyan, Peter"https://www.zbmath.org/authors/?q=ai:szmolyan.peter"Cao, Ming"https://www.zbmath.org/authors/?q=ai:cao.mingThe authors analyze a biochemical oscillator 3-dimensional model that describes the developmental stage of a myxobacteria. Observations from numerical simulations show that the corresponding ordinary differential system displays stable and robust oscillations. The existence of an attracting limit cycle is proved using geometric singular perturbation theory and blow-up method. It corresponds to a relaxation oscillation of an auxiliary system, whose singular perturbation nature originates from the small Michaelis-Menten constants of the biochemical model.
Reviewer: Joan Torregrosa (Barcelona)Mathematical modeling of the impact of temperature variations and immigration on malaria prevalence in Nigeriahttps://www.zbmath.org/1483.340682022-05-16T20:40:13.078697Z"Ukwajunor, Eunice E."https://www.zbmath.org/authors/?q=ai:ukwajunor.eunice-e"Akarawak, Eno E. E."https://www.zbmath.org/authors/?q=ai:akarawak.eno-e-e"Abiala, Israel Olutunji"https://www.zbmath.org/authors/?q=ai:abiala.israel-olutunjiDynamics of a vector-host model under switching environmentshttps://www.zbmath.org/1483.340692022-05-16T20:40:13.078697Z"Watts, Harrison"https://www.zbmath.org/authors/?q=ai:watts.harrison"Mishra, Arti"https://www.zbmath.org/authors/?q=ai:mishra.arti"Nguyen, Dang H."https://www.zbmath.org/authors/?q=ai:nguyen.dang-hai"Tuong, Tran D."https://www.zbmath.org/authors/?q=ai:tuong.tran-dinhSummary: In this paper, the stochastic vector-host model has been proposed and analysed using nice properties of piecewise deterministic Markov processes (PDMPs). A threshold for the stochastic model is derived whose sign determines whether the disease will eventually disappear or persist. We show mathematically the existence of scenarios where switching plays a significant role in surprisingly reversing the long-term properties of deterministic systems.Global stability in a three-species Lotka-Volterra cooperation model with seasonal successionhttps://www.zbmath.org/1483.340702022-05-16T20:40:13.078697Z"Xie, Xizhuang"https://www.zbmath.org/authors/?q=ai:xie.xizhuang"Niu, Lin"https://www.zbmath.org/authors/?q=ai:niu.linSummary: In this paper, we focus on a three-species Lotka-Volterra cooperation model with seasonal succession. The Floquet multipliers of all nonnegative periodic solutions of such a time-periodic system are estimated via the stability analysis of equilibria. By Brouwer fixed point theorem and the connecting orbits theorem, it is proved that there admits a unique positive periodic solution under appropriate conditions. Furthermore, sharp global asymptotical stability criteria for extinction and coexistence are established. Compared to the classical three-species Lotka-Volterra cooperation model, the introduction of seasonal succession may lead to species' extinction. Finally, some numerical examples are given to illustrate the effectiveness of our theoretical results.Complex dynamics of a SIRS epidemic model with the influence of hospital bed numberhttps://www.zbmath.org/1483.340712022-05-16T20:40:13.078697Z"Xu, Yancong"https://www.zbmath.org/authors/?q=ai:xu.yancong"Wei, Lijun"https://www.zbmath.org/authors/?q=ai:wei.lijun"Jiang, Xiaoyu"https://www.zbmath.org/authors/?q=ai:jiang.xiaoyu"Zhu, Zirui"https://www.zbmath.org/authors/?q=ai:zhu.ziruiSummary: In this paper, the nonlinear dynamics of a SIRS epidemic model with vertical transmission rate of neonates, nonlinear incidence rate and nonlinear recovery rate are investigated. We focus on the influence of public available resources (especially the number of hospital beds) on disease control and transmission. The existence and stability of equilibria are analyzed with the basic reproduction number as the threshold value. The conditions for the existence of transcritical bifurcation, Hopf bifurcation, saddle-node bifurcation, backward bifurcation and the normal form of Bogdanov-Takens bifurcation are obtained. In particular, the coexistence of limit cycle and homoclinic cycle, and the coexistence of stable limit cycle and unstable limit cycle are also obtained. This study indicates that maintaining enough number of hospital beds is very crucial to the control of the infectious diseases no matter whether the immunity loss population are involved or not. Finally, numerical simulations are also given to illustrate the theoretical results.Bifurcation and dynamic analyses of non-monotonic predator-prey system with constant releasing rate of predatorshttps://www.zbmath.org/1483.340722022-05-16T20:40:13.078697Z"Zhou, Hao"https://www.zbmath.org/authors/?q=ai:zhou.hao"Tang, Biao"https://www.zbmath.org/authors/?q=ai:tang.biao"Zhu, Huaiping"https://www.zbmath.org/authors/?q=ai:zhu.huaiping"Tang, Sanyi"https://www.zbmath.org/authors/?q=ai:tang.sanyiSummary: In this paper, we systematical study the rich dynamics and complex bifurcations of a non-monotonic predator-prey system with a constant releasing rate for the predator. We prove that the system can have at most three positive equilibria, and can undergo a sequence of bifurcations, including transcritical, saddle-node, Hopf, degenerate Hopf, double limit cycle, saddle-node homoclinic bifurcation (or homoclinic loop with a saddle-node), cusp bifurcation of codimension 2, and Bogdanov-Takens bifurcation of codimension 2 and 3. And the system can generate very rich dynamics, such as the existence of a semi-stable limit cycle, multiple coexistent periodic orbits, homoclinic loops, etc. Moreover, our results show that the dynamical behaviors highly rely on the constant releasing rate of predators and the initial conditions. That is, there exists a critical value of the constant releasing rate of predators such that (i) when the constant releasing rate is greater than the critical value, the prey goes to extinction for all admissible initial populations of both species; (ii) when the constant releasing rate is less than the critical value, the prey can always coexist with the predator. Numerical simulations are presented to verify the main results.Synchronization of nonlinearly coupled complex networks: distributed impulsive methodhttps://www.zbmath.org/1483.340732022-05-16T20:40:13.078697Z"Ding, Dong"https://www.zbmath.org/authors/?q=ai:ding.dong"Tang, Ze"https://www.zbmath.org/authors/?q=ai:tang.ze"Wang, Yan"https://www.zbmath.org/authors/?q=ai:wang.yan.3"Ji, Zhicheng"https://www.zbmath.org/authors/?q=ai:ji.zhichengSummary: This paper is devoted to discussing the exponential synchronization of a class of nonlinearly coupled complex networks (NCCNs) with time-varying delays and stochastic disturbance. In consideration of the complex networks would subject to certain impulse disturbances, a kind of distributed controller combining with pinning control and impulsive control schemes is designed. Based on the concept of average impulsive interval and comparison principles, sufficient conditions for achieving the exponential synchronization are derived. In addition, the exponential convergence velocities are obtained, respectively, with considering different functions of the impulsive inputs. Finally, a numerical simulation is presented to demonstrate the feasibility of the theoretical deduction.Composite synchronization of four exciters driven by induction motors in a vibration systemhttps://www.zbmath.org/1483.340742022-05-16T20:40:13.078697Z"Kong, Xiangxi"https://www.zbmath.org/authors/?q=ai:kong.xiangxi"Zhou, Chong"https://www.zbmath.org/authors/?q=ai:zhou.chong"Wen, Bangchun"https://www.zbmath.org/authors/?q=ai:wen.bangchunSummary: In this paper, a newly composite synchronization scheme is proposed to ensure the straight line vibration form of a linear vibration system driven by four exciters. Composite synchronization is a combination of self-synchronization and controlled synchronization. Firstly, controlled synchronization of two pairs of homodromous coupling exciters with zero phase differences is implemented by using the master-slave control structure and the adaptive sliding mode control algorithm. On basis of controlled synchronization, self-synchronization of two coupling exciters rotating in the opposite directions is studied. Based on the perturbation method, the synchronization and stability conditions of composite synchronization are obtained. The theoretical results indicate that composite synchronization of four exciters with zero phase differences can be implemented with different supply frequencies and the straight line vibration form of the linear vibration system also can be obtained. Some simulations are conducted to verify the feasibility of the proposed composite synchronization scheme. The effects of some structural parameters on composite synchronization of four exciters are discussed. Finally, some experiments are operated to validate the effectiveness of the proposed composite synchronization scheme.Inferring the connectivity of coupled oscillators and anticipating their transition to synchrony through lag-time analysishttps://www.zbmath.org/1483.340752022-05-16T20:40:13.078697Z"Leyva, Inmaculada"https://www.zbmath.org/authors/?q=ai:leyva.inmaculada"Masoller, Cristina"https://www.zbmath.org/authors/?q=ai:masoller.cristinaSummary: The synchronization phenomenon is ubiquitous in nature. In ensembles of coupled oscillators, explosive synchronization is a particular type of transition to phase synchrony that is first-order as the coupling strength increases. Explosive sychronization has been observed in several natural systems, and recent evidence suggests that it might also occur in the brain. A natural system to study this phenomenon is the Kuramoto model that describes an ensemble of coupled phase oscillators. Here we calculate bi-variate similarity measures (the cross-correlation, \(\rho_{ij}\), and the phase locking value, \(\text{PLV}_{ij})\) between the phases, \(\varphi_i(t)\) and \(\varphi_j(t)\), of pairs of oscillators and determine the lag time between them as the time-shift, \(\tau_{ij}\), which gives maximum similarity (i.e., the maximum of \(\rho_{ij}(\tau)\) or \(\text{PLV}_{ij}(\tau))\). We find that, as the transition to synchrony is approached, changes in the distribution of lag times provide an earlier warning of the synchronization transition (either gradual or explosive). The analysis of experimental data, recorded from Rossler-like electronic chaotic oscillators, suggests that these findings are not limited to phase oscillators, as the lag times display qualitatively similar behavior with increasing coupling strength, as in the Kuramoto oscillators. We also analyze the statistical relationship between the lag times between pairs of oscillators and the existence of a direct connection between them. We find that depending on the strength of the coupling, the lags can be informative of the network connectivity.Asymptotics of the solution to a two-band two-point boundary value problemhttps://www.zbmath.org/1483.340762022-05-16T20:40:13.078697Z"Tursunov, D. A."https://www.zbmath.org/authors/?q=ai:tursunov.dilmurat-abdillazhanovich"Omaralieva, G. A."https://www.zbmath.org/authors/?q=ai:omaralieva.g-aThe authors constructed a complete asymptotic approximation of the solution with respect to a small parameter of the Dirichlet boundary value problem for a singularly perturbed linear inhomogeneous second-order ordinary differential equation
\[
\varepsilon^4 y''_{\varepsilon}(x) + x^2p(x)y'_{\varepsilon}(x)-\varepsilon q(x)y_{\varepsilon}(x) = f(x),\ x\in(0,1),
\]
\[
y_{\varepsilon}(0) = a,\ y_{\varepsilon}(1) = b,
\]
where \(0 < \varepsilon\ll 1,\) \(p(x)>0,\) \(q(x)>0\) and \(f(x)\) are smooth functions on \([0,1]\) and \(a,b\in\mathbb{R}\) are known constants.
The problem under consideration differs from the previously investigated problems in that in the vicinity of the left boundary point \(x = 0\) there is a two-layer boundary layer, and the solution of the corresponding unperturbed problem is not a smooth function. Therefore, it is impossible to solve the problem using the classical method of boundary functions. First, a formal asymptotic approximation of the problem under study was constructed by generalized and classical methods of boundary functions, then, using the maximum principle, an estimate was obtained for the residual function of the constructed series. The resulting series is asymptotic in the sense of Erdei.
Reviewer: Robert Vrabel (Trnava)Asymptotic solution of the Cauchy problem for the first-order equation with perturbed Fredholm operatorhttps://www.zbmath.org/1483.340772022-05-16T20:40:13.078697Z"Uskov, Vladimir Igorevich"https://www.zbmath.org/authors/?q=ai:uskov.vladimir-igorevichSummary: We consider the Cauchy problem for a first-order differential equation in a Banach space. The equation contains a small parameter in the highest derivative and a Fredholm operator perturbed by an operator addition on the right-hand side. Systems with small parameter in the highest derivative describe the motion of a viscous flow, the behavior of thin and flexible plates and shells, the process of a supersonic viscous gas flow around a blunt body, etc. The presence of a boundary layer phenomenon is revealed; in this case, even a small additive has a strong influence on the behavior of the solution. Asymptotic expansion of the solution in powers of small parameter is constructed by means of the Vasil'yeva-Vishik-Lyusternik method. Asymptotic property of the expansion is proved. To construct the regular part of the expansion, the equation decomposition method is used. It is consisted in a step-by-step transition to similar problems of decreasing dimensions.Quadratic slow-fast systems on the planehttps://www.zbmath.org/1483.340782022-05-16T20:40:13.078697Z"Meza-Sarmiento, Ingrid S."https://www.zbmath.org/authors/?q=ai:meza-sarmiento.ingrid-s"Oliveira, Regilene"https://www.zbmath.org/authors/?q=ai:oliveira.regilene-d-s"da Silva, Paulo R."https://www.zbmath.org/authors/?q=ai:da-silva.paulo-ricardoIn this paper, singularly perturbed quadratic polynomial differential systems of the form \[\varepsilon \dot x = P_{\varepsilon}(x, y) = P(x, y, \varepsilon),\quad \dot y = Q_{\varepsilon}(x, y) = Q(x, y, \varepsilon)\] with \(x, y \in {\mathbb R}\), \(\varepsilon \geq 0\) are considered. It is assumed that \((P_{\varepsilon}, Q_{\varepsilon}) = 1\) for \(\varepsilon > 0\). It is shown that there are 10 classes of equivalence for these systems. The dynamics of these 10 classes on the Poincare disc when \(\varepsilon = 0\) are described. For \(\varepsilon > 0\), the possible local behavior of the solutions near of a finite and infinite equilibrium point under suitable conditions is presented. More specifically, if \(p_0\) is a finite equilibrium point then the local behavior for \(\varepsilon > 0\) is obtained using Fenichel theory. If \(p_0\) is an infinite equilibrium point, then there exists \(\mathcal{K} \subset\mathcal{M}_0\) normally hyperbolic and \(p_0 \in\mathcal{M}_0^{'} \cap\mathcal{K}\), where \(\mathcal{M}_0\) is the critical manifold of the singular perturbation problem under consideration.
Reviewer: Vasile Dragan (Bucureşti)Stochastic resonance in a high-order time-delayed feedback tristable dynamic system and its applicationhttps://www.zbmath.org/1483.340792022-05-16T20:40:13.078697Z"Shi, Peiming"https://www.zbmath.org/authors/?q=ai:shi.peiming"Zhang, Wenyue"https://www.zbmath.org/authors/?q=ai:zhang.wenyue"Han, Dongying"https://www.zbmath.org/authors/?q=ai:han.dongying"Li, Mengdi"https://www.zbmath.org/authors/?q=ai:li.mengdiSummary: A stochastic resonance (SR) tristable system based on a high-order time-delayed feedback is investigated and the feasibility of the system for weak fault signature extraction is discussed. The potential function, the mean first-passage time (MFPT) and the signal-to-noise ratio (SNR) are used to evaluate the model. Firstly, the potential function and stationary probability function (PDF) of the system are derived, and then the influence of the time delay parameters on the MFPT of the particles is analyzed. Secondly, the influences of time-delyed strength \(e\) and delyed length \(\tau\) on the SR system from the perspective of the transition of the particles in the potential wells are discussed, and then the SNR and the effect of the parameters on the SNR are derived. In addition, the high-order time-delayed feedback tristable stochastic resonance (HTFTSR) system is used to deal with faulty bearing data and is compared with traditional tristable stochastic resonance (TSR). The result shows that the nonlinear system model can accurately identify the fault frequency and improve the energy of the characteristic signal under the appropriate system parameters.On the Lyapunov-Perron reducible Markovian master equationhttps://www.zbmath.org/1483.340802022-05-16T20:40:13.078697Z"Szczygielski, Krzysztof"https://www.zbmath.org/authors/?q=ai:szczygielski.krzysztofConvergence rates for boundedly regular systemshttps://www.zbmath.org/1483.340812022-05-16T20:40:13.078697Z"Csetnek, Ernö Robert"https://www.zbmath.org/authors/?q=ai:csetnek.erno-robert"Eberhard, Andrew"https://www.zbmath.org/authors/?q=ai:eberhard.andrew-c|eberhard.andrew"Tam, Matthew K."https://www.zbmath.org/authors/?q=ai:tam.matthew-kLet \(\mathcal{H}\) be a real Hilbert space; an operator \(T \colon \mathcal{H} \to \mathcal{H}\) is called Hölder regular on \(U \subseteq \mathcal{H}\) if there exist \(\kappa > 0\) and \(\gamma \in (0,1)\) such that
\[
d(y, \mathrm{Fix }\,T) \leq \kappa \parallel y - T(y) \parallel^\gamma \quad \forall y \in U.
\]
If \(T\) is Hölder regular on each bounded subset of \(\mathcal{H}\) then it is called boundedly Hölder regular. If the corresponding property is true for \(\gamma = 1,\) the operator \(T\) is called boundedly linearly regular.
The main result. Let \(T\) be nonexpansive with Fix\(\, T \neq \emptyset\) and \(\lambda \colon [0, +\infty) \to [0,1]\) be Lebesgue measurable function with \(\lambda^\star := \inf_{t\geq0} \lambda (t) >0.\) Let \(x\) be the unique strong global solution of the equation
\[
\dot{x}(t) = \lambda (t)(T(x(t)) - x(t)), \quad x(0) = x_0.
\]
If \(T\) is boundedly linearly regular, then there exists \(\overline{x} \in \mathrm{Fix }\,T\) such that for almost all \(t \in [0,+\infty)\) the following estimate holds:
\[
\parallel x(t) - \bar{x} \parallel \leq 2 \exp \Big(- \frac{\lambda^\star}{2\kappa^2}t\Big)\, d(x_0, \mathrm{Fix }\,T).
\]
An analogous result is also valid for the case of a boundedly Hölder regular operator \(T.\)
Reviewer: Valerii V. Obukhovskij (Voronezh)\((\omega,\mathbb{T})\)-periodic solutions of impulsive evolution equationshttps://www.zbmath.org/1483.340822022-05-16T20:40:13.078697Z"Fečkan, Michal"https://www.zbmath.org/authors/?q=ai:feckan.michal"Liu, Kui"https://www.zbmath.org/authors/?q=ai:liu.kui"Wang, JinRong"https://www.zbmath.org/authors/?q=ai:wang.jinrongSummary: In this paper, we study \((\omega,\mathbb{T})\)-periodic impulsive evolution equations via the operator semigroups theory in Banach spaces \(X\), where \(\mathbb{T}: X\rightarrow X\) is a linear isomorphism. Existence and uniqueness of \((\omega,\mathbb{T})\)-periodic solutions results for linear and semilinear problems are obtained by Fredholm alternative theorem and fixed point theorems, which extend the related results for periodic impulsive differential equations.Long-time behavior of a gradient system governed by a quasiconvex functionhttps://www.zbmath.org/1483.340832022-05-16T20:40:13.078697Z"Rahimi Piranfar, Mohsen"https://www.zbmath.org/authors/?q=ai:piranfar.mohsen-rahimi"Khatibzadeh, Hadi"https://www.zbmath.org/authors/?q=ai:khatibzadeh.hadiThis is an excellent paper on the long time behavior of the solutions to the second order equation \[ u^{\prime \prime}(t)=\nabla \phi (u(t)), \ \ t\ge 0, \] where \(\phi\) is a differentiable quasiconvex function defined on a real Hilbert space \(H\) with a nonempty set of minimizers. More precisely, if in addition \(\nabla \phi\) is Lipschitz continuous on bounded sets and \(\Vert u(t) \Vert \) is bounded, then \(u(t)\) converges weakly as \(t\longrightarrow \infty\) to a point \(p\in (\nabla \phi )^{-1}(0)\), and if \(p\notin \mathrm{argmin}\, \phi\), then the convergence is strong; on the other hand, if \(\Vert u(t)\Vert\) is is unbounded, then \(\Vert u(t)\Vert \longrightarrow \infty\) as \(t\longrightarrow \infty\). Thus a previous analysis of the second author and the reviewer [J. Convex Anal. 26, No. 4, 1175--1186 (2019; Zbl 1435.34061)] is here successfully extended from the pseudoconvex case to the quasiconvex one. The discrete version of the above equation is also investigated in detail.
Reviewer: Gheorghe Moroşanu (Cluj-Napoca)Weak solvability for parabolic variational inclusions and application to quasi-variational problemshttps://www.zbmath.org/1483.340842022-05-16T20:40:13.078697Z"Kenmochi, Nobuyuki"https://www.zbmath.org/authors/?q=ai:kenmochi.nobuyuki"Niezgódka, Marek"https://www.zbmath.org/authors/?q=ai:niezgodka.marekThe authors investigate weak solvability of the class of parabolic variational evolution inclusions
\[
u^{\prime}(t)+\partial\varphi^{t}\left( p;u(t)\right) \ni f(t)\text{, }0<t<T
\]
\[
u(0)=u_{0}
\]
in a real Hilbert space, with nonlocal parameter \(p\), where \(\partial \varphi^{t}\left( p;\cdot\right) \) is the subdifferential of a time-dependent nonnegative convex function \(z\mapsto\varphi^{t}\left( p;z\right) \) with nonlocal dependence on \(p\), and \(f\in L^{2}\left( (0,T);H\right) \). \ The parameter \(p\) comes from a set \(X_{0}\) which is a bounded and closed subset of \(C\left( \left[ 0,T\right] ;X\right) \) where \(X\) is a real Banach space. \ Existence and uniqueness of a weak solution, and also continuous dependence on \(p\), \(f\) and \(u_{0}\), are proven under several technical assumptions. Specific examples are given for \(\varphi^{t}\left( p;\cdot\right) \). \ Existence of weak solutions for quasi-variational problems of the form
\[
u^{\prime}(t)+\partial\varphi^{t}\left( p;u(t)\right) \ni f(t)\text{, }0<t<T
\]
\[
u(0)=u_{0}
\]
\[
p=\Lambda_{p_{0}}u
\]
are also studied using these results and Schauder's fixed point theorem, where \(\Lambda_{p_{0}}\) is a feedback system which is an operator from a subset of \(C\left( \left[ 0,T\right] ;H\right) \) into \(X_{0}\). \ Weak solvability for perturbations of these problems is also considered and the paper closes with an application.
Reviewer: Daniel C. Biles (Nashville)Mild solution and approximate controllability of second-order retarded systems with control delays and nonlocal conditionshttps://www.zbmath.org/1483.340852022-05-16T20:40:13.078697Z"Haq, Abdul"https://www.zbmath.org/authors/?q=ai:haq.abdul"Sukavanam, N."https://www.zbmath.org/authors/?q=ai:sukavanam.nagarajanSummary: This work studies the approximate controllability of a class of second-order retarded semilinear differential equations with nonlocal conditions and with delays in control. First, we deduce the existence of mild solutions using cosine family and fixed point approach. For this, the nonlinear function is supposed to be locally Lipschitz. Controllability of the system is shown using an approximate and iterative technique. The results are illustrated using an example.A novel amplitude control method for constructing nested hidden multi-butterfly and multiscroll chaotic attractorshttps://www.zbmath.org/1483.340862022-05-16T20:40:13.078697Z"Wu, Qiujie"https://www.zbmath.org/authors/?q=ai:wu.qiujie"Hong, Qinghui"https://www.zbmath.org/authors/?q=ai:hong.qinghui"Liu, Xiaoyang"https://www.zbmath.org/authors/?q=ai:liu.xiaoyang"Wang, Xiaoping"https://www.zbmath.org/authors/?q=ai:wang.xiaoping"Zeng, Zhigang"https://www.zbmath.org/authors/?q=ai:zeng.zhigangSummary: A novel amplitude control method (ACM) is proposed to construct multiple self-excited or hidden attractors by scaling partial or total variables without changing their dynamic and topological properties. Various attractors including nested attractor, axisymmetric attractor, and centrosymmetric attractor can be obtained by multiplying signals with different amplitudes. An universal pulse control module is designed to realize the amplitude scale. Different number of scrolls can be adjusted by regulating the pulse signals without redesigning the nonlinear circuit. The classical Lorenz system and Jerk system are employed as examples to generate nested hidden multi-butterfly and multiscroll attractors. Some novel properties of ACM, such as nested morphology, amplitude modulation, and constant Lyapunov exponential spectrum, are analyzed theoretically and simulated numerically. The circuit design and PSpice simulation results are implemented to verify the availability and feasibility of the proposed approach.Smooth solutions of linear functional differential equations of neutral typehttps://www.zbmath.org/1483.340872022-05-16T20:40:13.078697Z"Cherepennikov, V. B."https://www.zbmath.org/authors/?q=ai:cherepennikov.valery-b"Kim, A. V."https://www.zbmath.org/authors/?q=ai:kim.alexandra-v|kim.arkadii-vladimirovichConsider a scalar functional differential equation of the form
\[
\dot{y}(t)+p\dot{y}(t/q)=ay(t-1)+f(t),\quad t\in[0,+\infty)
\]
with initial condition \(y(t)=x^N(t)\), \(t\in[-1,0]\). We assume that \(q>1\), \(f(t)=\sum_{n=0}^F f_nt^n\) and \(x^N(t)=\sum_{n=0}^N x_nt^n\).
Suppose that \(x^N(t)\) satisfies the equation \(\dot{x}(t)+p\dot{x}(t/q)=ax(t-1)+f(t)+f_Nt^N,\) \(t\in\mathbb{R}\).
If \(p\ne-q^n\) for all integer \(n>0\), then the solution of the initial-value problem on the segment \([0,T]\), \(T>1\) has at the connection point of the solution continuous derivatives of degree not less than \(N\).
Some numerical examples are given.
For the entire collection see [Zbl 1467.34001].
Reviewer: Nikita V. Artamonov (Moskva)On the spectral properties and positivity of solutions of a periodic boundary value problem for a second-order functional differential equationhttps://www.zbmath.org/1483.340882022-05-16T20:40:13.078697Z"Alves, Manuel J."https://www.zbmath.org/authors/?q=ai:alves.manuel-joachim"Labovskiĭ, Sergeĭ Mikhaĭlovich"https://www.zbmath.org/authors/?q=ai:labovskii.sergei-mikhailovichSummary: For a functional-differential operator
\[\mathcal{L} u = (1/\rho)\left(-(pu')'+\int_0^l u(s)d_s r(x,s)\right)\]
with symmetry, the completeness and orthogonality of the eigenfunctions is shown. The positivity conditions of the Green function of the periodic boundary value problem are obtained.Traces for Sturm-Liouville operators with constant delays on a star graphhttps://www.zbmath.org/1483.340892022-05-16T20:40:13.078697Z"Wang, Feng"https://www.zbmath.org/authors/?q=ai:wang.feng.1|wang.feng.2|wang.feng.4|wang.feng.3"Yang, Chuan-Fu"https://www.zbmath.org/authors/?q=ai:yang.chuanfuSummary: In this work we consider the spectral problems for Sturm-Liouville operators with constant delays on a star graph. First the asymptotics for the large eigenvalues of these operators are derived. Secondly the regularized trace formulae of these operators are established with the method of complex analysis.To estimating linear functionals values over solutions of systems with aftereffecthttps://www.zbmath.org/1483.340902022-05-16T20:40:13.078697Z"Maksimov, Vladimir Petrovich"https://www.zbmath.org/authors/?q=ai:maksimov.vladimir-pSummary: For a wide class of linear functional differential systems with Volterra operators, a constructive technique is proposed to obtain estimates of linear functionals values over solutions in conditions of uncertainty of external perturbations. It can be applied to solutions of boundary value problems with arbitrary number of boundary conditions as well as to description of attainability sets in control problems with respect to given on-target functionals. External perturbations are constrained by a given linear inequalities system on the main time segment. The technique is based on the results of general theory of functional differential equations about the solvability of boundary value problems with general linear boundary conditions and the representation of solutions. The problem under consideration is reduced to the generalized moment problem. Therewith the results on the properties of the Cauchy matrix to systems with aftereffect are of essential importance. The general form of functionals allows one to cover many cases being topical in applications such as multipoint, integral ones, as well as hybrids of those.Oscillation tests for differential equations with deviating argumentshttps://www.zbmath.org/1483.340912022-05-16T20:40:13.078697Z"Chatzarakis, G. E."https://www.zbmath.org/authors/?q=ai:chatzarakis.george-e"Purnaras, I. K."https://www.zbmath.org/authors/?q=ai:purnaras.ioannis-k"Stavroulakis, I. P."https://www.zbmath.org/authors/?q=ai:stavroulakis.ioannis-pThe authors continue their previous studies and provide effective conditions guaranteeing that all solutions to first-order delayed (resp. advanced) differential equations are oscillatory. Illustrative examples are presented as well.
Reviewer: Jiří Šremr (Brno)Delay engineered solitary states in complex networkshttps://www.zbmath.org/1483.340922022-05-16T20:40:13.078697Z"Schülen, Leonhard"https://www.zbmath.org/authors/?q=ai:schulen.leonhard"Ghosh, Saptarshi"https://www.zbmath.org/authors/?q=ai:ghosh.saptarshi"Kachhvah, Ajay Deep"https://www.zbmath.org/authors/?q=ai:kachhvah.ajay-deep"Zakharova, Anna"https://www.zbmath.org/authors/?q=ai:zakharova.anna"Jalan, Sarika"https://www.zbmath.org/authors/?q=ai:jalan.sarikaSummary: We present a technique to engineer solitary states by means of delayed links in a network of neural oscillators and in coupled chaotic maps. Solitary states are intriguing partial synchronization patterns, where a synchronized cluster coexists with solitary nodes displaced from this cluster and distributed randomly over the network. We induce solitary states in the originally synchronized network of identical nodes by introducing delays in the links for a certain number of selected network elements. It is shown that the extent of displacement and the position of solitary elements can be completely controlled by the choice (values) and positions (locations) of the incorporated delays, reshaping the delay engineered solitary states in the network.Oscillatory properties of solutions of higher order nonlinear functional differential equationshttps://www.zbmath.org/1483.340932022-05-16T20:40:13.078697Z"Sokhadze, Z."https://www.zbmath.org/authors/?q=ai:sokhadze.zazaSummary: Oscillatory properties of solutions of the functional differential equation
\[
u^{(n)}(t) = f(u)(t)
\]
and its particular cases
\[
\begin{aligned}
&u^{(n)}(t) = g \big( t, u(\tau_1(t)), \dots , u(\tau_m(t)) \big), \\
&u^{(n)}(t) = \sum_{k=1}^m g_k (t) \mathrm{ln} (1 + |u(\tau_k (t))|) \mathrm{sgn}(u(\tau_k (t)))
\end{aligned}
\]
are investigated. Here, \(f\) is an operator acting from the space \(C([a, +\infty[)\) to the space \(L_{\mathrm{loc}}(\mathbb{R}_+)\), \(a \leq 0\), \(g : \mathbb{R}_+ \times \mathbb{R}^m \to \mathbb{R}\) is a function satisfying the local Carathéodory conditions,
\[
g_k \in L_{\mathrm{loc}}(\mathbb{R}_+)\quad (k = 1, \dots , m),
\]
and \(\tau_k : \mathbb{R}_+ \to R \; (k = 1, \dots , m)\) are continuous functions such that
\[
\tau_k (t) \leq t \text{ for } t \in \mathbb{R}_+ \text{,} \quad \underset{n\to+\infty} {\lim} \tau_k (t) = +\infty\quad (k = 1, \dots , m).
\]Unbounded oscillation of fourth order functional differential equationshttps://www.zbmath.org/1483.340942022-05-16T20:40:13.078697Z"Tripathy, Arun Kumar"https://www.zbmath.org/authors/?q=ai:tripathy.arun-kumar"Mohanta, Rashmi Rekha"https://www.zbmath.org/authors/?q=ai:mohanta.rashmi-rekhaSummary: In this paper, sufficient conditions for oscillation of unbounded solutions of a class of fourth order neutral delay differential equations of the form
\[
(r(t)(y(t)+p(t)y(t-\tau))'')''+q(t)G(y(t-\alpha))-h(t)H(y(t-\sigma))=0
\]
are discussed under the assumption
\[
\int\limits^{\infty}_0\frac{t}{r(t)}\mathrm{d}t=\infty
\]Periodic solutions for some differential nonlinear systems with several delayshttps://www.zbmath.org/1483.340952022-05-16T20:40:13.078697Z"Gabsi, Hocine"https://www.zbmath.org/authors/?q=ai:gabsi.hocine"Ardjouni, Abdelouaheb"https://www.zbmath.org/authors/?q=ai:ardjouni.abdelouaheb"Djoudi, Ahcene"https://www.zbmath.org/authors/?q=ai:djoudi.ahceneSummary: By means of continuation theorem of coincidence degree theory and Krasnoselskii-Burton's fixed point theorem we study some differential nonlinear systems of several delays with a deviating argument having the form \[\begin{cases}\frac{dx(t)}{dt}=\beta|x(t-\tau(t))|^\alpha x(t)+f(t,u(t-\sigma(t)))+p(t),\\ \frac{du(t)}{dt}=a(t)g(u(t))+G(t,x(t-\tau)),u(t-\sigma(t))),\end{cases}\] where \(\alpha\) and \(\beta\) are two parameters with \(0<\alpha<1\). We give sufficient conditions on \(\beta,\alpha,f,g\) and \(G\) to offer, what we hope, an existence criteria of periodic solutions of above system. Some new results on the existence of periodic solutions are obtained. We end by giving an example to illustrate our claim.Periodic solutions for second order totally nonlinear iterative differential equationshttps://www.zbmath.org/1483.340962022-05-16T20:40:13.078697Z"Guerfi, Abderrahim"https://www.zbmath.org/authors/?q=ai:guerfi.abderrahim"Ardjouni, Abdelouaheb"https://www.zbmath.org/authors/?q=ai:ardjouni.abdelouahebSummary: Sufficient conditions in this paper are presented for the existence of periodic solutions of a second order totally nonlinear iterative differential equation. The equivalent integral equation of the given equation defines a fixed point mapping written as a sum of a large contraction and a compact map. The main tool used here is Krasnoselskii-Burton's fixed point technique.Positive periodic solutions for \(p\)-Laplacian neutral differential equations with a singularityhttps://www.zbmath.org/1483.340972022-05-16T20:40:13.078697Z"Li, Zhiyan"https://www.zbmath.org/authors/?q=ai:li.zhiyan"Kong, Fanchao"https://www.zbmath.org/authors/?q=ai:kong.fanchaoSummary: In this paper, we study the positive periodic solutions of a kind of \(p\)-Laplacian neutral differential equation with a singularity. By applying the continuation theorem and some analytic techniques, we shall establish several new criteria for the existence of positive periodic solutions for the considered problem. Some recent results in the literature are generalized and improved. Three examples are given to illustrate the effectiveness of our results.Bifurcations in a fractional-order neural network with multiple leakage delayshttps://www.zbmath.org/1483.340982022-05-16T20:40:13.078697Z"Huang, Chengdai"https://www.zbmath.org/authors/?q=ai:huang.chengdai"Liu, Heng"https://www.zbmath.org/authors/?q=ai:liu.heng"Shi, Xiangyun"https://www.zbmath.org/authors/?q=ai:shi.xiangyun"Chen, Xiaoping"https://www.zbmath.org/authors/?q=ai:chen.xiaoping"Xiao, Min"https://www.zbmath.org/authors/?q=ai:xiao.min"Wang, Zhengxin"https://www.zbmath.org/authors/?q=ai:wang.zhengxin"Cao, Jinde"https://www.zbmath.org/authors/?q=ai:cao.jindeSummary: This paper expatiates the stability and bifurcation for a fractional-order neural network (FONN) with double leakage delays. Firstly, the characteristic equation of the developed FONN is circumspectly researched by employing inequable delays as bifurcation parameters. Simultaneously the bifurcation criteria are correspondingly extrapolated. Then, unequal delays-spurred-bifurcation diagrams are primarily delineated to confirm the precision and correctness for the values of bifurcation points. Furthermore, it lavishly illustrates from the evidence that the stability performance of the proposed FONN can be demolished with the presence of leakage delays in accordance with comparative studies. Eventually, two numerical examples are exploited to underpin the feasibility of the developed theory. The results derived in this paper have perfected the retrievable outcomes on bifurcations of FONNs embodying unique leakage delay, which can nicely serve a benchmark deliberation and provide a comparatively credible guidance for the influence of multiple leakage delays on bifurcations of FONNs.Global stability for a class of functional differential equations with distributed delay and non-monotone bistable nonlinearityhttps://www.zbmath.org/1483.340992022-05-16T20:40:13.078697Z"Kuniya, Toshikazu"https://www.zbmath.org/authors/?q=ai:kuniya.toshikazu"Touaoula, Tarik Mohammed"https://www.zbmath.org/authors/?q=ai:touaoula.tarik-mohamedIn this paper, the following functional differential equation is considered:
\[
x'(t) = -f(x(t))+\int^{\tau}_0h(a)g(x(t-\tau))da, t>0; \; x(t)=\phi(t), -\tau\leq t\leq 0,
\]
where \(f\) and \(g\) are functions from \(\mathbb{R}\) to \(\mathbb{R}\) satisfying \(f(0) = g(0) = 0\) and the equation \(f(x) = g(x)\) has two positive roots \(x_2 > x_1 >0\). Under the assumption \(\int^{\tau}_0h(a)da =1\), the above equation has equilibria \(0, x_1\) and \(x_2\). Then, various criteria are established for \(0\) and \(x_2\) to be globally asymptotically stable by using Lyapunov functional methods. Note that the global asymptotic stability of \(0\) and that of \(x_2\) have restrictions on the magnitude of \(x(t)\). Comparing this work with those in the references, this deals with more general cases and it is a valuable extension to the existing theory.
Reviewer: Zhanyuan Hou (London)On the stability of a system of two linear hybrid functional differential systems with aftereffecthttps://www.zbmath.org/1483.341002022-05-16T20:40:13.078697Z"Simonov, Pëtr Mikhaĭlovich"https://www.zbmath.org/authors/?q=ai:simonov.petr-mSummary: We consider a system of two hybrid vector equations containing linear difference (defined on a discrete set) and functional differential (defined on a half-axis) parts. To study it, a model system of two vector equations is chosen, one of which is linear difference with aftereffect (LDEA), and the other is a linear functional differential with aftereffect (LFDEA). Two equivalent representations of this system are shown: the first representation in the form of LFDEA, the second -- in the form of LDEA. This allows us to study the stability issues of the system under consideration using the well-known results on the stability of LFDEA and LDEA.
Using the results of the article [\textit{S.A. Gusarenko}, Interuniversity Collection of Scientific Papers, 1989, 3--9, Perm Polytechnic Institute, Perm (In Russian)], two examples are shown when a joint system of four equations will be stable with respect to the right side. In the first example, we use the LFDEA for which sufficient conditions for the sign-definiteness of the elements of the \(2 \times 2\) Cauchy matrix function are known (in terms of the LFDEA coefficients). In the second example, LFDEA is given such that LFDEA is a system of linear ordinary differential equations (LODE) of the second order. In both cases, estimates of the components of the Cauchy matrix function are known. An exponential estimate with a negative exponent is given for the components of the Cauchy matrix function of LDEA.Stabilization of complex-valued stochastic functional differential systems on networks via impulsive controlhttps://www.zbmath.org/1483.341012022-05-16T20:40:13.078697Z"Wang, Pengfei"https://www.zbmath.org/authors/?q=ai:wang.pengfei"Li, Shaoyu"https://www.zbmath.org/authors/?q=ai:li.shaoyu"Su, Huan"https://www.zbmath.org/authors/?q=ai:su.huanSummary: In this paper, we consider the stabilization problem of complex-valued stochastic functional differential systems on networks (CSFDNs) via impulsive control. By the aid of complex version Itô's formula, the CSFDNs is studied in complex domain directly without splitting their real and imaginary parts. Then by combining Lyapunov-Razumikhin method and graph-theoretical technique, we give several novel stability criteria for CSFDNs. These criteria show that the impulses and connectivity of CSFDNs play important roles in the stability analysis of CSFDNs. Subsequently, the stability of complex-valued stochastic networks with time-varying delay is investigated. Finally, an illustrative example of complex-valued stochastic neural networks is presented to show the validness of the main results.Optimal control problems for a neutral integro-differential system with infinite delayhttps://www.zbmath.org/1483.341022022-05-16T20:40:13.078697Z"Huang, Hai"https://www.zbmath.org/authors/?q=ai:huang.hai"Fu, Xianlong"https://www.zbmath.org/authors/?q=ai:fu.xianlongSummary: This work devotes to the study on problems of optimal control and time optimal control for a neutral integro-differential evolution system with infinite delay. The main technique is the theory of resolvent operators for linear neutral integro-differential evolution systems constructed recently in literature. We first establish the existence and uniqueness of mild solutions and discuss the compactness of the solution operator for the considered control system. Then, we investigate the existence of optimal controls for the both cases of bounded and unbounded admissible control sets under some assumptions. Meanwhile, the existence of time optimal control to a target set is also considered and obtained by limit arguments. An example is given at last to illustrate the applications of the obtained results.On using coupled fixed-point theorems for mild solutions to coupled system of multipoint boundary value problems of nonlinear fractional hybrid pantograph differential equationshttps://www.zbmath.org/1483.341032022-05-16T20:40:13.078697Z"Iqbal, Muhammad"https://www.zbmath.org/authors/?q=ai:iqbal.muhammad-azhar|iqbal.muhammad-waqas|iqbal.muhammad-sajid|iqbal.muhammad-faisal|iqbal.muhammad-sohail|iqbal.muhammad-naveed|iqbal.muhammad-asad|iqbal.muhammad-zafar|iqbal.muhammad-kashif|iqbal.muhammad-javed|iqbal.muhammad-mutahir"Shah, Kamal"https://www.zbmath.org/authors/?q=ai:shah.kamal"Khan, Rahmat Ali"https://www.zbmath.org/authors/?q=ai:khan.rahmat-aliIn this paper, authors consider a coupled systems of multipoint boundary value problems of fractional order hybrid differential equations. The perturbation is taken as nonliner and of second type. The fractional derivative is taken of Caputo's type. The unique solution of the boundary value problem is established under certain hypothesis. The fixed point technique is used to establish the results. Mainly Burton and couple type fixed point theorems are used. The proportional type delay that represent Pantograph equations is considered. Authors give two examples for better illustrations.
Reviewer: Syed Abbas (Mandi)Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach spacehttps://www.zbmath.org/1483.341042022-05-16T20:40:13.078697Z"Kumar, Ankit"https://www.zbmath.org/authors/?q=ai:kumar.ankit"Jeet, Kamal"https://www.zbmath.org/authors/?q=ai:jeet.kamal"Vats, Ramesh Kumar"https://www.zbmath.org/authors/?q=ai:vats.ramesh-kumarSummary: This paper aims to establish sufficient conditions for the exact controllability of the nonlocal Hilfer fractional integro-differential system of Sobolev-type using the theory of propagation family \(\{P(t), \; t\geq0\}\) generated by the operators \(A\) and \(R\). For proving the main result we do not impose any condition on the relation between the domain of the operators \(A\) and \(R\). We also do not assume that the operator \(R\) has necessarily a bounded inverse. The main tools applied in our analysis are the theory of measure of noncompactness, fractional calculus, and Sadovskii's fixed point theorem. Finally, we provide an example to show the application of our main result.Existence and uniqueness of mild solutions for quasi-linear fractional integro-differential equationshttps://www.zbmath.org/1483.341052022-05-16T20:40:13.078697Z"Ramos, Priscila Santos"https://www.zbmath.org/authors/?q=ai:ramos.priscila-santos"Sousa, J. Vanterler da C."https://www.zbmath.org/authors/?q=ai:vanterler-da-costa-sousa.jose"de Oliveira, E. Capelas"https://www.zbmath.org/authors/?q=ai:de-oliveira.edmundo-capelasSummary: We discuss the existence and uniqueness of mild solutions for a class of quasi-linear fractional integro-differential equations with impulsive conditions via Hausdorff measures of noncompactness and fixed point theory in Banach space. Mild solution controllability is discussed for two particular cases.Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaceshttps://www.zbmath.org/1483.341062022-05-16T20:40:13.078697Z"Singh, Soniya"https://www.zbmath.org/authors/?q=ai:singh.soniya"Arora, Sumit"https://www.zbmath.org/authors/?q=ai:arora.sumit"Mohan, Manil T."https://www.zbmath.org/authors/?q=ai:mohan.manil-t"Dabas, Jaydev"https://www.zbmath.org/authors/?q=ai:dabas.jaydevSummary: In this paper, we consider the second order semilinear impulsive differential equations with state-dependent delay. First, we consider a linear second order system and establish the approximate controllability result by using a feedback control. Then, we obtain sufficient conditions for the approximate controllability of the considered system in a separable, reflexive Banach space via properties of the resolvent operator and Schauder's fixed point theorem. Finally, we apply our results to investigate the approximate controllability of the impulsive wave equation with state-dependent delay.Faedo-Galerkin approximation of mild solutions of fractional functional differential equationshttps://www.zbmath.org/1483.341072022-05-16T20:40:13.078697Z"Vanterler da Costa Sousa, José"https://www.zbmath.org/authors/?q=ai:vanterler-da-costa-sousa.jose"Fečkan, Michal"https://www.zbmath.org/authors/?q=ai:feckan.michal"de Oliveira, Edmundo Capelas"https://www.zbmath.org/authors/?q=ai:de-oliveira.edmundo-capelasSummary: In the paper, we discuss the existence and uniqueness of mild solutions of a class of fractional functional differential equations in Hilbert space separable using the Banach fixed point theorem technique. In this sense, Faedo-Galerkin approximation to the solution is studied and demonstrated some convergence results.Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivativeshttps://www.zbmath.org/1483.341082022-05-16T20:40:13.078697Z"Zeng, Biao"https://www.zbmath.org/authors/?q=ai:zeng.biaoSummary: The goal of this paper is to provide systematic approaches to study the feedback control systems governed by fractional impulsive delay evolution equations involving Caputo fractional derivatives in separable reflexive Banach spaces. This work is a continuation of previous work. We firstly give an existence result of mild solutions for the equations by applying the Banach's fixed point theorem and the Leray-Schauder alternative fixed point theorem. Next, by using the Filippove theorem and the Cesari property, we obtain the existence result of feasible pairs for the feedback control system. Finally, some applications are given to illustrate our main results.Controllability of impulsive fractional integro-differential evolution equationshttps://www.zbmath.org/1483.341092022-05-16T20:40:13.078697Z"Gou, Haide"https://www.zbmath.org/authors/?q=ai:gou.haide"Li, Yongxiang"https://www.zbmath.org/authors/?q=ai:li.yongxiangSummary: In this paper, we are concerned with the controllability for a class of impulsive fractional integro-differential evolution equation in a Banach space. Sufficient conditions of the existence of mild solutions and approximate controllability for the concern problem are presented by considering the term \(u'(\cdot)\) and finding a control \(v\) such that the mild solution satisfies \(u(b)=u_b\) and \(u'(b)=u'_b\). The discussions are based on Mönch fixed point theorem as well as the theory of fractional calculus and \((\alpha,\beta)\)-resolvent operator. Finally, an example is given to illustrate the feasibility of our results.Existence, uniqueness and stability of fractional impulsive functional differential inclusionshttps://www.zbmath.org/1483.341102022-05-16T20:40:13.078697Z"da C. Sousa, J. Vanterler"https://www.zbmath.org/authors/?q=ai:vanterler-da-costa-sousa.jose"Kucche, Kishor D."https://www.zbmath.org/authors/?q=ai:kucche.kishor-dSummary: In the paper, we discuss necessary and sufficient conditions to obtain the existence, uniqueness and stability of solutions of fractional impulsive functional differential equations towards the \(\psi \)-Liouville-Caputo fractional derivative, through fixed point theorem, Arzela-Ascoli theorem and multivalued analysis theory.Ulam's type stabilities for conformable fractional differential equations with delayhttps://www.zbmath.org/1483.341112022-05-16T20:40:13.078697Z"Wang, Sen"https://www.zbmath.org/authors/?q=ai:wang.sen"Jiang, Wei"https://www.zbmath.org/authors/?q=ai:jiang.wei.1"Sheng, Jiale"https://www.zbmath.org/authors/?q=ai:sheng.jiale"Li, Rui"https://www.zbmath.org/authors/?q=ai:li.rui.2|li.rui|li.rui.3|li.rui.1|li.rui.4Summary: In this paper, we investigate the existence and uniqueness of solutions and Ulam's type stabilities including the well-known Ulam-Hyers stability and the newly extended Ulam-Hyers conformable exponential stability for two classes of fractional differential equations with the conformable fractional derivative and the time delay. The Banach contraction principle, the technique of Picard operator, the Gronwall integral inequalities, and generalized iterated integral inequality in the sense of conformable fractional integral are the main tools for deriving our main results. Finally, several illustrative examples will be presented to demonstrate our work.Existence results for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary conditionshttps://www.zbmath.org/1483.341122022-05-16T20:40:13.078697Z"Zuo, Mingyue"https://www.zbmath.org/authors/?q=ai:zuo.mingyue"Hao, Xinan"https://www.zbmath.org/authors/?q=ai:hao.xinan"Liu, Lishan"https://www.zbmath.org/authors/?q=ai:liu.lishan"Cui, Yujun"https://www.zbmath.org/authors/?q=ai:cui.yujunSummary: In this paper, we are concerned with the existence and uniqueness of solutions for impulsive fractional integro-differential equation of mixed type with constant coefficient and antiperiodic boundary condition. Our results are based on the Banach contraction mapping principle and the Krasnoselskii fixed point theorem. Some examples are also given to illustrate our results.Existence and stability of solutions to neutral conformable stochastic functional differential equationshttps://www.zbmath.org/1483.341132022-05-16T20:40:13.078697Z"Xiao, Guanli"https://www.zbmath.org/authors/?q=ai:xiao.guanli"Wang, JinRong"https://www.zbmath.org/authors/?q=ai:wang.jinrong"O'Regan, D."https://www.zbmath.org/authors/?q=ai:oregan.donalSummary: This paper studies conformable stochastic functional differential equations of neutral type. Firstly, the existence and uniqueness theorem of a solution is established. Secondly, the moment estimation and exponential stability results are given. Thirdly, the Ulam type stability in mean square is discussed. Finally, two examples are given to illustrate our results.Periodic solutions for a nonautonomous mathematical model of hematopoietic stem cell dynamicshttps://www.zbmath.org/1483.341142022-05-16T20:40:13.078697Z"Adimy, Mostafa"https://www.zbmath.org/authors/?q=ai:adimy.mostafa"Amster, Pablo"https://www.zbmath.org/authors/?q=ai:amster.pablo"Epstein, Julián"https://www.zbmath.org/authors/?q=ai:epstein.julianAuthors' abstract: The main purpose of this paper is to study the existence of periodic solutions for a nonautonomous differential-difference system describing the dynamics of hematopoietic stem cell (HSC) population under some external periodic regulatory factors at the cellular cycle level. The starting model is a nonautonomous system of two age-structured partial differential equations describing the HSC population in quiescent \((G_0)\) and proliferating (\(G_1\), \(S\), \(G_2\) and \(M\)) phase. We are interested in the effects of periodically time varying coefficients due for example to circadian rhythms or to the periodic use of certain drugs, on the dynamics of HSC population. The method of characteristics reduces the age-structured model to a nonautonomous differential-difference system. We prove under appropriate conditions on the parameters of the system, using topological degree techniques and fixed point methods, the existence of periodic solutions of our model.
Reviewer: Jiří Šremr (Brno)Asymptotic convergence in delay differential equations arising in epidemiology and physiologyhttps://www.zbmath.org/1483.341152022-05-16T20:40:13.078697Z"El-Morshedy, Hassan A."https://www.zbmath.org/authors/?q=ai:el-morshedy.hassan-a"Ruiz-Herrera, Alfonso"https://www.zbmath.org/authors/?q=ai:ruiz-herrera.alfonsoThis paper investigates the dynamics of a general broad class of delay differential equation, especially for the presence of oscillations. On the other hand, this paper devotes to proving that the global attraction towards a nontrivial equilibrium could be reduced to the nonexistence of solutions of a certain system of inequalities. Their theoretical results are easily applicable and cover situations in which the global attraction critically depends on the delay compared with the precious works.
For the application of the theoretical results, this paper emphasizes two examples, that is, epidemic models with awareness and the production of platelets. In epidemiology, the different types of population awareness and the time delays of individuals' responses to available information about the disease play a critical role in its spread. Using the theoretical results, they describe qualitative properties of the behavioral responses that prevent the presence of sustainable oscillations in the number of infected individuals. On the other hand, for physiological models, they discuss the influence of some biological parameters in certain anomalies in the production of platelets.
Above all, the idea of this work is novel with the theoretical analysis and application. Meanwhile, this paper focus on the dynamics of a general broad class of delay differential equation, whose results have generality. Consequently, this paper can be regarded as a significant work in mathematical biology.
Reviewer: Xiao Wang (Changsha)Dynamics behaviors of a stage-structured pest management model with time delay and impulsive effectshttps://www.zbmath.org/1483.341162022-05-16T20:40:13.078697Z"Yang, Jiangtao"https://www.zbmath.org/authors/?q=ai:yang.jiangtaoSummary: In this paper, we study a delayed stage-structured pest management model with periodic releasing infective pest, spraying pesticide and harvesting crop at three different fixed moments. By using impulsive-type comparison results, small-amplitude perturbation techniques, and the global attractivity of a one-order time-delay system, sufficient condition ensuring the global attractivity of the susceptible pest-eradication periodic solution is derived. Furthermore, the permanence of the model is also derived. An example and its numerical simulations are given to verify the effectiveness of the theoretical results.Inverse problem for the Sturm-Liouville equation with piecewise entire potential and piecewise constant weight on a curvehttps://www.zbmath.org/1483.341172022-05-16T20:40:13.078697Z"Golubkov, Andrey Alexandrovich"https://www.zbmath.org/authors/?q=ai:golubkov.andrey-alexandrovichSummary: A Sturm-Liouville equation with a piecewise entire potential and a non-zero piecewise constant weight function on a curve of an arbitrary shape lying on the complex plane is considered. For such equation, the inverse spectral problem is posed with respect to the ratio of elements of one column or one row of the transfer matrix along the curve. The uniqueness of the solution to the problem is proved with the help of the method of unit transfer matrix using the study of asymptotic solutions of the Sturm-Liouville equation for large values of the absolute value of the spectral parameter. The obtained results allowed to consider inverse problem for a previously unexplored class of Sturm-Liouville equations with three unknown coefficients on a segment of the real axis.On entire solutions of a class of second-order algebraic differential equationshttps://www.zbmath.org/1483.341182022-05-16T20:40:13.078697Z"Yanchenko, A. Ya."https://www.zbmath.org/authors/?q=ai:yanchenko.aleksandr-yakovlevichLet \(\mathbb{C}(z)\) be the field of rational functions over \(\mathbb{C}\), and let \(\mathbb{C}[\omega_0,\ldots,\omega_n]\) denote the set of polynomials of variables \(\omega_0,\ldots,\omega_n\) over the field of complex numbers. Furthermore, let \(b_0,b_1,b_2\in\mathbb{C}[z]\) with \(b_2\not\equiv 0\) and let \(A\in\mathbb{C}[z,\omega_0,\omega_1,\omega_2]\) such that \(A[z,0,\omega_1,-(b_1/b_2)\omega_1]\not\in\mathbb{C}(z)\). In the paper under review it is shown that if \(y=f(z)\) is a finite-order entire solution of \[ b_2 y'' + b_1 y' + b_0y + y A(z,y,y',y'') =0, \] then there exist \(A_1,A_2\in\mathbb{C}[z]\) and \(B_1,B_2\in\mathbb{C}(z)\) such that \[ f(z)=B_1 e^{A_1} + B_2 e^{A_2}. \]
Reviewer: Risto Korhonen (Joensuu)Converse growth estimates for ODEs with slowly growing solutionshttps://www.zbmath.org/1483.341192022-05-16T20:40:13.078697Z"Gröhn, Janne"https://www.zbmath.org/authors/?q=ai:grohn.janneLet \(f_1\), \(f_2\) be linearly independent solutions of the second-order linear differential equation
\[
f"(z)+A(z)f(z) = 0
\]
where the coefficient \(A(z)\) is a holomorphic function in the open unit disk \(\mathbb{D}\) of the complex plane \(\mathbb{C}\). The author studies miscellaneous properties of the above equation and their relationship to the subharmonic auxiliary function
\[
u = - \log\left(\frac{f_1}{f_2}\right)^{\#}
\]
where \(^{\#}\) stands for spherical derivative i.e.
\[
w^{\#} \overset{\mathrm{def}}{=} \frac{|w'|}{1+|w|^2}.
\]
In total fourteen theorems are proved in the paper. They could be roughly classified in six categories:
\begin{itemize}
\item Theorems concerning bounded solutions;
\item Useful identities;
\item Blaschke-oscillatory equations;
\item Nevanlinna interpolating sequences;
\item Point-wise growth restricted solutions;
\item Prescribed fixed points solutions.
\end{itemize}
The paper is roughly divided into two parts. In the first part the the author states theorems, motivation behind his ideas as well as the relationship to the known results in the existing literature. A few theorems are sharper versions of the existing results, the others are completely new. The advantages of spherical derivative over Bank-Laine approach and arguments based on the Schwarzian derivative are also discussed. The proofs are presented in the second part of the paper. They are clear and easy to follow. The author exhibits encyclopedic knowledge of existing literature both of linear differential equations of the second order in the complex domain as well as complex function theory.
Reviewer: Predrag Punosevac (Pittsburgh)Growth of solutions of non-homogeneous linear differential equations and its applicationshttps://www.zbmath.org/1483.341202022-05-16T20:40:13.078697Z"Pramanik, Dilip Chandra"https://www.zbmath.org/authors/?q=ai:pramanik.dilip-chandra"Biswas, Manab"https://www.zbmath.org/authors/?q=ai:biswas.manabLet \(H\subset \mathbb{C}\) be set with positive upper density, let \(a_0(z),\ldots,a_k(z)\), \(b(z)\) and \(c(z)\) be entire functions and let \(0\leq q < p\). In the paper under review it is shown that if there exists a constant \(\eta >0\) such that \[ |a_j(z)|\leq e^{q|z|^\eta}, \qquad |b(z)| \geq e^{p|z|^\eta}, \qquad |c(z)| \leq e^{q|z|^\eta}, \] for all \(z\in H\), then all meromorphic solutions \(f\not\equiv 0\) of \[ a_k(z) f^{(k)} + \cdots + a_1(z)f' + a_0(z) f = b(z) f + c(z) \] are of infinite order. The paper is concluded by two results on sharing value problems related to the equation above.
Reviewer: Risto Korhonen (Joensuu)Resolving singularities and monodromy reduction of Fuchsian connectionshttps://www.zbmath.org/1483.341212022-05-16T20:40:13.078697Z"Chiang, Yik-Man"https://www.zbmath.org/authors/?q=ai:chiang.yik-man"Ching, Avery"https://www.zbmath.org/authors/?q=ai:ching.avery"Tsang, Chiu-Yin"https://www.zbmath.org/authors/?q=ai:tsang.chiu-yinLet \(L(y)=0\) be the linear differential equation of Fuchsian class over \(\mathbb{C}(z)\). Under which conditions can solutions of this equation be expressed through hypergeometric functions? The answer to this question is important for many equations related to applied physical problems. Specifically for the Heun's differential equation \(\frac{d^{^{2}}y}{dz^{2}}+(\frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\epsilon}{z-t})\frac{dy}{dz}+\frac{\alpha\beta z-q}{z(z-1)(z-t)}=0\).
The general idea of solving such problems is to remove apparent singularities (see [\textit{M. A. Barkatou} and \textit{S. S. Maddah}, in: Proceedings of the 40th international symposium on symbolic and algebraic computation, ISSAC 2015, Bath, UK, July 6--9, 2015. New York, NY: Association for Computing Machinery (ACM). 53--60 (2015; Zbl 1346.68268)]). ``However, in order to overcome the ambiguity of the interplay between the local and global aspects of solutions of Fuchsian equations typically using classical language, it is the purpose of this article to apply sheaf theoretic language to study geometric aspects of Fuchsian connections where one of their singularities becomes apparent (i.e. being resolved).'' The proof of the effectiveness of the approach proposed by the authors may be their recover Takemura's eigenvalue inclusion theorem [\textit{K. Takemura}, J. Phys. A, Math. Theor. 45, No. 8, Article ID 085211, 14 p. (2012; Zbl 1247.34132)] and obtaining of new hypergeometric expansions of solutions to Heun's equations.
Reviewer: Mykola Grygorenko (Kyïv)Solutions of Painlevé II on real intervals: novel approximating sequenceshttps://www.zbmath.org/1483.341222022-05-16T20:40:13.078697Z"Bracken, Anthony J."https://www.zbmath.org/authors/?q=ai:bracken.anthony-jAn approximation method is proposed for a boundary value problem for the second Painlevé equation with Neumann boundary conditions
\[
y^{\prime\prime}(z) =2y(z)^3 +z y(z) +C, \quad y'(a)=0, \quad y'(b)=0, \quad (a<z<b).
\]
In the first step, a generalization of the second Painlevé equation is considered. The generalized equation on \(E(x)\) contains \(E(0), E(1)\) as nonlinear terms. In the expansion series \( E(x)=E_1(x) +E_2(x)+\cdots \), each \(E_n\) satisfies a nonhomogeneous Airy equation. By a suitable change of variables, a curious approximation series \(y_E^{(n)}\) is defined. \(y_E^{(n)}\) is a solution of a nonlinear equation with Neumann boundary conditions on an interval \([a_n, b_n\)], accompanied with a constant \(C_n\). When \(n \to \infty\), \(y_E^{(n)}\) converges to the solution \(y(z)\) and \(a_n, b_n\) and \(C_n\) also converges to \(a, b\) and \(C\), respectively, if the nonlinear term \(|y(z)|^3\) is small. A numerical example is also shown.
It is not clear why this method works well although the second Painlevé equation is closely related to the Airy function.
Reviewer: Yousuke Ohyama (Tokushima)Polynomial solutions of fractional polynomial differential equations generated by the second Painlevé equationhttps://www.zbmath.org/1483.341232022-05-16T20:40:13.078697Z"Khakimova, Zilya Nail'evna"https://www.zbmath.org/authors/?q=ai:khakimova.zilya-nailevna"Timofeeva, Larisa Nikolaevna"https://www.zbmath.org/authors/?q=ai:timofeeva.larisa-nikolaevna"Zaĭtsev, Oleg Valentinovich"https://www.zbmath.org/authors/?q=ai:zaitsev.oleg-valentinovichIn the paper under review, a pseudogroup of symmetries of Painleve-II equation is constructed. It transforms a solution space of one Painleve-II equation to the solution space of another Painleve-II equation (with other values of parameters).
The constructed pseudogroup is applied to rational solutions of a Painleve-II equation that exist for every integer value of parameter.
Reviewer: Dmitry Artamonov (Moskva)Some remark on oscillation of second order impulsive delay dynamic equations on time scaleshttps://www.zbmath.org/1483.341242022-05-16T20:40:13.078697Z"Chhatria, Gokula Nanda"https://www.zbmath.org/authors/?q=ai:chhatria.gokula-nandaSummary: This article deals with the oscillation criteria for a very extensively studied second order impulsive delay dynamic equations on time scale by using the Riccati transformation technique. Some examples are given to show the effect of impulse and to illustrate our main results.Change and variations. A history of differential equations to 1900https://www.zbmath.org/1483.350032022-05-16T20:40:13.078697Z"Gray, Jeremy"https://www.zbmath.org/authors/?q=ai:gray.jeremy-jThis is the fourth book of the author in the Springer Undergratuate Mathematics Series, after a history of geometry in the 19th century [\textit{J. Gray}, Worlds out of nothing. A course in the history of geometry in the 19th century. London: Springer (2011; Zbl 1205.01013)], a history of analysis in the 19th century [\textit{J. Gray}, The real and the complex: a history of analysis in the 19th century. Cham: Springer (2015; Zbl 1330.01001)] and a history of abstract algebra [\textit{J. Gray}, A history of abstract algebra. From algebraic equations to modern algebra. Cham: Springer (2018; Zbl 1411.01005)]. It presents a history of ODEs, PDEs and calculus of variations from the origins to around 1900.
The following topics are, among others, discussed in the book: Debeaune's inverse tangent problem; Bernoulli's brachistochrones; D'Alambert's vibrating string, Euler's equations of fluid mechanics; D'Alambert's method of characteristics; Euler-Lagrange equations; Maupertuis' principle of least action; separation of variables and Fourier's series solutions to the heat equation; Poincaré's non-Euclidean geometry; Kovalevskaya's theorem and counter-example; Green's theorems and functions; Schwarz's alternating method; the telegraphist's equation and the trans-atlantic cable; Riemann's shock wave paper; the plateau problem and minimal surfaces; Hamilton's and Jacobi's equations; the rigorisation of calculus of variations; Hilbert's problems about PDEs and calculus of variations; du-Bois-Reymond's paper on classification of second-order PDEs; Hadamard's notion of well-posedness.
Also, historical and mathematical exercises are included as well as advices to students on how to write essays. Finally, the author presents translations into English of original papers by Cauchy, Riemann, Schwarz, Darboux and Picard.
Reviewer: Lutz Recke (Berlin)Fractional oscillon equations; solvability and connection with classical oscillon equationshttps://www.zbmath.org/1483.350242022-05-16T20:40:13.078697Z"Bezerra, Flank D. M."https://www.zbmath.org/authors/?q=ai:bezerra.flank-david-morais"Figueroa-López, Rodiak N."https://www.zbmath.org/authors/?q=ai:figueroa-lopez.rodiak-n"Nascimento, Marcelo J. D."https://www.zbmath.org/authors/?q=ai:nascimento.marcelo-jose-diasSummary: In this paper we are concerned with the asymptotic behavior of nonautonomous fractional approximations of oscillon equation
\[
u_{tt} - \mu (t) \Delta u+ \omega (t)u_t = f(u),\, x \in \Omega,\, t \in \mathbb{R},
\]
subject to Dirichlet boundary condition on \(\partial \Omega\), where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\), \(N \geq 3\), the function \(\omega\) is a time-dependent damping, \(\mu\) is a time-dependent squared speed of propagation, and \(f\) is a nonlinear functional. Under structural assumptions on \(\omega\) and \(\mu\) we establish the existence of time-dependent attractor for the fractional models in the sense of \textit{A. N. Carvalho} et al. [Attractors for infinite-dimensional non-autonomous dynamical systems. Berlin: Springer (2013; Zbl 1263.37002)], and \textit{F. Di Plinio} et al. [Discrete Contin. Dyn. Syst. 29, No. 1, 141--167 (2011; Zbl 1223.37100)].Balanced viscosity solutions to a rate-independent coupled elasto-plastic damage systemhttps://www.zbmath.org/1483.350722022-05-16T20:40:13.078697Z"Crismale, Vito"https://www.zbmath.org/authors/?q=ai:crismale.vito"Rossi, Riccarda"https://www.zbmath.org/authors/?q=ai:rossi.riccardaIn nonlinear elasticity, rate-independent systems are idealized models where internal oscillations and viscous dissipations are neglected, since the (slower) scale of external loadings is dominant. On the other hand, in the latter time scale the system presents time discontinuities, corresponding to fast transitions between equilibria. In such transitions, a major role is played by the viscous dissipations.
A well-known method to study e.g.\ damage models is to consider a system where the flow rule for the damage variable is viscously regularized; next, one passes to the limit as the viscosity parameter tends to zero. The time discontinuities of the resulting evolution can be interpolated by means of transitions governed again by viscosity.
In this paper the authors study a model for damage coupled with plasticity, affected by viscosity both in the damage evolution and in the elastoplastic evolution. Moreover, a further dissipation source may come from a hardening process. Viscosity and hardening provide regularizing terms in the PDE system.
In the rate-independent idealization, one would neglect both viscosity and hardening. To rigorously see this, the authors consider a singularly perturbed PDE system where the terms related to viscosity and hardening are modulated by small parameters tending to zero. By tuning the speed of the convergence of such coefficients, one may model a system where the elastic and the plastic strain converge to rate-independent evolution with the same rate, or with a faster rate, than the damage variable.
Specifically, the system analyzed by the authors features three coefficients: a hardening parameter \(\mu\), a viscosity parameter \(\varepsilon\) related to damage, and a viscosity coefficient \(\varepsilon\nu\) related to plasticity. In fact, \(\nu\) is a rate parameter that modulates the rate of convergence of the damage variable with respect to the plastic strain. The authors study the convergence of the system as \(\varepsilon\to0\) while \(\nu,\mu\) are fixed, or as \(\varepsilon,\nu\to0\), or as all parameters \(\varepsilon,\nu,\mu\) converge to zero. In the limit, they obtain different notions of solutions, showing in the time discontinuities a single-rate or a multi-rate character. Studying various notions of rate-independent solutions is important in order to understand which of them captures the behavior of the system for small viscosities.
Reviewer: Giuliano Lazzaroni (Firenze)Wavefronts for degenerate diffusion-convection reaction equations with sign-changing diffusivityhttps://www.zbmath.org/1483.351232022-05-16T20:40:13.078697Z"Berti, Diego"https://www.zbmath.org/authors/?q=ai:berti.diego"Corli, Andrea"https://www.zbmath.org/authors/?q=ai:corli.andrea"Malaguti, Luisa"https://www.zbmath.org/authors/?q=ai:malaguti.luisaSummary: We consider in this paper a diffusion-convection reaction equation in one space dimension. The main assumptions are about the reaction term, which is monostable, and the diffusivity, which changes sign once or even more than once; then, we deal with a forward-backward parabolic equation. Our main results concern the existence of globally defined traveling waves, which connect two equilibria and cross both regions where the diffusivity is positive and regions where it is negative. We also investigate the monotony of the profiles and show the appearance of sharp behaviors at the points where the diffusivity degenerates. In particular, if such points are interior points, then the sharp behaviors are new and unusual.Modeling, approximation, and time optimal temperature control for binder removal from ceramicshttps://www.zbmath.org/1483.351612022-05-16T20:40:13.078697Z"Chicone, Carmen"https://www.zbmath.org/authors/?q=ai:chicone.carmen-c"Lombardo, Stephen J."https://www.zbmath.org/authors/?q=ai:lombardo.stephen-j"Retzloff, David G."https://www.zbmath.org/authors/?q=ai:retzloff.david-gSummary: The process of binder removal from green ceramic components -- a reaction-gas transport problem in porous media -- has been analyzed with a number of mathematical techniques: 1) non-dimensionalization of the governing decomposition-reaction ordinary differential equation (ODE) and of the reaction gas-permeability partial differential equation (PDE); 2) development of a pseudo steady state approximation (PSSA) for the PDE, including error analysis via \(L^2\) norm and singular perturbation methods; 3) derivation and analysis of a discrete model approximation; and 4) development of a time optimal control strategy to minimize processing time with temperature and pressure constraints. Theoretical analyses indicate the conditions under which the PSSA and discrete models are viable approximations. Numerical results indicate that under a range of conditions corresponding to practical binder burnout conditions, utilization of the optimal temperature protocol leads to shorter cycle times as compared to typical industrial practice.Convergence analysis of asymptotic preserving schemes for strongly magnetized plasmashttps://www.zbmath.org/1483.352692022-05-16T20:40:13.078697Z"Filbet, Francis"https://www.zbmath.org/authors/?q=ai:filbet.francis"Rodrigues, L. Miguel"https://www.zbmath.org/authors/?q=ai:rodrigues.luis-miguel"Zakerzadeh, Hamed"https://www.zbmath.org/authors/?q=ai:zakerzadeh.hamedSummary: The present paper is devoted to the convergence analysis of a class of asymptotic preserving particle schemes [the first and second authors, SIAM J. Numer. Anal. 54, No. 2, 1120--1146 (2016; Zbl 1342.35392)] for the Vlasov equation with a strong external magnetic field. In this regime, classical Particle-in-Cell methods are subject to quite restrictive stability constraints on the time and space steps, due to the small Larmor radius and plasma frequency. The asymptotic preserving discretization that we are going to study removes such a constraint while capturing the large-scale dynamics, even when the discretization (in time and space) is too coarse to capture fastest scales. Our error bounds are explicit regarding the discretization, stiffness parameter, initial data and time.Second-grade fluid model with Caputo-Liouville generalized fractional derivativehttps://www.zbmath.org/1483.353282022-05-16T20:40:13.078697Z"Sene, Ndolane"https://www.zbmath.org/authors/?q=ai:sene.ndolaneSummary: In this paper, we propose a novel method for obtaining the solution of the fractional differential equation in the class of second-grade fluids models. The technique described in this paper is called the double integral method. The method generates, in general, an approximate solution for the fractional diffusion equations, the energy equations, or the heat equations. In our study, we use the generalized fractional derivative in Caputo-Liouville's sense. For the illustrations of our method, we propose the graphical representations of the approximates solutions obtained by using the double integral method. We propose interpretations and physical discussions of the solutions obtained with the double integral method.A new class of Lyapunov functions for stability analysis of singular dynamical systems. Elements of \(p\)-regularity theoryhttps://www.zbmath.org/1483.370382022-05-16T20:40:13.078697Z"Evtushenko, Yu. G."https://www.zbmath.org/authors/?q=ai:evtushenko.yuri-g"Tret'yakov, A. A."https://www.zbmath.org/authors/?q=ai:tretyakov.alexey-aSummary: A new approach is proposed for studying the stability of dynamical systems in the case when traditional Lyapunov functions are ineffective or not applicable for research at all. The main tool used to analyze degenerate systems is the so-called \(p\)-factor Lyapunov function, which makes it possible to reduce the original problem to a new one based on constructions of \(p\)-regularity theory. An example of a meaningful application of the considered method is given.New multi-scroll attractors obtained via Julia set mappinghttps://www.zbmath.org/1483.370452022-05-16T20:40:13.078697Z"Atangana, Abdon"https://www.zbmath.org/authors/?q=ai:atangana.abdon"Bouallegue, Ghaith"https://www.zbmath.org/authors/?q=ai:bouallegue.ghaith"Bouallegue, Kais"https://www.zbmath.org/authors/?q=ai:bouallegue.kaisSummary: The Cobra attractor have attracted very recently and the model has been investigated using classical differential operators with integer and non-integer order. The model, in the case of fractional differential operator, is able to replicate indeed the Cobra for some values of fractional order. On the other hand, Julia set has been used for many purposes, in this paper; we develop a procedure that combines some chaotic attractors with the Julia set mapping to obtain multi-roll attractors. Using our algorithm, we obtained for the first time a lung of human being.Energy analysis of Sprott-A system and generation of a new Hamiltonian conservative chaotic system with coexisting hidden attractorshttps://www.zbmath.org/1483.370462022-05-16T20:40:13.078697Z"Jia, Hongyan"https://www.zbmath.org/authors/?q=ai:jia.hongyan"Shi, Wenxin"https://www.zbmath.org/authors/?q=ai:shi.wenxin"Wang, Lei"https://www.zbmath.org/authors/?q=ai:wang.lei.7|wang.lei.9|wang.lei.15|wang.lei.17|wang.lei.18|wang.lei.6|wang.lei.11|wang.lei.5|wang.lei.19|wang.lei.16|wang.lei.8|wang.lei.14|wang.lei|wang.lei.4"Qi, Guoyuan"https://www.zbmath.org/authors/?q=ai:qi.guoyuanSummary: The paper firstly investigates energy cycle of the Sprott-A system by transforming the Sprott-A system into the Kolmogorov-type system. We found the dynamics of the Sprott-A system are influenced by the change along the energy exchange between the conservative energy and the external supplied energy. And the action of the external supplied torque is the main reason that the Sprott-A system generates chaos. Secondly, based on energy analysis of the Sprott-A system, a new four-dimension (4-D) chaotic system is obtained. The new 4-D chaotic system is a conservative system with a constant Hamiltonian energy. Besides, it is also a no-equilibrium system, this means that the new 4-D chaotic system can exhibit hidden characteristics. Further, the coexisting hidden attractors are found when selecting different initial points. Finally, the new 4-D chaotic system is implemented by FPGA, and the coexisting attractors observed are consistent with those found in numerical analysis, which in experiment verifies the existence of coexisting hidden attractors of the new 4-D chaotic system from physical point of view.A new megastable nonlinear oscillator with infinite attractorshttps://www.zbmath.org/1483.370472022-05-16T20:40:13.078697Z"Leutcho, Gervais Dolvis"https://www.zbmath.org/authors/?q=ai:leutcho.gervais-dolvis"Jafari, Sajad"https://www.zbmath.org/authors/?q=ai:jafari.sajad"Hamarash, Ibrahim Ismael"https://www.zbmath.org/authors/?q=ai:hamarash.ibrahim-ismael"Kengne, Jacques"https://www.zbmath.org/authors/?q=ai:kengne.jacques"Tabekoueng Njitacke, Zeric"https://www.zbmath.org/authors/?q=ai:njitacke.zeric-tabekoueng"Hussain, Iqtadar"https://www.zbmath.org/authors/?q=ai:hussain.iqtadarSummary: Dynamical systems with megastable properties are very rare in the literature. In this paper, we introduce a new two-dimensional megastable dynamical system with a line of equilibria, having an infinite number of stable states. By modifying this new system with temporally-periodic forcing term, a new two-dimensional non-autonomous nonlinear oscillator capable to generate an infinite number of coexisting limit cycle attractors, torus attractors and, strange attractors is constructed. The analog implementation of the new megastable oscillator is investigated to further support numerical analyses and henceforth validate the mathematical model.Two-parameter unfolding of a parabolic point of a vector field in \(\mathbb{C}\) fixing the originhttps://www.zbmath.org/1483.370592022-05-16T20:40:13.078697Z"Rousseau, Christiane"https://www.zbmath.org/authors/?q=ai:rousseau.christianeThe author studies the bifurcation of a family of polynomial vector fields \(\dot{z}=z(z^k+\epsilon_1 z+\epsilon_0)\) in the complex plane with two parameters. The powerful tool of the ``periodgon'' is used to describe the bifurcation diagrams. The periodgon is a new invariant introduced by the author and \textit{A.Chéritat}
[``Generic 1-parameter pertubations of a vector field with a singular point of codimension \(k\)'', Preprint, \url{arXiv:1701.03276}]
to characterize a polynomial vector field up to a rotation of order \(k\) if it is monic and centered. Later, it has been generalized in [\textit{M. Klimeš} and the author, Conform. Geom. Dyn. 22, 141--184 (2018; Zbl 1403.37057)] to all generic polynomial vector fields. With the help of this tool, the bifurcations of parabolic points and homoclinic loops through infinity are investigated. For a generic 2-parameter unfolding of a parabolic point of codimension \(k\) preserving the origin, the author raises an interesting question of whether there exists a unique normal form in which the parameters are uniquely defined (i.e., canonically defined) if the origin is fixed in the unfolding.
Reviewer: Kwok-wai Chung (Hong Kong)Solvable dynamical systems in the plane with polynomial interactionshttps://www.zbmath.org/1483.370732022-05-16T20:40:13.078697Z"Calogero, Francesco"https://www.zbmath.org/authors/?q=ai:calogero.francesco-a"Payandeh, Farrin"https://www.zbmath.org/authors/?q=ai:payandeh.farrinSummary: In this paper we report a few examples of algebraically solvable dynamical systems characterized by \(2\) coupled Ordinary Differential Equations which read as follows:
\[
\dot{x}_n=P^{\left( n\right) }\left( x_1,x_2\right),\quad n=1,2,
\]
with \(P^{(n)}(x_1,x_2)\) specific polynomials of relatively low degree in the \(2\) dependent variables \(x_1\equiv x_1(t)\) and \(x_2\equiv x_2(t)\). These findings are obtained via a new twist of a recent technique to identify dynamical systems solvable by algebraic operations, themselves explicitly identified as corresponding to the time evolutions of the zeros of polynomials the coefficients of which evolve according to algebraically solvable (systems of) evolution equations.
For the entire collection see [Zbl 1456.14003].Stability analysis of fractional order mathematical model of tumor-immune system interactionhttps://www.zbmath.org/1483.371102022-05-16T20:40:13.078697Z"Öztürk, Ilhan"https://www.zbmath.org/authors/?q=ai:ozturk.ilhan"Özköse, Fatma"https://www.zbmath.org/authors/?q=ai:ozkose.fatmaSummary: In this paper, a fractional-order model of tumor-immune system interaction has been considered. In modeling dynamics, the total population of the model is divided into three subpopulations: macrophages, activated macrophages and tumor cells. The effects of fractional derivative on the stability and dynamical behaviors of the solutions are investigated by using the definition of the Caputo fractional operator that provides convenience for initial conditions of the differential equations. The existence and uniqueness of the solutions for the fractional derivative is examined and numerical simulations are presented to verify the analytical results. In addition, our model is used to describe the kinetics of growth and regression of the B-lymphoma \(BCL_1\) in the spleen of mice. Numerical simulations are given for different choices of fractional order \(\alpha\) and the obtained results are compared with the experimental data. The best approach to reality is observed around \(\alpha=0.80\). One can conclude that fractional model best fit experimental data better than the integer order model.On existence, uniqueness and Ulam's stability results for boundary value problems of fractional iterative integrodifferential equationshttps://www.zbmath.org/1483.450092022-05-16T20:40:13.078697Z"Kendre, S. D."https://www.zbmath.org/authors/?q=ai:kendre.subhash-dhondiba"Unhale, S. I."https://www.zbmath.org/authors/?q=ai:unhale.subhash-ishwarSummary: The author's aim in the given paper is to study local existence, uniqueness, Ulam-Hyers stability and generalized Ulam-Hyers stability of solutions for boundary value problems of fractional iterative integrodifferential equations. The successive approximation method is applied for the numerical solution of boundary value problems of fractional iterative integrodifferential equations.Fractional glassy relaxation and convolution modules of distributionshttps://www.zbmath.org/1483.460382022-05-16T20:40:13.078697Z"Kleiner, T."https://www.zbmath.org/authors/?q=ai:kleiner.tillmann"Hilfer, R."https://www.zbmath.org/authors/?q=ai:hilfer.rudolfSummary: Solving fractional relaxation equations requires precisely characterized domains of definition for applications of fractional differential and integral operators. Determining these domains has been a long-standing problem. Applications in physics and engineering typically require extension from domains of functions to domains of distributions. In this work, convolution modules are constructed for given sets of distributions that generate distributional convolution algebras. Convolutional inversion of fractional equations leads to a broad class of multinomial Mittag-Leffler type distributions. A~comprehensive asymptotic analysis of these is carried out. Combined with the module construction the asymptotic analysis yields domains of distributions, that guarantee existence and uniqueness of solutions to fractional differential equations. The mathematical results are applied to anomalous dielectric relaxation in glasses. An analytic expression for the frequency dependent dielectric susceptibility is applied to broadband spectra of glycerol. This application reveals a temperature independent and universal dynamical scaling exponent.Fine structure of the dichotomy spectrumhttps://www.zbmath.org/1483.470112022-05-16T20:40:13.078697Z"Pötzsche, Christian"https://www.zbmath.org/authors/?q=ai:potzsche.christianSummary: The dichotomy spectrum is a crucial notion in the theory of dynamical systems, since it contains information on stability and robustness properties. However, recent applications in nonautonomous bifurcation theory showed that a detailed insight into the fine structure of this spectral notion is necessary. On this basis, we explore a helpful connection between the dichotomy spectrum and operator theory. It relates the asymptotic behavior of linear nonautonomous difference equations to the point, surjectivity and Fredholm spectra of weighted shifts. This link yields several dynamically meaningful subsets of the dichotomy spectrum, which not only allows to classify and detect bifurcations, but also simplifies proofs for results on the long term behavior of difference equations with explicitly time-dependent right-hand side.Friedrichs extension of operators defined by even order Sturm-Liouville equations on time scaleshttps://www.zbmath.org/1483.470192022-05-16T20:40:13.078697Z"Zemánek, Petr"https://www.zbmath.org/authors/?q=ai:zemanek.petr"Hasil, Petr"https://www.zbmath.org/authors/?q=ai:hasil.petrSummary: In this paper we characterize the Friedrichs extension of operators associated with the 2\(n\)th order Sturm-Liouville dynamic equations on time scales with using the time reversed symplectic systems and its recessive system of solutions. A~nontrivial example is also provided.On the Bari basis properties of the root functions of non-self adjoint \(q\)-Sturm-Liouville problemshttps://www.zbmath.org/1483.470322022-05-16T20:40:13.078697Z"Allahverdiev, B. P."https://www.zbmath.org/authors/?q=ai:allahverdiev.bilender-pasaoglu"Tuna, H."https://www.zbmath.org/authors/?q=ai:tuna.huseyin|tuna.huseinSummary: This paper deals with the dissipative regular \(q\)-Sturm-Liouville problem. We prove that the system of root functions of this operator forms a Bari bases in the space \(L_q^2(I)\) by using the asymptotic behavior at infinity for its eigenvalues.Dilations, models and spectral problems of non-self-adjoint Sturm-Liouville operatorshttps://www.zbmath.org/1483.470432022-05-16T20:40:13.078697Z"Allahverdiev, Bilender P."https://www.zbmath.org/authors/?q=ai:allahverdiev.bilender-pasaogluSummary: In this study, we investigate the maximal dissipative singular Sturm-Liouville operators acting in the Hilbert space \(L_{r}^{2}(a,b)\)\ \( (-\infty \leq a<b\leq \infty)\), that [are] the extensions of a minimal symmetric operator\ with defect index (\(2,2\)) (in limit-circle case at singular end points \(a\)\ and \(b\)).\ We examine two classes of dissipative operators with separated boundary conditions and we establish, for each case, a self-adjoint dilation\ of the dissipative operator as well as its incoming and outgoing spectral representations, which enables us to define the scattering matrix of the dilation. Moreover, we construct a functional model of the dissipative operator and identify its characteristic function in terms of the Weyl function of a self-adjoint operator. We present several theorems on completeness of the system of root functions of the dissipative perators and verify them.On higher regularized traces of a differential operator with bounded operator coefficient given in a finite intervalhttps://www.zbmath.org/1483.470792022-05-16T20:40:13.078697Z"Sezer, Yonca"https://www.zbmath.org/authors/?q=ai:sezer.yonca"Bakşi, Özlem"https://www.zbmath.org/authors/?q=ai:baksi.ozlem"Karayel, Serpil"https://www.zbmath.org/authors/?q=ai:karayel.serpil-sengulSummary: In this work, we find a higher regularized trace formula for a regular Sturm-Liouville differential operator with operator coefficient.Forward-backward approximation of nonlinear semigroups in finite and infinite horizonhttps://www.zbmath.org/1483.470882022-05-16T20:40:13.078697Z"Contreras, Andrés"https://www.zbmath.org/authors/?q=ai:contreras.andres-a"Peypouquet, Juan"https://www.zbmath.org/authors/?q=ai:peypouquet.juanThe authors consider the problem
\[
\begin{aligned}
-&\dot{u}(t)\in\left( A+B\right) u(t) \text{ for a.e. }t>0,\\
&u(0)=u_{0}\in D(A),
\end{aligned}
\]
in a class of Banach spaces, where \(A\) is \(m\)-accretive and \(B\) is coercive. First, the approximation of solutions is investigated. Solutions are approximated by trajectories constructed by interpolation of sequences generated using forward-backward iteration and these are shown to converge uniformly on a finite time interval, proving existence and uniqueness of solutions. Second, asymptotic equivalence results are given that connect the behaviour of forward-backward iterations as the number of iterations goes to infinity with the behaviour of the solution as time goes to infinity, for step sizes that are sufficiently small. These results are based on a certain inequality which the authors trace back to \textit{E. Hille} [Fysiogr. Sällsk. Lund Förh. 21, No. 14, 130--142 (1951; Zbl 0044.32902)].
Reviewer: Daniel C. Biles (Nashville)Differential equation approximations of stochastic network processes: an operator semigroup approachhttps://www.zbmath.org/1483.471252022-05-16T20:40:13.078697Z"Bátkai, András"https://www.zbmath.org/authors/?q=ai:batkai.andras"Kiss, Istvan Z."https://www.zbmath.org/authors/?q=ai:kiss.istvan-z"Sikolya, Eszter"https://www.zbmath.org/authors/?q=ai:sikolya.eszter"Simon, Péter L."https://www.zbmath.org/authors/?q=ai:simon.peter-lSummary: The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view, the stochastic model is identified by its master equation, which is a system of linear ODEs with large state space size (\(N\)). We derive a single non-linear ODE (called mean-field approximation) for the expected value that yields a good approximation as \(N\) tends to infinity. Using only elementary semigroup theory, we can prove the order \(\mathcal{O}(1/N)\) convergence of the solution of the system to that of the mean-field equation. The proof holds also for cases that are somewhat more general than the usual density dependent one. Moreover, for Markov chains where the transition rates satisfy some sign conditions, a~new approach using a countable system of ODEs for proving convergence to the mean-field limit is proposed.A robust pseudospectral method for numerical solution of nonlinear optimal control problemshttps://www.zbmath.org/1483.490102022-05-16T20:40:13.078697Z"Mehrpouya, Mohammad Ali"https://www.zbmath.org/authors/?q=ai:mehrpouya.mohammad-ali"Peng, Haijun"https://www.zbmath.org/authors/?q=ai:peng.haijunSummary: In the present paper, a robust pseudospectral method for efficient numerical solution of nonlinear optimal control problems is presented. In the proposed method, at first, based on the Pontryagin's minimum principle, the first-order necessary conditions of optimality which are led to the Hamiltonian boundary value problem are derived. Then, utilizing a pseudospectral method for discretization, the nonlinear optimal control problem is converted to a system of nonlinear algebraic equations. However, the need to have a good initial guess may lead to a challenging problem for solving the obtained system of nonlinear equations. So, an optimization approach is introduced to simplify the need of a good initial guess. Numerical findings of some benchmark examples are presented at the end and computational features of the proposed method are reported.An ODE reduction method for the semi-Riemannian Yamabe problem on space formshttps://www.zbmath.org/1483.530592022-05-16T20:40:13.078697Z"Fernández, Juan Carlos"https://www.zbmath.org/authors/?q=ai:fernandez.juan-carlos"Palmas, Oscar"https://www.zbmath.org/authors/?q=ai:palmas.oscarThe authors prove the existence of blowing-up and globally defined solutions of Yamabe-type partial differential equations on semi-Euclidean space and on the pseudosphere of dimension at least 3. In the proof they use isoparametric functions which allow the reduction to a generalized Emden-Fowler ordinary differential equation.
Reviewer: Hans-Bert Rademacher (Leipzig)Around Efimov's differential test for homeomorphismhttps://www.zbmath.org/1483.530782022-05-16T20:40:13.078697Z"Alexandrov, Victor"https://www.zbmath.org/authors/?q=ai:alexandrov.victor-aThere is a famous result due to Efimov, more precisely the following Theorem: No surface can be \(C^2\)-immersed in Euclidean 3-space so as to be complete in the induced Riemannian metric, with Gauss curvature \(K \le \) constant \(< 0\).
The paper under review starts with a mini-survey of results related to the previous theorem.
Among other things, Efimov established that the condition \(K \le\) constant \(< 0\) is not the only obstacle for the immersibility of a complete surface of negative curvature; he showed that a rather slow change of Gauss curvature is another obstacle. In all those numerous articles, he used to a large extent one and the same method based on the study of the spherical image of a surface. At that study, an essential role belongs to statements that, under some conditions, a locally homeomorphic mapping \(f : \mathbb{R}^2 \to \mathbb{R}^2\) is a global homeomorphism and \(f(\mathbb{R}^2)\) is a convex domain in \(\mathbb{R}^2\).
Two other theorems of Efimov are recalled in the present paper and the author gives an overview on the analogues of these theorems, their generalizations and applications. The article is devoted to presentation of results motivated by the theory of surfaces, the theory of global inverse function, the Jacobian Conjecture, and the global asymptotic stability of dynamical systems, respectively.
Reviewer: Adela-Gabriela Mihai (Bucureşti)Applications of some fixed point theorems for fractional differential equations with Mittag-Leffler kernelhttps://www.zbmath.org/1483.540232022-05-16T20:40:13.078697Z"Afshari, Hojjat"https://www.zbmath.org/authors/?q=ai:afshari.hojjat"Baleanu, Dumitru"https://www.zbmath.org/authors/?q=ai:baleanu.dumitru-iSummary: Using some fixed point theorems for contractive mappings, including \(\alpha\)-\(\gamma\)-Geraghty-type contraction, \(\alpha\)-type \(F\)-contraction, and some other contractions in \(\mathcal{F}\)-metric space, this research intends to investigate the existence of solutions for some Atangana-Baleanu fractional differential equations in the Caputo sense [\textit{A. Atangana} and \textit{D. Baleanu}, ``New fractional derivative with non-local and non-singular kernel. Theory and application to heat transfer model'', Therm. Sci. 20, No. 2, 763--769 (2016; \url{doi:10.2298/TSCI160111018A})].Nonlinear \(F\)-contractions on \(b\)-metric spaces and differential equations in the frame of fractional derivatives with Mittag-Leffler kernelhttps://www.zbmath.org/1483.540252022-05-16T20:40:13.078697Z"Alqahtani, Badr"https://www.zbmath.org/authors/?q=ai:alqahtani.badr"Fulga, Andreea"https://www.zbmath.org/authors/?q=ai:fulga.andreea"Jarad, Fahd"https://www.zbmath.org/authors/?q=ai:jarad.fahd"Karapınar, Erdal"https://www.zbmath.org/authors/?q=ai:karapinar.erdalSummary: In this manuscript, we aim to refine and characterize nonlinear \(F\)-contractions in a more general framework of \(b\)-metric spaces. We investigate the existence and uniqueness of such contractions in this setting. We discuss the solutions to differential equations in the setting of fractional derivatives involving Mittag-Leffler kernels (Atangana-Baleanu fractional derivative) by using nonlinear \(F\)-contractions that indicate the genuineness of the presented result.Singularities of singular solutions of first-order differential equations of clairaut typehttps://www.zbmath.org/1483.580102022-05-16T20:40:13.078697Z"Saji, Kentaro"https://www.zbmath.org/authors/?q=ai:saji.kentaro"Takahashi, Masatomo"https://www.zbmath.org/authors/?q=ai:takahashi.masatomoThe work of this paper is a part of an ongoing research on understanding singularities of envelopes for differential equations of Clairaut type. Let us explain the main notions and concepts. First, consider the ordinary differential equation \[ F(x,y,p)=0, \tag{1} \] where \(p\) stands for the derivative \(dy/dx\) and \(F\) is defined in a domain of the space \(J^1(\mathbb{R},\mathbb{R})\) that consists of 1-jets of functions \(y(x)\), i.e., the space with coordinates \(x,y,p\) equipped with the contact 1-form \(pdx - dy = 0\). Assume that equation (1) defines a smooth surface in \(J^1(\mathbb{R},\mathbb{R})\), then the contact structure cuts the vector field \[ X = F_p \partial_x + pF_p \partial_y + (F_x+pF_y) \partial_p \] on this surface. Integral curves of the field \(X\) are 1-jet extensions of solutions of equation (1). The canonical projection \(\pi(x,y,p) = (x,y)\) restricted to the surface \(\{F=0\}\) has singular points on the set \(\{F=F_p=0\}\) called the criminant, and the projection of the criminant is the discriminant set of equation (1). Generically, the criminant and the discriminant set are curves, the field \(X\) vanishes at isolated points of the criminant, and the discriminant set is the locus of singularities of solution of (1) (almost all of which are 3:2-cusps). However, there exist a special class of equations (1) called Clairaut type. It is defined by the condition that the function \(F_x+pF_y\) vanishes on the criminant identically. Under light additional conditions, in this case the discriminant set is the envelop of solutions of (1), and consequently, it is a solution as well. The basic examples are \(p^2 = y\) and classical Clairaut's equation itself: \[ f(p) = xp-y, \ \ f''(p) \not\equiv 0. \tag{2} \] The discriminant set of (2) is the dual Legendrian curve to the graph \(y=f(x)\), it is the envelop of its tangent lines \(xc-y=f(c)\), \(c=const\), which are also solutions of (2). It is regular at points where \(f''(p) \neq 0\) are it is singular if \(f''(p)=0\). For instance, it has 3:2-cusps at points where \(f''(p)=0\), \(f'''(p) \neq 0\). This is the simplest example of singularities of envelopes.
Second, the authors investigate the partial differential equation \[ F(x_1, x_2,y,p_1, p_2)=0, \tag{3} \] where \(p_i\) stands for the derivative \(dy/dx_i\) and \(F\) is defined in a domain of the space \(J^1(\mathbb{R}^2,\mathbb{R})\) that consists of 1-jets of functions \(y(x_1,x_2)\), i.e., the space with coordinates \(x_1, x_2,y,p_1, p_2\) equipped with the contact 1-form \(p_1 dx_1+ p_2 dx_2 - dy = 0\). Similarly to the above, there exists a special class of equations (3) called Clairaut type. The authors show that (under light additional conditions) Clairaut type equations (3) have envelops of solutions (which are solutions as well) and establish the list of typical singularities of their envelops: cuspidal edge, swallowtail, cuspidal butterfly, cuspidal lips/beaks, etc (frontal singularities).
Third, the authors investigate the system of equations \[ F(x_1, x_2,y,p_1, p_2)=0, \ \ G(x_1, x_2,y,p_1, p_2)=0, \tag{4} \] where the functions \(F, G\) are defined in a domain of the space \(J^1(\mathbb{R}^2,\mathbb{R})\), all notations are similar to (3) and the Poisson bracket \([F,G]\) on the manifold \(\{F=G=0\}\) is identically zero. The authors introduce a natural notion of Clairaut type systems (4) and establish the list of typical singularities of their envelops similar to those for (3).
Reviewer: Alexey O. Remizov (Moskva)The birth of random evolutionshttps://www.zbmath.org/1483.600382022-05-16T20:40:13.078697Z"Hersh, Reuben"https://www.zbmath.org/authors/?q=ai:hersh.reubenFrom the text: The theory of random evolutions was born in Albuquerque in the late 1960s, flourished and matured in the 1970s, sprouted a robust daughter in Kiev in the 1980s, and is today a tool or method, applicable in a variety of ``real-world'' ventures.On solutions of stochastic equations with current and osmotic velocitieshttps://www.zbmath.org/1483.600802022-05-16T20:40:13.078697Z"Gliklikh, Yuri E."https://www.zbmath.org/authors/?q=ai:gliklikh.yuri-eSummary: The main aim of this paper is to collect the results connected with existence of solutions of equations with current and osmotic velocities, published in several articles, and supply all constructions and results with complete proofs. Some new properties of both types of equations and their interrelation are described.
For the entire collection see [Zbl 1470.46003].Stochastic spikes and Poisson approximation of one-dimensional stochastic differential equations with applications to continuously measured quantum systemshttps://www.zbmath.org/1483.600822022-05-16T20:40:13.078697Z"Kolb, Martin"https://www.zbmath.org/authors/?q=ai:kolb.martin"Liesenfeld, Matthias"https://www.zbmath.org/authors/?q=ai:liesenfeld.matthiasSummary: Motivated by the recent contribution [\textit{M. Bauer} and \textit{D. Bernard}, Ann. Henri Poincaré 19, No. 3, 653--693 (2018; Zbl 1391.60073)], we study the scaling limit behavior of a class of one-dimensional stochastic differential equations which has a unique attracting point subject to a small additional repulsive perturbation. Problems of this type appear in the analysis of continuously monitored quantum systems. We extend the results of Bauer and Bernard [loc. cit.] and prove a general result concerning the convergence to a homogeneous Poisson process using only classical probabilistic tools.Harnack and shift Harnack inequalities for degenerate (functional) stochastic partial differential equations with singular driftshttps://www.zbmath.org/1483.600842022-05-16T20:40:13.078697Z"Lv, Wujun"https://www.zbmath.org/authors/?q=ai:lv.wujun"Huang, Xing"https://www.zbmath.org/authors/?q=ai:huang.xingSummary: The existence and uniqueness of the mild solutions for a class of degenerate functional stochastic partial differential equations (SPDEs) are obtained, where the drift is assumed to be Hölder-Dini continuous. Moreover, the non-explosion of the solution is proved under some reasonable conditions. In addition, the Harnack inequality is derived by the method of coupling by change of measure. Finally, the shift Harnack inequality is obtained for the equations without delay, which is new even in the non-degenerate case. An example is presented in the final part of the paper.Transport analysis of infinitely deep neural networkhttps://www.zbmath.org/1483.620722022-05-16T20:40:13.078697Z"Sonoda, Sho"https://www.zbmath.org/authors/?q=ai:sonoda.sho"Murata, Noboru"https://www.zbmath.org/authors/?q=ai:murata.noboruSummary: We investigated the feature map inside deep neural networks (DNNs) by tracking the transport map. We are interested in the role of depth -- why do DNNs perform better than shallow models? -- and the interpretation of DNNs -- what do intermediate layers do? Despite the rapid development in their application, DNNs remain analytically unexplained because the hidden layers are nested and the parameters are not faithful. Inspired by the integral representation of shallow NNs, which is the continuum limit of the width, or the hidden unit number, we developed the flow representation and transport analysis of DNNs. The flow representation is the continuum limit of the depth, or the hidden layer number, and it is specified by an ordinary differential equation (ODE) with a vector field. We interpret an ordinary DNN as a transport map or an Euler broken line approximation of the flow. Technically speaking, a dynamical system is a natural model for the nested feature maps. In addition, it opens a new way
to the
coordinate-free treatment of DNNs by avoiding the redundant parametrization of DNNs. Following Wasserstein geometry, we analyze a flow in three aspects: dynamical system, continuity equation, and Wasserstein gradient flow. A key finding is that we specified a series of transport maps of the denoising autoencoder (DAE), which is a cornerstone for the development of deep learning. Starting from the shallow DAE, this paper develops three topics: the transport map of the deep DAE, the equivalence between the stacked DAE and the composition of DAEs, and the development of the double continuum limit or the integral representation of the flow representation. As partial answers to the research questions, we found that deeper DAEs converge faster and the extracted features are better; in addition, a deep Gaussian DAE transports mass to decrease the Shannon entropy of the data distribution. We expect that further investigations on these questions lead to the development of an interpretable and principled alternatives
to DNNs.Theory and numerical approximations of fractional integrals and derivativeshttps://www.zbmath.org/1483.650072022-05-16T20:40:13.078697Z"Li, Changpin"https://www.zbmath.org/authors/?q=ai:li.changpin"Cai, Min"https://www.zbmath.org/authors/?q=ai:cai.min|cai.min.1The book provides a survey of various relevant concepts for generalizing the idea of a differential operator to operators of non-integer order, commonly known as fractional derivatives. Moreover, it discusses techniques for the numerical handling of these operators.
The first part of the book is devoted to a study of the analytical properties of Riemann-Liouville, Caputo and Riesz derivatives and fractional Laplacians. A few other concepts are also mentioned briefly. The relations of these operators to fractional differential equations are discussed as well. In addition, this part of the text contains a brief discussion of a few potential applications of such equations such as, e.g. continuous-time random walks. Essentially, many parts of the material displayed here do not go beyond what has been provided in the classical book of \textit{St. G. Samko} et al. [Fractional integrals and derivatives: theory and applications. Transl. from the Russian. New York, NY: Gordon and Breach (1993; Zbl 0818.26003)], but the presentation by Li and Cai appears to be a bit less technical.
The second (and larger) part of the book is then devoted to the numerical side of the topic. The authors present a large selection of (very different) possible approaches for numerically dealing with fractional derivatives and integrals and explain some of their properties. Unfortunately, the practical usefulness of the error analysis (where provided) is somewhat limited because the authors either impose very restrictive smoothness conditions that cannot be assumed to be satisfied in general or they do not mention the required smoothness conditions precisely at all.
Reviewer: Kai Diethelm (Schweinfurt)Computing the density function of complex models with randomness by using polynomial expansions and the RVT technique. Application to the SIR epidemic modelhttps://www.zbmath.org/1483.650152022-05-16T20:40:13.078697Z"Calatayud, Julia"https://www.zbmath.org/authors/?q=ai:calatayud.julia"Carlos Cortés, Juan"https://www.zbmath.org/authors/?q=ai:cortes.juan-carlos"Jornet, Marc"https://www.zbmath.org/authors/?q=ai:jornet.marcSummary: This paper concerns the computation of the probability density function of the stochastic solution to general complex systems with uncertainties formulated via random differential equations. In the existing literature, the uncertainty quantification for random differential equations is based on the approximation of statistical moments by simulation or spectral methods, or on the computation of the exact density function via the random variable transformation (RVT) method when a closed-form solution is available. However, the problem of approximating the density function in a general setting has not been published yet. In this paper, we propose a hybrid method based on stochastic polynomial expansions, the RVT technique, and multidimensional integration schemes, to obtain accurate approximations to the solution density function. A problem-independent algorithm is proposed. The algorithm is tested on the SIR (susceptible-infected-recovered) epidemiological model, showing significant improvements compared to the previous literature.The homotopy method for the complete solution of quadratic two-parameter eigenvalue problemshttps://www.zbmath.org/1483.650592022-05-16T20:40:13.078697Z"Dong, Bo"https://www.zbmath.org/authors/?q=ai:dong.boSummary: We propose a homotopy method to solve the quadratic two-parameter eigenvalue problems, which arise frequently in the analysis of the asymptotic stability of the delay differential equation. Our method does not require to form coupled generalized eigenvalue problems with Kronecker product type coefficient matrices and thus can avoid the increasing of the computational cost and memory storage. Numerical results and the applications in the delay differential equations are presented to illustrate the effectiveness and efficiency of our method. It appears that our method tends to be more effective than the existing methods in terms of speed, accuracy and memory storage as the problem size grows.Shifted fractional Jacobi collocation method for solving fractional functional differential equations of variable orderhttps://www.zbmath.org/1483.651082022-05-16T20:40:13.078697Z"Abdelkawy, M. A."https://www.zbmath.org/authors/?q=ai:abdelkawy.mohamed-a"Lopes, António M."https://www.zbmath.org/authors/?q=ai:lopes.antonio-m"Babatin, Mohammed M."https://www.zbmath.org/authors/?q=ai:babatin.mohammed-mSummary: Functional differential equations have been widely used for modeling real-world phenomena in distinct areas of science. However, classical calculus can not provide always the best description of some complex phenomena, namely those observed in biological systems and medicine. This paper proposes a new numerical method for solving variable order fractional functional differential equations (VO-FFDE). Firstly, the shifted fractional Jacobi collocation method (SF-JC) is applied to solve the VO-FFDE with initial conditions. Then, the SF-JC is applied to the VO-FFDE with boundary conditions. Several numerical examples with different types of VO-FFDE demonstrate the superiority of the proposed method.A novel method for singularly perturbed delay differential equations of reaction-diffusion typehttps://www.zbmath.org/1483.651092022-05-16T20:40:13.078697Z"Chakravarthy, P. Pramod"https://www.zbmath.org/authors/?q=ai:chakravarthy.p-pramod"Kumar, Kamalesh"https://www.zbmath.org/authors/?q=ai:kumar.kamaleshSummary: In this paper, we consider a boundary value problem for a singularly perturbed delay differential equation of reaction-diffusion type. A fitted operator finite difference scheme based on Numerov's method is constructed. An extensive amount of computational work has been carried out to demonstrate the applicability of the proposed method.Uniform convergence method for a delay differential problem with layer behaviourhttps://www.zbmath.org/1483.651102022-05-16T20:40:13.078697Z"Cimen, Erkan"https://www.zbmath.org/authors/?q=ai:cimen.erkan"Amiraliyev, Gabil M."https://www.zbmath.org/authors/?q=ai:amiraliyev.gabil-mSummary: Difference method on a piecewise uniform mesh of Shishkin type, for a singularly perturbed boundary-value problem for a linear second-order delay differential equation is examined. It is proved that it gives essentially a first-order parameter-uniform convergence in the discrete maximum norm. Furthermore, numerical results are presented in support of the theory.New numerical simulations for some real world problems with Atangana-Baleanu fractional derivativehttps://www.zbmath.org/1483.651112022-05-16T20:40:13.078697Z"Gao, Wei"https://www.zbmath.org/authors/?q=ai:gao.wei.3"Ghanbari, Behzad"https://www.zbmath.org/authors/?q=ai:ghanbari.behzad"Baskonus, Haci Mehmet"https://www.zbmath.org/authors/?q=ai:baskonus.haci-mehmetSummary: In this work, we introduce ABC-Caputo operator with ML kernel and its main characteristics are discussed. Viral diseases models for AIDS and Zika are considered, and finally, as third model, the macroeconomic model involving ABC fractional derivatives is investigated, respectively. It is presented that the AB Caputo derivatives satisfy the Lipschitz condition along with superposition property. The numerical methods for solving the fractional models are presented by means of ABC fractional derivative in a detailed manner. Finally the simulation results obtained in this paper according to the suitable values of parameters are also manifested.Legendre wavelet solution of high order nonlinear ordinary delay differential equationshttps://www.zbmath.org/1483.651122022-05-16T20:40:13.078697Z"Gümgüm, Sevin"https://www.zbmath.org/authors/?q=ai:gumgum.sevin"Ersoy Özdek, Demet"https://www.zbmath.org/authors/?q=ai:ersoy-ozdek.demet"Özaltun, Gökçe"https://www.zbmath.org/authors/?q=ai:ozaltun.gokceSummary: The purpose of this paper is to illustrate the use of the Legendre wavelet method in the solution of high-order nonlinear ordinary differential equations with variable and proportional delays. The main advantage of using Legendre polynomials lies in the orthonormality property, which enables a decrease in the computational cost and runtime. The method is applied to five differential equations up to sixth order, and the results are compared with the exact solutions and other numerical solutions when available. The accuracy of the method is presented in terms of absolute errors. The numerical results demonstrate that the method is accurate, effectual and simple to apply.Computational methods for the fractional optimal control HIV infectionhttps://www.zbmath.org/1483.651152022-05-16T20:40:13.078697Z"Abd Elal, Leila F."https://www.zbmath.org/authors/?q=ai:elal.leila-f-abd"Sweilam, Nasser H."https://www.zbmath.org/authors/?q=ai:sweilam.nasser-hassan"Nagy, Abdelhameed M."https://www.zbmath.org/authors/?q=ai:nagy.abdelhameed-m"Almaghrebi, Yousef S."https://www.zbmath.org/authors/?q=ai:almaghrebi.yousef-sSummary: In this paper two numerical methods are used to study the nonlinear fractional optimal control problem (FOCP) for the human immunodeficiency virus (HIV) model. The objective functional is based on a combination of maximizing benefit relied on uninfected cells count and minimizing the systemic cost of chemotherapy. The state equations are given as a system of fractional order differential equations (FODEs). The fractional derivatives are described in the Caputo sense. The Pontriagyn maximum principle (PMP) is used to obtain a necessary optimality condition for the FOCP. The optimality system is derived and we introduce an iterative optimal control method (IOCM) to solved it numerically, comparisons between IOCM and the generalized Euler method (GEM) are given. Numerical experiment is presented to demonstrate the validity and applicability of the proposed technique. we can conclude that IOCM is preferable because the uninfected cells are increasing using the proposed method than GEM, moreover the infected cells are decreasing in better way than GEM.On three-dimensional variable order time fractional chaotic system with nonsingular kernelhttps://www.zbmath.org/1483.651172022-05-16T20:40:13.078697Z"Hashemi, M. S."https://www.zbmath.org/authors/?q=ai:hashemi.mir-sajjad"Inc, Mustafa"https://www.zbmath.org/authors/?q=ai:inc.mustafa"Yusuf, Abdullahi"https://www.zbmath.org/authors/?q=ai:yusuf.abdullahi-aSummary: We use the Adams-Bashforth-Moulton scheme (ABMS) to determine the approximate solution of a variable order fractional three-dimensional chaotic process. The derivative is defined in the fractional sense of variable order Atangana-Baleanu-Caputo (ABC). Numerical examples show that to solve these variable-order fractional differential equations easily and efficiently, the Adams-Bashforth-Moulton method can be implemented. Lastly, simulation results demonstrate the proposed robust control's effectiveness.Numerical regularity map for fundamental one-dimensional fractional differential equations with Hölder continuous solutionshttps://www.zbmath.org/1483.651182022-05-16T20:40:13.078697Z"Kato, Mana"https://www.zbmath.org/authors/?q=ai:kato.mana"Fujiwara, Hiroshi"https://www.zbmath.org/authors/?q=ai:fujiwara.hiroshi"Imai, Hitoshi"https://www.zbmath.org/authors/?q=ai:imai.hitoshiSummary: In the paper, fundamental one-dimensional fractional differential equations in the sense of Caputo with Hölder continuous solutions are solved numerically. A large number of numerical experiments results in a numerical regularity map which visualizes relationship between the Hölder exponent of the solution, the convergence rate of schemes, and the order of the derivative. The map reveals that the rate of convergence changes depending on whether the derivative order is greater than or less than one.Solution of the fractional Bratu-type equation via fractional residual power series methodhttps://www.zbmath.org/1483.651192022-05-16T20:40:13.078697Z"Khalouta, Ali"https://www.zbmath.org/authors/?q=ai:khalouta.ali"Kadem, Abdelouahab"https://www.zbmath.org/authors/?q=ai:kadem.abdelouahabSummary: In this paper, we present numerical solution for the fractional Bratu-type equation via fractional residual power series method (FRPSM). The fractional derivatives are described in Caputo sense. The main advantage of the FRPSM in comparison with the existing methods is that the method solves the nonlinear problems without using linearization, discretization, perturbation or any other restriction. Three numerical examples are given and the results are numerically and graphically compared with the exact solutions. The solutions obtained by the proposed method are in complete agreement with the solutions available in the literature. The results reveal that the FRPSM is a very effective, simple and efficient technique to handle a wide range of fractional differential equations.Numerical results for ordinary and partial differential equations describing motions of elastic materialshttps://www.zbmath.org/1483.651202022-05-16T20:40:13.078697Z"Kosugi, Chiharu"https://www.zbmath.org/authors/?q=ai:kosugi.chiharu"Aiki, Toyohiko"https://www.zbmath.org/authors/?q=ai:aiki.toyohiko"Anthonissen, Martijn"https://www.zbmath.org/authors/?q=ai:anthonissen.martijn-johannes-hermanus"Okumura, Makoto"https://www.zbmath.org/authors/?q=ai:okumura.makotoSummary: We discuss an ordinary differential equation system proposed in [\textit{T. Aiki} and \textit{C. Kosugi}, Adv. Math. Sci. Appl. 29, No. 2, 459--494 (2020; Zbl 07377911)] as a mathematical model for shrinking and stretching motions of elastic materials. Also, a numerical scheme due to the structure-preserving numerical method was constructed. Our aim of this paper is to compare the numerical results for periodic solutions by several methods in order to investigate their accuracy. We note that a proof for existence of periodic solutions of the ODE system is given. Finally, we derive a partial differential equation model from the ODE system and show numerical results for the PDE model.Reliable numerical modelling of malaria propagation.https://www.zbmath.org/1483.651222022-05-16T20:40:13.078697Z"Faragó, István"https://www.zbmath.org/authors/?q=ai:farago.istvan"Mincsovics, Miklós Emil"https://www.zbmath.org/authors/?q=ai:mincsovics.miklos-emil"Mosleh, Rahele"https://www.zbmath.org/authors/?q=ai:mosleh.raheleSummary: We investigate biological processes, particularly the propagation of malaria. Both the continuous and the numerical models on some fixed mesh should preserve the basic qualitative properties of the original phenomenon. Our main goal is to give the conditions for the discrete (numerical) models of the malaria phenomena under which they possess some given qualitative property, namely, to be between zero and one. The conditions which guarantee this requirement are related to the time-discretization step-size. We give a sufficient condition for some explicit methods. For implicit methods we prove that the above property holds unconditionally.Robust numerical method for singularly perturbed differential equations with large delayhttps://www.zbmath.org/1483.651232022-05-16T20:40:13.078697Z"Abdulla, Murad Ibrahim"https://www.zbmath.org/authors/?q=ai:abdulla.murad-ibrahim"Duressa, Gemechis File"https://www.zbmath.org/authors/?q=ai:duressa.gemechis-file"Debela, Habtamu Garoma"https://www.zbmath.org/authors/?q=ai:debela.habtamu-garomaSummary: In this paper, a singularly perturbed differential equation with a large delay is considered. The considered problem contains a large delay parameter on the reaction term. The solution of the problem exhibits the interior layer due to the delay parameter and the strong right boundary layer due to the small perturbation parameter \(\varepsilon\). The resulting singularly perturbed problem is solved using the fitted non-polynomial spline method. The stability and parameter uniform convergence of the proposed method is proved. To validate the applicability of the scheme, two model problems of the variable coefficient are considered for numerical experimentation.Bernoulli wavelet method for numerical solution of anomalous infiltration and diffusion modeling by nonlinear fractional differential equations of variable orderhttps://www.zbmath.org/1483.651262022-05-16T20:40:13.078697Z"Chouhan, Devendra"https://www.zbmath.org/authors/?q=ai:chouhan.devendra"Mishra, Vinod"https://www.zbmath.org/authors/?q=ai:mishra.vinod-kumar"Srivastava, H. M."https://www.zbmath.org/authors/?q=ai:srivastava.hari-mohanSummary: In this paper, generalized fractional-order Bernoulli wavelet functions based on the Bernoulli wavelets are constructed to obtain the numerical solution of problems of anomalous infiltration and diffusion modeling by a class of nonlinear fractional differential equations with variable order. The idea is to use Bernoulli wavelet functions and operational matrices of integration. Firstly, the generalized fractional-order Bernoulli wavelets are constructed. Secondly, operational matrices of integration are derived and utilize to convert the fractional differential equations (FDE) into a system of algebraic equations. Finally, some numerical examples are presented to demonstrate the validity, applicability and accuracy of the proposed Bernoulli wavelet method.Theoretical and computational results for mixed type Volterra-Fredholm fractional integral equationshttps://www.zbmath.org/1483.652092022-05-16T20:40:13.078697Z"Amin, Rohul"https://www.zbmath.org/authors/?q=ai:amin.rohul"Alrabaiah, Hussam"https://www.zbmath.org/authors/?q=ai:alrabaiah.hussam"Mahariq, Ibrahim"https://www.zbmath.org/authors/?q=ai:mahariq.ibrahim"Zeb, Anwar"https://www.zbmath.org/authors/?q=ai:zeb.anwarA computational algorithm for the numerical solution of nonlinear fractional integral equationshttps://www.zbmath.org/1483.652102022-05-16T20:40:13.078697Z"Amin, Rohul"https://www.zbmath.org/authors/?q=ai:amin.rohul"Senu, Norazak"https://www.zbmath.org/authors/?q=ai:senu.norazak"Hafeez, Muhammad Bilal"https://www.zbmath.org/authors/?q=ai:hafeez.muhammad-bilal"Arshad, Noreen Izza"https://www.zbmath.org/authors/?q=ai:arshad.noreen-izza"Ahmadian, Ali"https://www.zbmath.org/authors/?q=ai:ahmadian.ali"Salahshour, Soheil"https://www.zbmath.org/authors/?q=ai:salahshour.soheil"Sumelka, Wojciech"https://www.zbmath.org/authors/?q=ai:sumelka.wojciechCubic B-spline approximation for linear stochastic integro-differential equation of fractional orderhttps://www.zbmath.org/1483.652212022-05-16T20:40:13.078697Z"Mirzaee, Farshid"https://www.zbmath.org/authors/?q=ai:mirzaee.farshid"Alipour, Sahar"https://www.zbmath.org/authors/?q=ai:alipour.saharSummary: In this paper, the cubic B-spline collocation method is used for solving the stochastic integro-differential equation of fractional order. we show that stochastic integro-differential equation of fractional order is equivalent to a modified stochastic integral equation. Then we apply the proposed method to obtain a numerical scheme of the modified stochastic integral equation. Using this method, the problem solving turns into a linear system solution of equations. Also, the convergence analysis of this numerical approach has been discussed. In the end, examples are given to test the accuracy and the implementation of the method. The results are compared with the results obtained by other methods to verify that this method is accurate and efficient.Modeling and analyzing the dynamic spreading of epidemic malware by a network eigenvalue methodhttps://www.zbmath.org/1483.680182022-05-16T20:40:13.078697Z"Liu, Wanping"https://www.zbmath.org/authors/?q=ai:liu.wanping"Zhong, Shouming"https://www.zbmath.org/authors/?q=ai:zhong.shou-mingSummary: This paper mainly focuses on studying the influence of network characteristics on malware spreading. Firstly, a generalized model with weakly-protected and strongly-protected susceptible nodes is developed by considering the possibility of an intruded node converting back to a weakly-protected susceptible one. The dynamics of the generalized compartmental model is intensively discussed and analyzed, deriving several sufficient conditions for its global stability. Following this work, a novel node-based model is newly proposed to describe malware propagation over an arbitrary connected network including synthesized and real networks. From a microscopic perspective, we establish the novel model by introducing several different variables for each node which describe the probabilities of a node locating at respective states. Our theoretical analysis shows that the largest eigenvalue of the propagating network is a key factor determining malware prevalence. Specifically, the range of the leading eigenvalue can be split into three subintervals in which malware approaches extinction very quickly, or tends to extinction, or persists, depending on into which subinterval the largest eigenvalue of the propagating network falls. Theoretically, the trivial equilibrium of our new node-based model is clearly proved to be exponentially globally stable when the maximum eigenvalue is less than a threshold. We also illustrate the predictive effectiveness of our model by designing some numerical simulations on some regular and scale-free networks. Consequently, we conclude that malware prevalence can be effectively prevented by properly adjusting the spreading network, e.g., reducing the number of nodes and deleting some edges, so that its maximum eigenvalue falls into the appropriate subinterval.A compartmental model to explore the interplay between virus epidemics and honeynet potencyhttps://www.zbmath.org/1483.680192022-05-16T20:40:13.078697Z"Ren, Jianguo"https://www.zbmath.org/authors/?q=ai:ren.jianguo"Xu, Yonghong"https://www.zbmath.org/authors/?q=ai:xu.yonghongSummary: Honeynet technology is an active approach that is used to capture novel viruses and provide feedback on a matching immunization strategy. A compartmental model is formulated and analyzed to explore the interplay between virus epidemics and potency of a heterogeneous honeynet. Theoretical analysis of the model shows the conditions under which the minimum amount and best location in configuring a honeynet are determined. Furthermore, the honeypot with more system vulnerabilities is beneficial for mitigating the virus epidemic to a lower level, whereas the honeynet with a lower power law index is better for acquiring the virus samples. A number of numerical examples are presented to illustrate the theoretical analysis. On the basis of the results, some ideas for imposing restrictions on the spread of virus or improving the design of a honeynet are suggested.ParaPlan: a tool for parallel reachability analysis of planar polygonal differential inclusion systemshttps://www.zbmath.org/1483.682052022-05-16T20:40:13.078697Z"Sandler, Andrei"https://www.zbmath.org/authors/?q=ai:sandler.andrei"Tveretina, Olga"https://www.zbmath.org/authors/?q=ai:tveretina.olgaSummary: We present the ParaPlan tool which provides the reachability analysis of planar hybrid systems defined by differential inclusions (SPDI). It uses the parallelized and optimized version of the algorithm underlying the SPeeDI tool
[\textit{E. Asarin} et al., Lect. Notes Comput. Sci. 2404, 354--358 (2002; Zbl 1010.68791)].
The performance comparison demonstrates the speed-up of up to 83 times with respect to the sequential implementation on various benchmarks. Some of the benchmarks we used are randomly generated with the novel approach based on the partitioning of the plane with Voronoi diagrams.
For the entire collection see [Zbl 1436.68017].The spatial Hill four-body problem. I: An exploration of basic invariant setshttps://www.zbmath.org/1483.700332022-05-16T20:40:13.078697Z"Burgos-García, Jaime"https://www.zbmath.org/authors/?q=ai:burgos-garcia.jaime"Bengochea, Abimael"https://www.zbmath.org/authors/?q=ai:bengochea.abimael"Franco-Pérez, Luis"https://www.zbmath.org/authors/?q=ai:franco-perez.luisSummary: In this work we perform a first study of basic invariant sets of the spatial Hill's four-body problem, where we have used both analytical and numerical approaches. This system depends on a mass parameter \(\mu\) in such a way that the classical Hill's problem is recovered when \(\mu=0\). Regarding the numerical work, we perform a numerical continuation, for the Jacobi constant \(C\) and several values of the mass parameter \(\mu\) by applying a classical predictor-corrector method, together with a high-order Taylor method considering variable step and order and automatic differentiation techniques, to specific boundary value problems related with the reversing symmetries of the system. The solution of these boundary value problems defines initial conditions of symmetric periodic orbits. Some of the results were obtained departing from periodic orbits within Hill's three-body problem. The numerical explorations reveal that a second distant disturbing body has a relevant effect on the stability of the orbits and bifurcations among these families. We have also found some new families of periodic orbits that do not exist in the classical Hill's three-body problem; these families have some desirable properties from a practical point of view.Dynamics of a tourism sustainability model with distributed delayhttps://www.zbmath.org/1483.760102022-05-16T20:40:13.078697Z"Kaslik, Eva"https://www.zbmath.org/authors/?q=ai:kaslik.eva"Neamţu, Mihaela"https://www.zbmath.org/authors/?q=ai:neamtu.mihaelaSummary: This paper generalizes the existing minimal mathematical model of a given generic touristic site by including a distributed time-delay to reflect the whole past history of the number of tourists in their influence on the environment and capital flow. A stability and bifurcation analysis is carried out on the coexisting equilibria of the model, with special emphasis on the positive equilibrium. Considering general delay kernels and choosing the average time-delay as bifurcation parameter, a Hopf bifurcation analysis is undertaken in the neighborhood of the positive equilibrium. This leads to the theoretical characterization of the critical values of the average time delay which are responsible for the occurrence of oscillatory behavior in the system. Extensive numerical simulations are also presented, where the influence of the investment rate and competition parameter on the qualitative behavior of the system in a neighborhood of the positive equilibrium is also discussed.Nonlinear propagation of leaky TE-polarized electromagnetic waves in a metamaterial Goubau linehttps://www.zbmath.org/1483.780022022-05-16T20:40:13.078697Z"Smolkin, Eugene"https://www.zbmath.org/authors/?q=ai:smolkin.eugene"Smirnov, Yury"https://www.zbmath.org/authors/?q=ai:smirnov.yury-gSummary: Propagation of leaky TE-polarized electromagnetic waves in the Goubau line (a perfectly conducting cylinder covered by a concentric dielectric layer) filled with nonlinear metamaterial medium is studied. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of the Green function of an auxiliary boundary value problem on an interval. The existence of propagating nonlinear leaky TE waves for the chosen nonlinearity (Kerr law) is proved using the method of contraction. For the numerical solution, a method based on solving an auxiliary Cauchy problem (a version of the shooting method) is proposed. New propagation regimes are discovered.Occurrence of vibrational resonance in an oscillator with an asymmetric Toda potentialhttps://www.zbmath.org/1483.780042022-05-16T20:40:13.078697Z"Kolebaje, Olusola"https://www.zbmath.org/authors/?q=ai:kolebaje.olusola"Popoola, O. O."https://www.zbmath.org/authors/?q=ai:popoola.oyebola-o"Vincent, U. E."https://www.zbmath.org/authors/?q=ai:vincent.uchechukwu-eSummary: Vibrational resonance (VR) is a phenomenon wherein the response of a nonlinear oscillator driven by biharmonic forces with two different frequencies, \(\omega\) and \(\varOmega\), such that \(\varOmega \gg \omega\), is enhanced by optimizing the parameters of high-frequency driving force. In this paper, an counterintuitive scenario in which a biharmonically driven nonlinear oscillator does not vibrate under the well known VR conditions is reported. This behaviour was observed in a system with an integrable and asymmetric Toda potential driven by biharmonic forces in the usual VR configuration. It is shown that with constant dissipation and in the presence of biharmonic forces, VR does not take place, whereas with nonlinear displacement-dependent periodic dissipation multiple VR can be induced at certain values of high-frequency force parameters. Theoretical analysis are validated using numerical computation and Simulink implementation in MATLAB. Finally, the regime in parameter space of the dissipation for optimum occurrence of multiple VR in the Toda oscillator was estimated. This result would be relevant for experimental applications of dual-frequency driven laser models where the Toda potential is extensively employed.Remarks on extension of convex functions and application to evolution inclusions generated by \(-\Delta\beta\)https://www.zbmath.org/1483.800042022-05-16T20:40:13.078697Z"Kenmochi, Nobuyuki"https://www.zbmath.org/authors/?q=ai:kenmochi.nobuyukiThe author considers the evolution inclusion \(u_{t}-\Delta \beta (u)\ni f\) in \(\Omega \times (0,T)\), where \(\Omega \) is a bounded and smooth domain of \( \mathbb{R}^{N}\), \(1\leq N<\infty \), \(\beta \) is a maximal monotone graph in \( L^{2}(\Omega )\), \(f\in L^{2}(\Omega \times (0,T))\).\ The boundary condition \( \beta (u)\ni h\in W^{1,2}(0,T;H^{1/2}(\partial \Omega ))\) is imposed on \( \partial \Omega \times (0,T)\).\ The initial condition \(u(\cdot ,0)=u_{0}\in L^{2}(\Omega )\) is added on \(\Omega \). The author introduces a proper, lower semicontinuous and convex function \(\widehat{\beta }\) on \(\mathbb{R}\) such that \(\widehat{\beta }(0)=0\) and \(\partial _{\mathbb{R}}\widehat{\beta } =\beta \), where \(\partial _{\mathbb{R}}\widehat{\beta }\) is the subdifferential of \(\widehat{\beta }\) on \(\mathbb{R}\). He also introduces a proper, lower semicontinuous and convex function \(\varphi _{h,0}^{t}\) on \( L^{2}(\Omega )\) given as \(\varphi _{h,0}^{t}=\int_{\Omega }(\widehat{\beta } (z(x))-u_{h}(x,t)z(x))dx\) if \(z\in L^{2}(\Omega )\) and \(\widehat{\beta } (z)\in L^{1}(\Omega )\), \(\varphi _{h,0}^{t}=\infty \) otherwise, \(u_{h}\) is such that \(u_{h}(t)\in L^{2}(\Omega )\) and \(u_{h}(t)=h(t)\) a.e. on \(\partial \Omega \), and an extension \(\varphi _{h}^{t}\) of \(\varphi _{h,0}^{t}\) through \(\varphi _{h}^{t}(z)=\varphi _{h,0}^{t}(z)\) if \(z\in L^{2}(\Omega )\) and \(\varphi _{h}^{t}(z)=\infty \) otherwise. The main purpose of the paper is to solve the above inclusion problem without any growth assumption on \( \widehat{\beta }\). The author first proves results concerning extensions of proper, lower semicontinuous and convex functions and he gives examples based on the Moreau-Yosida approximation of a proper, lower semicontinuous and convex function. He also proves existence results for evolution inclusions generated by \(-\Delta \beta \). Coming back to the original evolution inclusion problem, the author defines the notion of weak solution to this problem as a function \(u\in C([0,T];H^{-1}(\Omega ))\) such that \( u(0)=u_{0}\), \(u\) is weakly continuous from \([0,T]\) into \(L^{2}(\Omega )\), \( u^{\prime }\in L_{\mathrm{loc}}^{2}((0,T];H^{-1}(\Omega ))\), there exists \(\widetilde{ u}\in L_{\mathrm{loc}}^{2}((0,T];H^{1}(\Omega ))\) such that \(\widetilde{u}\in \beta (u)\) a.e. in \(\Omega \times (0,T)\), \(\widetilde{u}(t)-S_{0}(h(t))\in H_{0}^{1}(\Omega )\) a.e. \(t\in (0,T)\) and \(u\) satisfies a weak formulation of the above evolution inclusion problem. Under further hypotheses on the data, the author proves the existence of a unique weak solution to this problem, on which he proves estimates. The proof is based on the construction of approximate evolution inclusion equations, for which the author quotes from one of his previous papers the existence of a unique solution. He then proves uniform estimates on the approximate solutions. The last step of the proof consists to pass to the limit on the approximations which are introduced.
Reviewer: Alain Brillard (Riedisheim)A new approach to solve the Schrodinger equation with an anharmonic sextic potentialhttps://www.zbmath.org/1483.810632022-05-16T20:40:13.078697Z"Nanni, Luca"https://www.zbmath.org/authors/?q=ai:nanni.lucaSummary: In this study, the N-dimensional radial Schrodinger equation with an anharmonic sextic potential is solved by the extended Nikirov-Uranov method. We prove that the radial function can be factorised as the product between an exponential function and a polynomial function solution of the biconfluent Heun equation. The approach investigated in this article aims to be an alternative to other known methods of solving, as it has the advantage of dealing with simple, first-order differential and algebraic equations and avoiding numerous and laborious coordinate transformations and series expansions.The stationary optomechanical entanglement between an optical cavity field and a cubic anharmonic oscillatorhttps://www.zbmath.org/1483.810682022-05-16T20:40:13.078697Z"Huang, Sumei"https://www.zbmath.org/authors/?q=ai:huang.sumei"Wu, Yunqi"https://www.zbmath.org/authors/?q=ai:wu.yunqi"Chen, Aixi"https://www.zbmath.org/authors/?q=ai:chen.aixiSummary: Currently, the creation of the quantum entanglement is still one of the most challenging goals. Here, we theoretically investigate the stationary entanglement between an optical cavity field mode and a cubic nonlinear vibrating mirror in an optomechanical system. We show that the mechanical nonlinearity gives rise to the enhancement of the maximum optomechanical entanglement, and shifts the maximum entanglement towards high effective cavity detuning values. We find that the mechanical nonlinearity makes the optomechanical entanglement more robust against the thermal noise of the surrounding environment.Quantum graphs on radially symmetric antitreeshttps://www.zbmath.org/1483.810742022-05-16T20:40:13.078697Z"Kostenko, Aleksey"https://www.zbmath.org/authors/?q=ai:kostenko.aleksey-s"Nicolussi, Noema"https://www.zbmath.org/authors/?q=ai:nicolussi.noemaIn the present study the authors mainly focused their attention on antitrees from the perspective of quantum graphs and discussed a detailed spectral analysis of the Kirchhoff Laplacian on radially symmetric antitrees. Antitrees come into sight in the investigation of discrete Laplacians and attracted a noteworthy attention especially after the work of \textit{K.-T. Sturm} [J. Reine Angew. Math. 456, 173--196 (1994; Zbl 0806.53041)]. Also, Kostenko and Nicolussi considered the approach intorudced by [\textit{V. A. Mikhailets}, Funct. Anal. Appl. 30, No. 2, 144--146 (1996; Zbl 0874.34069); translation from Funkts. Anal. Prilozh. 30, No. 2, 90--93 (1996); \textit{B. Muckenhoupt}, Stud. Math. 44, 31--38 (1972; Zbl 0236.26015)] for radially symmetric trees and used some ideas from [\textit{J. Breuer} and \textit{N. Levi}, Ann. Henri Poincaré 21, No. 2, 499--537 (2020; Zbl 1432.05061)], where discrete Laplacians on radially symmetric ``weighted'' graphs have been analyzed. To summarize in general terms, in this paper, after recalling some necessary definitions and presenting an hypothesis, the authors studied characterization of self-adjointness and a complete description of self-adjoint extensions, spectral gap estimates and spectral types (discrete, singular and absolutely continuous spectrum). Next, they demonstrated their main results by considering two special classes of antitrees: (i) antitrees with exponentially increasing sphere numbers and (ii) antitrees with polynomially increasing sphere numbers.
Reviewer: Mustafa Salti (Mersin)Non-conformal attractor in boost-invariant plasmashttps://www.zbmath.org/1483.811432022-05-16T20:40:13.078697Z"Chattopadhyay, Chandrodoy"https://www.zbmath.org/authors/?q=ai:chattopadhyay.chandrodoy"Jaiswal, Sunil"https://www.zbmath.org/authors/?q=ai:jaiswal.sunil-prasad"Du, Lipei"https://www.zbmath.org/authors/?q=ai:du.lipei"Heinz, Ulrich"https://www.zbmath.org/authors/?q=ai:heinz.ulrich"Pal, Subrata"https://www.zbmath.org/authors/?q=ai:pal.subrataSummary: We study the dissipative evolution of (0+1)-dimensionally expanding media with Bjorken symmetry using the Boltzmann equation for massive particles in relaxation-time approximation. Breaking conformal symmetry by a mass induces a non-zero bulk viscous pressure in the medium. It is shown that even a small mass (in units of the local temperature) drastically modifies the well-known attractor for the shear Reynolds number previously observed in massless systems. For generic nonzero particle mass, neither the shear nor the bulk viscous pressure relax quickly to a non-equilibrium attractor; they approach the hydrodynamic limit only late, at small values of the inverse Reynolds numbers.
Only the longitudinal pressure, which is a combination of thermal, shear and bulk viscous pressures, continues to show early approach to a far-off-equilibrium attractor, driven by the rapid longitudinal expansion at early times. Second-order dissipative hydrodynamics based on a gradient expansion around locally isotropic thermal equilibrium fails to reproduce this attractor.Noise-induced dynamics in a Josephson junction driven by trichotomous noiseshttps://www.zbmath.org/1483.820082022-05-16T20:40:13.078697Z"Jin, Yanfei"https://www.zbmath.org/authors/?q=ai:jin.yanfei"Wang, Heqiang"https://www.zbmath.org/authors/?q=ai:wang.heqiangSummary: Noise-induced dynamics is explored in a Josephson junction system driven by multiplicative and additive trichotomous noises in this paper. Under the adiabatic approximation, the analytical expression of average output current for the Josephson junction is obtained, which can be used to characterize stochastic resonance (SR). If only the additive trichotomous noise is considered, the large correlation time of additive noise can induce the suppression and the SR in the curve of average output current. When the effects of both multiplicative and additive trichotomous noises are considered, two pronounced peaks exist in the curves of average output current for large multiplicative noise amplitude and optimal additive noise intensity. That is, the stochastic multi-resonance phenomenon is observed in this system. Moreover, the curve of average output current appears a single peak as a function of multiplicative noise intensity, which disappears for the case of small fixed additive noise amplitude. Especially, the mean first-passage time (MFPT) as the function of additive trichotomous noise intensity displays a non-monotonic behavior with a maximum for the large multiplicative noise amplitude, which is called the phenomenon of the noise enhanced stability (NES).Non-local imprints of gravity on quantum theoryhttps://www.zbmath.org/1483.830242022-05-16T20:40:13.078697Z"Maziashvili, Michael"https://www.zbmath.org/authors/?q=ai:maziashvili.michael"Silagadze, Zurab K."https://www.zbmath.org/authors/?q=ai:silagadze.zurab-kSummary: During the last two decades or so much effort has been devoted to the discussion of quantum mechanics (QM) that in some way incorporates the notion of a minimum length. This upsurge of research has been prompted by the modified uncertainty relation brought about in the framework of string theory. In general, the implementation of minimum length in QM can be done either by modification of position and momentum operators or by restriction of their domains. In the former case we have the so called soccer-ball problem when the naive classical limit appears to be drastically different from the usual one. Starting with the latter possibility, an alternative approach was suggested in the form of a band-limited QM. However, applying momentum cutoff to the wave-function, one faces the problem of incompatibility with the Schrödinger equation. One can overcome this problem in a natural fashion by appropriately modifying Schrödinger equation. But incompatibility takes place for boundary conditions as well. Such wave-function cannot have any more a finite support in the coordinate space as it simply follows from the Paley-Wiener theorem. Treating, for instance, the simplest quantum-mechanical problem of a particle in an infinite potential well, one can no longer impose box boundary conditions. In such cases, further modification of the theory is in order. We propose a non-local modification of QM, which has close ties to the band-limited QM, but does not require a hard momentum cutoff. In the framework of this model, one can easily work out the corrections to various processes and discuss further the semi-classical limit of the theory.Stability analysis of geodesics and quasinormal modes of a dual stringy black hole via Lyapunov exponentshttps://www.zbmath.org/1483.830452022-05-16T20:40:13.078697Z"Giri, Shobhit"https://www.zbmath.org/authors/?q=ai:giri.shobhit"Nandan, Hemwati"https://www.zbmath.org/authors/?q=ai:nandan.hemwatiSummary: We investigate the stability of both timelike as well as null circular geodesics in the vicinity of a dual (3+1) dimensional stringy black hole (BH) spacetime by using an excellent tool so-called Lyapunov exponent. The proper time \((\tau)\) Lyapunov exponent \((\lambda_p)\) and coordinate time \((t)\) Lyapunov exponent \((\lambda_c)\) are explicitly derived to analyze the stability of equatorial circular geodesics for the stringy BH spacetime with \textit{electric charge} parameter \((\alpha )\) and \textit{magnetic charge} parameter \((Q)\). By computing
these exponents for both the cases of BH spacetime, it is observed that the coordinate time Lyapunov exponent of magnetically charged stringy BH for both timelike and null geodesics are independent of magnetic charge parameter \((Q)\). The variation of the ratio of Lyapunov exponents with radius of timelike circular orbits \((r_0/M)\) for both the cases of stringy BH are presented. The behavior of instability exponent for null circular geodesics with respect to charge parameters \((\alpha\) and \(Q)\) are also observed for both the cases of BH. Further, by establishing a relation between quasinormal modes (QNMs) and parameters related to null circular geodesics (like angular frequency and Lyapunov exponent), we deduced the QNMs (or QNM frequencies) for a massless scalar field perturbation around \textit{both} the cases of stringy BH spacetime in the eikonal limit. The variation of scalar field potential with charge parameters and angular momentum of perturbation \((l)\) are visually presented and discussed accordingly.An alternative to the Teukolsky equationhttps://www.zbmath.org/1483.830472022-05-16T20:40:13.078697Z"Hatsuda, Yasuyuki"https://www.zbmath.org/authors/?q=ai:hatsuda.yasuyukiSummary: We conjecture a new ordinary differential equation exactly isospectral to the radial component of the homogeneous Teukolsky equation. We find this novel relation by a hidden symmetry implied from a four-dimensional \(\mathcal{N}=2\) supersymmetric quantum chromodynamics. Our proposal is powerful both in analytical and in numerical studies. As an application, we derive high-order perturbative series of quasinormal mode frequencies in the slowly rotating limit. We also test our result numerically by comparing it with a known technique.Greybody factor for a rotating Bardeen black hole by perfect fluid dark matterhttps://www.zbmath.org/1483.830642022-05-16T20:40:13.078697Z"Sharif, M."https://www.zbmath.org/authors/?q=ai:sharif.muhammad-a-r|sharif.mhd-saeed|sharif.masoud"Shaukat, Sulaman"https://www.zbmath.org/authors/?q=ai:shaukat.sulamanSummary: In this paper, the greybody factor is studied analytically for a rotating regular Bardeen black hole surrounded by perfect fluid dark matter. Firstly, we examine the behavior of effective potential by using the radial equation of motion developed from the Klein-Gordon equation. We then consider tortoise coordinate to convert the radial equation into Schrödinger form equation. We solve the radial equation of motion and obtain two different asymptotic solutions in terms of hypergeometric function measured at distinct regimes so called near and far-field horizons. These solutions are smoothly matched over the whole radial coordinate in an intermediate regime to check their viability. Finally, we measure the absorption probability for massless scalar field and examine the effect of perfect fluid dark matter. It is concluded that both the effective potential and greybody factor increase with perfect fluid dark matter.Climate change effects on fractional order prey-predator modelhttps://www.zbmath.org/1483.860042022-05-16T20:40:13.078697Z"Sekerci, Yadigar"https://www.zbmath.org/authors/?q=ai:sekerci.yadigarSummary: The key issue in ecology is how environmental changes associated with global climate change, specifically rising temperatures influence relationships of species. Predation is one of the main focus for understanding ecosystem responses to climate change. In this work, fractional order prey-predator system is considered by singular and nonsingular fractional operators within Caputo, Caputo-Fabrizio (CF) and Atangana-Baleanu-Caputo (ABC) sense. The predation rate is considered by a function of time, the function of the temperature refer to climate change, which explains how rising temperatures lead to predation. Extensive numerical simulations are performed to provide details of the underlying structure of the system. It is observed that the population fluctuate more in CF model than Caputo and ABC cases in time. Real-world observations are supported by obtained numerical observations of fluctuations in prey and predator populations under the effect of increasing temperature.Evolution of fractional-order chaotic economic systems based on non-degenerate equilibrium pointshttps://www.zbmath.org/1483.860082022-05-16T20:40:13.078697Z"Zhang, Guoxing"https://www.zbmath.org/authors/?q=ai:zhang.guoxing"Qian, Pengxiao"https://www.zbmath.org/authors/?q=ai:qian.pengxiao"Su, Zhaoxian"https://www.zbmath.org/authors/?q=ai:su.zhaoxianSummary: The economic system is an irreversible entropy increase process which is constructed by many elements and is far away from the equilibrium point; and affected by various parameters change, it is quite common that its motion state appears chaotic phenomenon due to instability. The extremely complex and not completely random aperiodic motion form of chaotic phenomenon is strongly sensitive to initial conditions. The development of nonlinear science, especially the emergence and development of chaos and fractal theory, has gradually become a powerful tool for economists to study the complexity, uncertainty and nonlinearity of social economic systems; and some visionary economists began to apply the research results of nonlinear science to economics, which has produced nonlinear economics. On the basis of summarizing and analyzing previous research works, this paper first obtains the non-degenerate equilibrium point of some typical fractional-order chaotic economic systems and transforms the equilibrium points of those systems to the origin through coordinate transformation, and then analyzes the Jacobi matrixes of new systems obtained through coordinate translation, and the parameter conditions of bifurcation in the economic systems are finally given and the numerical simulation of the fractional-order chaotic economic system evolution is carried out through bifurcation diagram, phase diagram and time series diagram. The study results of this paper provide a reference for the further study of the evolution of fractional-order chaotic economic systems with non-degenerate equilibrium points.Chaotic dynamics and chaos control for the fractional-order geomagnetic field modelhttps://www.zbmath.org/1483.860102022-05-16T20:40:13.078697Z"Al-khedhairi, A."https://www.zbmath.org/authors/?q=ai:al-khedhairi.abdulrahman"Matouk, A. E."https://www.zbmath.org/authors/?q=ai:matouk.ahmed-e"Khan, I."https://www.zbmath.org/authors/?q=ai:khan.irfan|khan.inam-ullah|khan.israr-h|khan.i-s|khan.ihsan-ullah|khan.indadul|khan.imran-a|khan.irshadullah|khan.imad-m|khan.imdadullah|khan.imranullah|khan.irshad-ahmad|khan.izaz-ullah|khan.ishaque|khan.izhar-ali|khan.imran-f|khan.ilyas|khan.islam|khan.idrees-a|khan.izharul-h|khan.izhar-ahmed|khan.ishita-kamal|khan.imdad|khan.iftikhar-ahmed|khan.ishtiaque-ahmed|khan.ishtiaq-rasoolSummary: Fractional-order Geomagnetic Field model is considered in this work. A sufficient condition is used to prove that the solution of the fractional-order Geomagnetic Field model exists and is unique in a specific region. Conditions for continuous dependence on initial conditions in our model are discussed. In addition, the conditions of local stability of the model's five equilibrium points are obtained. Chaotic attractors are shown to exist in the proposed fractional model. Also, Lyapunov exponents of the fractional-order Geomagnetic Field model are calculated and computations of Lyapunov spectrum as functions of all the model's parameters and fractional-order are performed. Moreover, a novel linear control technique based on Lyapunov stability theory is introduced here to stabilize the chaotic states of the fractional-order Geomagnetic Field model to its five equilibrium points. Finally, to verify the validity of our theoretical results and the effectiveness of the control scheme, numerical simulations based on the Atangana-Baleanu fractional integral in Caputo-sense are done to produce the chaotic attractors.Transitions in consumption behaviors in a peer-driven stochastic consumer networkhttps://www.zbmath.org/1483.911202022-05-16T20:40:13.078697Z"Jungeilges, Jochen"https://www.zbmath.org/authors/?q=ai:jungeilges.jochen-a"Ryazanova, Tatyana"https://www.zbmath.org/authors/?q=ai:ryazanova.tatyana-vladimirovnaSummary: We study transition phenomena between attractors occurring in a stochastic network of two consumers. The consumption of each individual is strongly influenced by the past consumption of the other individual, while own consumption experience only plays a marginal role. From a formal point of view we are dealing with a special case of a nonlinear stochastic consumption model taking the form of a 2-dimensional non-invertible map augmented by additive and/or parametric noise. In our investigation of the stochastic transitions we rely on a mixture of analytical and numerical techniques with a central role given to the concept of the stochastic sensitivity function and the related technique of confidence domains. We find that in the case of parametric noise the stochastic sensitivity of fixed points and cycles considered is considerably higher than in the case of additive noise. Three types of noise induced transitions between attractors are identified: (i) Escape from a stochastic fixed point with converge to a stochastic \(k\)-cycle, (ii) escape from the stochastic \(k\)-cycle to a stochastic fixed point, and (iii) cases in which the consumption process moves between the respective stochastic attractors for ever. The noise intensities at which such transitions are likely to occur tend to be smaller in the case of parametric noise than with additive noise.Complex interplay between monetary and fiscal policies in a real economy modelhttps://www.zbmath.org/1483.911312022-05-16T20:40:13.078697Z"Cavalli, Fausto"https://www.zbmath.org/authors/?q=ai:cavalli.fausto"Naimzada, Ahmad K."https://www.zbmath.org/authors/?q=ai:naimzada.ahmad-k"Pecora, Nicolò"https://www.zbmath.org/authors/?q=ai:pecora.nicoloSummary: In this paper we consider a nonlinear model for the real economy described by a multiplier-accelerator setup. The model comprises the government sector, which influences the output dynamics by means of the fiscal policy, and the money market, where the money supply depends upon the fluctuations in the economic activity. Through rigorous analytical tools combined with numerical simulations, we investigate the stability conditions of the unique steady state and the emergence of different kinds of endogenous dynamics, which are the results of the action of the fiscal and the monetary policy through their reactivity degrees. Such policies, if properly tuned, can lead the economy toward the desired full employment target but, on the other hand, can also generate endogenous fluctuations in the pace of the economic activity, associated with the occurrence of closed invariant curves and multistability phenomena.Construction of indicator system of regional economic system impact factors based on fractional differential equationshttps://www.zbmath.org/1483.911422022-05-16T20:40:13.078697Z"Zhang, Jun"https://www.zbmath.org/authors/?q=ai:zhang.jun.9|zhang.jun|zhang.jun.1|zhang.jun.2|zhang.jun.3|zhang.jun.10|zhang.jun.6|zhang.jun.5|zhang.jun.7|zhang.jun.4|zhang.jun.8"Fu, Xiaoming"https://www.zbmath.org/authors/?q=ai:fu.xiaoming"Morris, Harry"https://www.zbmath.org/authors/?q=ai:morris.harrySummary: In the evaluation activities of various economic systems, the selection of the design of the indicator system is often subjective. In practice, these principles tend to focus on formalization, and there are big differences with the process of selecting specific indicators. The regional economic development has entered a critical period of transition to a high-quality development stage, but the contradiction of insufficient regional economic imbalance development is still a constraint factor for the high-quality development of the regional economy. Based on the fractional differential equation analysis method, it expounds the mechanism of high-quality development of regional economy from 10 dimensions such as GDP, tertiary industry added value, local fiscal revenue, scientific and technological innovation, and market mechanism. Combining the ideas of the new development concept to promote the high-quality development of regional economy, the system is evaluated by fractional differential equations. The best indicator system framework was selected to build an economic system impact factor indicator system. According to the SEM analysis results, the weights of each index are determined.Complexity evolution of chaotic financial systems based on fractional calculushttps://www.zbmath.org/1483.912252022-05-16T20:40:13.078697Z"Wen, Chunhui"https://www.zbmath.org/authors/?q=ai:wen.chunhui"Yang, Jinhai"https://www.zbmath.org/authors/?q=ai:yang.jinhaiSummary: Economics and finance are extremely complex nonlinear systems involving human subjects with many subjective factors. There are numerous attribute properties that cannot be described by the theory of integer-order calculus; so it is necessary to theoretically study the internal complexity of the economic and financial system using the method of bifurcation and chaos of fractional nonlinear dynamics. Fractional calculus can more accurately describe the existence characteristics of complex physical, financial or medical systems, and can truly reflect the actual state properties of these systems; therefore the application of fractional order in chaotic systems has great significance to study the mathematical analysis of nonlinear dynamic systems, and the use of fractional calculus theory to model the complexity evolution of fractional chaotic financial systems has attracted more and more scholars' attention. On the basis of summarizing and analyzing previous studies, this paper qualitatively analyzes the stability of equilibrium solution of fractional-order chaotic financial system, and explores the complexity evolution law of the financial system near the equilibrium point and the occurring conditions of asymptotic chaotic state near this equilibrium point, and simulate the complexity evolution of chaotic financial systems using the Admas-Bashforth-Moulton finite difference method for mapping, phase diagram and time series graph. The research results of this paper provide a reference for government to formulate relevant economic policies, decision-making or further research on the complexity evolution of fractional-order chaotic financial systems.Regarding new numerical solution of fractional schistosomiasis disease arising in biological phenomenahttps://www.zbmath.org/1483.920072022-05-16T20:40:13.078697Z"Veeresha, P."https://www.zbmath.org/authors/?q=ai:veeresha.pundikala"Baskonus, Haci Mehmet"https://www.zbmath.org/authors/?q=ai:baskonus.haci-mehmet"Prakasha, D. G."https://www.zbmath.org/authors/?q=ai:prakasha.doddabhadrappla-gowda"Gao, Wei"https://www.zbmath.org/authors/?q=ai:gao.wei.3"Yel, Gulnur"https://www.zbmath.org/authors/?q=ai:yel.gulnurSummary: In this paper, we study to find the numerical solution of fractional Schistosomiasis disease by using a numerical method. Fractional Schistosomiasis disease model is used to symbolize a parasitic disease caused by trematode flukes of the genus Schistosoma. The physical behaviour of results obtained by using \(q\)-\textit{homotopy analyses transform method (q-HATM)} in terms of plots for different fractional-order is captured. The results obtained by using considered method is more effective and easy to apply in order to examine the nature of multi-dimensional differential equations of fractional order arising in biological disease.Autapse-induced complicated oscillations of a ring FHN neuronal network with multiple delayed couplingshttps://www.zbmath.org/1483.920162022-05-16T20:40:13.078697Z"Mao, Xiaochen"https://www.zbmath.org/authors/?q=ai:mao.xiaochen"Zhou, Xiangyu"https://www.zbmath.org/authors/?q=ai:zhou.xiangyu|zhou.xiangyu.1"Shi, Tiantian"https://www.zbmath.org/authors/?q=ai:shi.tiantian"Qiao, Lei"https://www.zbmath.org/authors/?q=ai:qiao.lei.2|qiao.lei|qiao.lei.1In this paper a delayed FitzHugh-Nagumo (FHN) neural network with a single autaptic connection is studied. The authors present stability analysis of the problem under consideration. The authors prove delay-independent and delay-dependent stability and Hopf bifurcation. An example is considered in which multi-periodic oscillations and chaos are obtained via numerical simulations. These simulations show that the autaptic connection leads to a quiescent state, periodic and chaotic firing patterns, and multistability.
For the entire collection see [Zbl 1470.74004].
Reviewer: Angela Slavova (Sofia)Delay-induced synchronization in two coupled chaotic memristive Hopfield neural networkshttps://www.zbmath.org/1483.920252022-05-16T20:40:13.078697Z"Wang, Zhen"https://www.zbmath.org/authors/?q=ai:wang.zhen.2"Parastesh, Fatemeh"https://www.zbmath.org/authors/?q=ai:parastesh.fatemeh"Rajagopal, Karthikeyan"https://www.zbmath.org/authors/?q=ai:rajagopal.karthikeyan"Hamarash, Ibrahim Ismael"https://www.zbmath.org/authors/?q=ai:hamarash.ibrahim-ismael"Hussain, Iqtadar"https://www.zbmath.org/authors/?q=ai:hussain.iqtadarSummary: This paper is concerned with the synchronization of two coupled hyperbolic-type Hopfield neural networks with a memristive synaptic connection. The results show that the coupling with no time-delay cannot lead to complete synchronization and increasing the coupling strength causes anti-phase synchronization. While adding time-delays to the coupling term not only changes the dynamical behavior of the network but also facilitates reaching the full synchronization manifold. The network is investigated in two different cases of single and multiple time-delays. The calculated average synchronization error indicates that using two time-delays has a better effect on the synchronization of systems. However, the synchronization region is lessened by increasing the time-delay values.Generalized patterns from local and non local reactionshttps://www.zbmath.org/1483.920302022-05-16T20:40:13.078697Z"Cencetti, Giulia"https://www.zbmath.org/authors/?q=ai:cencetti.giulia"Battiston, Federico"https://www.zbmath.org/authors/?q=ai:battiston.federico"Carletti, Timoteo"https://www.zbmath.org/authors/?q=ai:carletti.timoteo"Fanelli, Duccio"https://www.zbmath.org/authors/?q=ai:fanelli.duccioSummary: A class of systems is considered, where immobile species associated to distinct patches, the nodes of a network, interact both locally and at a long-range, as specified by an (interaction) adjacency matrix. Non local interactions are treated in a mean-field setting which enables the system to reach a homogeneous consensus state, either constant or time dependent. We provide analytical evidence that such homogeneous solution can turn unstable under externally imposed disturbances, following a symmetry breaking mechanism which anticipates the subsequent outbreak of the patterns. The onset of the instability can be traced back, via a linear stability analysis, to a dispersion relation that is shaped by the spectrum of an unconventional \textit{reactive Laplacian}. The proposed mechanism prescinds from the classical Local Activation and Lateral Inhibition scheme, which sits at the core of the Turing recipe for diffusion driven instabilities. Examples of systems displaying a fixed-point or a limit cycle, in their uncoupled versions, are discussed. Taken together, our results pave the way for alternative mechanisms of pattern formation, opening new possibilities for modeling ecological, chemical and physical interacting systems.Vibrational mono-/bi-resonance and wave propagation in FitzHhugh-Nagumo neural systems under electromagnetic inductionhttps://www.zbmath.org/1483.920372022-05-16T20:40:13.078697Z"Ge, Mengyan"https://www.zbmath.org/authors/?q=ai:ge.mengyan"Lu, Lulu"https://www.zbmath.org/authors/?q=ai:lu.lulu"Xu, Ying"https://www.zbmath.org/authors/?q=ai:xu.ying"Mamatimin, Rozihajim"https://www.zbmath.org/authors/?q=ai:mamatimin.rozihajim"Pei, Qiming"https://www.zbmath.org/authors/?q=ai:pei.qiming"Jia, Ya"https://www.zbmath.org/authors/?q=ai:jia.yaSummary: In this paper, an modified FitzHugh-Nagumo (FHN) neural model was employed to investigate the vibrational resonance (VR) phenomenon, the collective behaviors, and the transmission of weak low-frequency (LF) signal driven by high-frequency (HF) stimulus under the action of different electromagnetic induction in single FHN neuron and feed-forward feedback network (FFN) system, respectively. For the single FHN system, by increasing the amplitude of HF stimulus, the phenomena of vibrational mono-/bi-resonance are observed, and the input weak signal and output of system are synchronized, and the information of the weak LF signal is amplified. For the FFN system, the phenomena of vibrational mono-/bi-resonances are also occurred, both frequency and amplitude of the HF stimulus play an important role in the vibrational bi-resonances and transmission of weak LF signal in the FHN neural FFN.A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivativehttps://www.zbmath.org/1483.920412022-05-16T20:40:13.078697Z"Baleanu, Dumitru"https://www.zbmath.org/authors/?q=ai:baleanu.dumitru-i"Jajarmi, Amin"https://www.zbmath.org/authors/?q=ai:jajarmi.amin"Mohammadi, Hakimeh"https://www.zbmath.org/authors/?q=ai:mohammadi.hakimeh"Rezapour, Shahram"https://www.zbmath.org/authors/?q=ai:rezapour.shahramSummary: In this research, we aim to propose a new fractional model for human liver involving Caputo-Fabrizio derivative with the exponential kernel. Concerning the new model, the existence of a unique solution is explored by using the Picard-Lindelöf approach and the fixed-point theory. In addition, the mathematical model is implemented by the homotopy analysis transform method whose convergence is also investigated. Eventually, numerical experiments are carried out to better illustrate the results. Comparative results with the real clinical data indicate the superiority of the new fractional model over the pre-existent integer-order model with ordinary time-derivatives.Mathematical modeling of tumor-immune system interactions: the effect of rituximab on breast cancer immune responsehttps://www.zbmath.org/1483.920432022-05-16T20:40:13.078697Z"Bitsouni, Vasiliki"https://www.zbmath.org/authors/?q=ai:bitsouni.vasiliki"Tsilidis, Vasilis"https://www.zbmath.org/authors/?q=ai:tsilidis.vasilisSummary: tBregs are a newly discovered subcategory of B regulatory cells, which are generated by breast cancer, resulting in the increase of Tregs and therefore in the death of NK cells. In this study, we use a mathematical and computational approach to investigate the complex interactions between the aforementioned cells as well as CD\(8^+\) T cells, CD\(4^+\) T cells and B cells. Furthermore, we use data fitting to prove that the functional response regarding the lysis of breast cancer cells by NK cells has a ratio-dependent form. Additionally, we include in our model the concentration of rituximab -- a monoclonal antibody that has been suggested as a potential breast cancer therapy -- and test its effect, when the standard, as well as experimental dosages, are administered.Modelling the risk of HIV infection for drug abusershttps://www.zbmath.org/1483.920442022-05-16T20:40:13.078697Z"Bloomquist, Angelica"https://www.zbmath.org/authors/?q=ai:bloomquist.angelica"Vaidya, Naveen K."https://www.zbmath.org/authors/?q=ai:vaidya.naveen-kSummary: Drugs of abuse, such as opiates, are one of the leading causes for transmission of HIV in many parts of the world. Drug abusers often face a higher risk of acquiring HIV because target cell (CD4+ T-cell) receptor expression differs in response to morphine, a metabolite of common opiates. In this study, we use a viral dynamics model that incorporates the T-cell expression difference to formulate the probability of infection among drug abusers. We quantify how the risk of infection is exacerbated in morphine conditioning, depending on the timings of morphine intake and virus exposure. With in-depth understanding of the viral dynamics and the increased risk for these individuals, we further evaluate how preventive therapies, including pre- and post-exposure prophylaxis, affect the infection risk in drug abusers. These results are useful to devise ideal treatment protocols to combat the several obstacles those under drugs of abuse face.Modeling of stable immune response managementhttps://www.zbmath.org/1483.920512022-05-16T20:40:13.078697Z"Rusakov, S."https://www.zbmath.org/authors/?q=ai:rusakov.s-a|rusakov.s-g|rusakov.sergei-vladimirovich"Chirkov, M."https://www.zbmath.org/authors/?q=ai:chirkov.m-v|chirkov.m-k"Volinsky, I."https://www.zbmath.org/authors/?q=ai:volinsky.irina-lSummary: A numerical solution of discrete control of immune response in infectious disease is considered. Ordinary differential equations with a delayed argument are used to describe immune processes. Immunotherapy is selected as the control factor, which consists in introduction of donor antibodies. The paper shows that immunotherapy allows for carrying out effective treatment in acute forms of infectious disease.Mathematical insights into neuroendocrine transdifferentiation of human prostate cancer cellshttps://www.zbmath.org/1483.920522022-05-16T20:40:13.078697Z"Turner, Leo"https://www.zbmath.org/authors/?q=ai:turner.leo"Burbanks, Andrew"https://www.zbmath.org/authors/?q=ai:burbanks.andrew-d"Cerasuolo, Marianna"https://www.zbmath.org/authors/?q=ai:cerasuolo.mariannaSummary: Prostate cancer represents the second most common cancer diagnosed in men and the fifth most common cause of death from cancer worldwide. In this paper, we consider a nonlinear mathematical model exploring the role of neuroendocrine transdifferentiation in human prostate cancer cell dynamics. Sufficient conditions are given for both the biological relevance of the model's solutions and for the existence of its equilibria. By means of a suitable Lyapunov functional the global asymptotic stability of the tumour-free equilibrium is proven, and through the use of sensitivity and bifurcation analyses we identify the parameters responsible for the occurrence of Hopf and saddle-node bifurcations. Numerical simulations are provided highlighting the behaviour discovered, and the results are discussed together with possible improvements to the model.P53-Mdm2 loop stability and oscillatory dynamics with Mdm2-induced delay effect in P53https://www.zbmath.org/1483.920682022-05-16T20:40:13.078697Z"Baba, Mohd Younus"https://www.zbmath.org/authors/?q=ai:baba.mohd-younus"Saleem, Mohammad"https://www.zbmath.org/authors/?q=ai:saleem.mohammad"Raheem, Abdur"https://www.zbmath.org/authors/?q=ai:raheem.abdurSummary: In this paper, we consider P53-Mdm2 negative feedback loop supposed to be the core circuit of genome. We study stability and the oscillatory dynamics of the loop. Many of the studies modeled this loop by delay-differential equations with P53-induced transcriptional delay in the production of Mdm2. We, however, highlight the importance of Mdm2- induced delay in the degradation of P53 protein. We consider two forms of P53 protein i.e., plain P53 and active P53 along with its principal antagonist protein Mdm2 to formulate a minimal model. Active P53 finds its inclusion in the loop in the presence of DNA damage represented by a Boolean variable `s'. The analysis of the model provides thresholds on delays using Nyquist criterion such that delays in the degradation of P53 lower than these thresholds guarantee stability of the loop in that all proteins plain P53, active P53 and Mdm2 approach to stable equilibrium state. The oscillatory dynamics in proteins, if any, would exist beyond these thresholds.Uniqueness of weakly reversible and deficiency zero realizations of dynamical systemshttps://www.zbmath.org/1483.920732022-05-16T20:40:13.078697Z"Craciun, Gheorghe"https://www.zbmath.org/authors/?q=ai:craciun.gheorghe"Jin, Jiaxin"https://www.zbmath.org/authors/?q=ai:jin.jiaxin"Yu, Polly Y."https://www.zbmath.org/authors/?q=ai:yu.polly-yThe authors address the relationship between chemical reaction networks obeying mass-action kinetics and the topological dynamics of corresponding systems of differential equations. A reaction network together with rate constants gives rise to a specific differential equation. The question is in how far a differential equation determines a relevant reaction network. In general, the differential equation does not select a particular network. The main result tells us that this is nevertheless the case for the simplest and most important class of networks, consisting of those that are weakly reversible and deficiency zero (Theorem 3.12). If different networks show the same dynamics then at most one belongs to this special class. The paper presents a thorough introduction to the pertinent theory together with a description of its development, equipped with numerous references to the literature.
Reviewer: Dieter Erle (Dortmund)An investigation on Michaelis-Menten kinetics based complex dynamics of tumor-immune interactionhttps://www.zbmath.org/1483.920742022-05-16T20:40:13.078697Z"Das, Parthasakha"https://www.zbmath.org/authors/?q=ai:das.parthasakha"Mukherjee, Sayan"https://www.zbmath.org/authors/?q=ai:mukherjee.sayan"Das, Pritha"https://www.zbmath.org/authors/?q=ai:das.prithaSummary: We investigate the dynamics of a three-dimensional tumor-immune interactions system. Local dynamics of the system has studied by the finding stability and Hopf bifurcation at biologically feasible equilibria. Further, chaotic phenomena have been investigated by measuring the asymptotic growth of the corresponding phase space trajectory with the various bifurcating parameters of the system. A dynamics of mean fluctuations of the tumor growth based on its local maxima have investigated under the same parameters. A significant correlation between the fluctuations and the tumor dynamics have verified by statistical analysis.Periodic solutions for a model of tumor volume with anti-angiogenic periodic treatmenthttps://www.zbmath.org/1483.920772022-05-16T20:40:13.078697Z"Díaz-Marín, Homero"https://www.zbmath.org/authors/?q=ai:diaz-marin.homero-g"Osuna, Osvaldo"https://www.zbmath.org/authors/?q=ai:osuna.osvaldoSummary: In this work, we consider the dynamics of a model for tumor volume growth under a drug periodic treatment targeting the process of angiogenesis within the vascularized cancer tissue. We give sufficient conditions for the existence and uniqueness of a global attractor consisting of a periodic solution. This conditions happen to be satisfied by values of the parameters tested for realistic experimental data. Numerical simulations are provided illustrating our findings.New approach for the model describing the deathly disease in pregnant women using Mittag-Leffler functionhttps://www.zbmath.org/1483.920782022-05-16T20:40:13.078697Z"Gao, Wei"https://www.zbmath.org/authors/?q=ai:gao.wei.3"Veeresha, P."https://www.zbmath.org/authors/?q=ai:veeresha.pundikala"Prakasha, D. G."https://www.zbmath.org/authors/?q=ai:prakasha.doddabhadrappla-gowda"Baskonus, Haci Mehmet"https://www.zbmath.org/authors/?q=ai:baskonus.haci-mehmet"Yel, Gulnur"https://www.zbmath.org/authors/?q=ai:yel.gulnurSummary: In this paper, numerical solution of the mathematical model describing the deathly disease in pregnant women with fractional order is investigated with the help of \(q\)-\textit{homotopy analysis transform method} \((q\)-HATM). This sophisticated and important model is consisted of a system of four equations, which illustrate a deathly disease spreading pregnant women called Lassa hemorrhagic fever disease. The fixed point theorem is considered so as to demonstrate the existence and uniqueness of the obtained numerical solution for the governing fractional model. The proposed method is also included the Laplace transform technique with \(q\)-homotopy analysis scheme, and fractional derivative defined with Atangana-Baleanu (AB) operator. In order to illustrate and validate the efficiency of the future technique, the projected model in the sense of fractional order is also considered. Moreover, the physical behaviors of the obtained numerical results are presented in terms of simulations for diverse fractional order.Modeling the virus-induced tumor-specific immune response with delay in tumor virotherapyhttps://www.zbmath.org/1483.920812022-05-16T20:40:13.078697Z"Li, Qian"https://www.zbmath.org/authors/?q=ai:li.qian"Xiao, Yanni"https://www.zbmath.org/authors/?q=ai:xiao.yanniSummary: It is urgently required to design novel cancer therapies which lead to permanent cancer eradication or cancer control. Oncolytic virotherapy is a promising cancer treatment strategy using genetically engineered viruses which can selectively infect, replicate in and kill tumor cells without harming normal cells. Due to the variable interactions between tumor cells and oncolytic viruses (OVs) as well as the accordingly immune response, the impact of tumor virotherapy on tumor control is prominent. We propose a novel mathematical modeling framework based on delay differential equation to study tumor virotherapy with mediated antitumor immunity by OVs, which incorporates complex tumor-virus-immune system interactions. We initially study the existence and local stability of equilibria, and theoretically and numerically investigate the local Hopf bifurcation from the positive equilibrium by considering the time delay as a bifurcation parameter. Further, we analyze the direction of Hopf bifurcation and the stability of bifurcating periodic solution. Main results show that the time delay can induce Hopf bifurcation and result in periodic oscillations, indicating that the delayed tumor-specific CTL response induced by OVs leads to complex dynamics and may significantly influence the development process of tumor growth. Our findings also provide insight into important aspects of the virotherapy, including the dependence of the efficacy on key factors. We find that enhancing the induction of antitumor immunity by OVs can reduce the complexity of dynamics by strengthening the stability and show considerable effects on combating tumor. Further, proper choice of OVs with relative strong ability to lyse tumor cells is beneficial to ultimately control the development process of tumor growth.Insights into the dynamics of ligand-induced dimerisation via mathematical modelling and analysishttps://www.zbmath.org/1483.920822022-05-16T20:40:13.078697Z"White, C."https://www.zbmath.org/authors/?q=ai:white.carla"Rottschäfer, V."https://www.zbmath.org/authors/?q=ai:rottschafer.vivi"Bridge, L. J."https://www.zbmath.org/authors/?q=ai:bridge.lloyd-jSummary: The vascular endothelial growth factor (VEGF) receptor (VEGFR) system plays a role in cancer and many other diseases. It is widely accepted that VEGFR receptors dimerise in response to VEGF binding. However, analysis of these mechanisms and their implications for drug development still requires further exploration. In this paper, we present a mathematical model representing the binding of VEGF to VEGFR and the subsequent ligand-induced dimerisation. A key factor in this work is the qualitative and quantitative effect of binding cooperativity, which describes the effect that the binding of a ligand to a receptor has on the binding of that ligand to a second receptor, and the dimerisation of these receptors. We analyse the ordinary differential equation system at equilibrium, giving analytical solutions for the total amount of ligand bound. For time-course dynamics, we use numerical methods to explore possible behaviours under various parameter regimes, while perturbation analysis is used to understand the intricacies of these behaviours. Our simulation results show an excellent fit to experimental data, towards validating the model.Dynamic behavior of a fractional order prey-predator model with group defensehttps://www.zbmath.org/1483.921072022-05-16T20:40:13.078697Z"Alidousti, Javad"https://www.zbmath.org/authors/?q=ai:alidousti.javad"Ghafari, Elham"https://www.zbmath.org/authors/?q=ai:ghafari.elhamSummary: In this paper, we consider a fractional order prey predator model with a prey and two predator species with the group defense capability. In this model, we use the Holling-IV functional response, called Monod-Haldane function, for interactions between prey and predator species. Boundedness of the solution will be proved. Local stability of system's equilibrium points will be investigated analytically and the required conditions for existence of Hopf bifurcation will be obtained. Finally, by using numerical methods, the validity of the obtained results and more dynamical behaviors of system, such as chaotic and periodic solutions will be assessed.Atangana-Baleanu fractional framework of reproducing kernel technique in solving fractional population dynamics systemhttps://www.zbmath.org/1483.921102022-05-16T20:40:13.078697Z"Hasan, Shatha"https://www.zbmath.org/authors/?q=ai:hasan.shatha"El-Ajou, Ahmad"https://www.zbmath.org/authors/?q=ai:el-ajou.ahmad"Hadid, Samir"https://www.zbmath.org/authors/?q=ai:hadid.samir-b"Al-Smadi, Mohammed"https://www.zbmath.org/authors/?q=ai:al-smadi.mohammed-h"Momani, Shaher"https://www.zbmath.org/authors/?q=ai:momani.shaher-mSummary: In this article, a class of population growth model, the fractional nonlinear logistic system, is studied analytically and numerically. This model is investigated by means of Atangana-Baleanu fractional derivative with a non-local smooth kernel in Sobolev space. Existence and uniqueness theorem for the fractional logistic equation is provided based on the fixed-point theory. In this orientation, two numerical techniques are implemented to obtain the approximate solutions; the reproducing-kernel algorithm is based on the Schmidt orthogonalization process to construct a complete normal basis, while the successive substitution algorithm is based on an appropriate iterative scheme. Convergence analysis associated with the suggested approaches is provided to demonstrate the applicability theoretically. The impact of the fractional derivative on population growth is discussed by a class of nonlinear logistical models using the derivatives of Caputo, Caputo-Fabrizio, and Atangana-Baleanu. Using specific examples, numerical simulations are presented in tables and graphs to show the effect of the fractional operator on the population curve as. The present results confirm the theoretical predictions and depict that the suggested schemes are highly convenient, quite effective and practically simplify computational time.Study on evolution of a predator-prey model in a polluted environmenthttps://www.zbmath.org/1483.921112022-05-16T20:40:13.078697Z"Liu, Bing"https://www.zbmath.org/authors/?q=ai:liu.bing.1"Wang, Xin"https://www.zbmath.org/authors/?q=ai:wang.xin|wang.xin.9|wang.xin.3|wang.xin.8|wang.xin.13|wang.xin.12|wang.xin.7|wang.xin.5|wang.xin.4|wang.xin.1|wang.xin.2|wang.xin.11|wang.xin.6|wang.xin.10"Song, Le"https://www.zbmath.org/authors/?q=ai:song.le"Liu, Jingna"https://www.zbmath.org/authors/?q=ai:liu.jingnaSummary: In this paper, we investigate the effects of pollution on the body size of prey about a predator-prey evolutionary model with a continuous phenotypic trait in a pulsed pollution discharge environment. Firstly, an eco-evolutionary predator-prey model incorporating the rapid evolution is formulated to investigate the effects of rapid evolution on the population density and the body size of prey by applying the quantitative trait evolutionary theory. The results show that rapid evolution can increase the density of prey and avoid population extinction, and with the worsening of pollution, the evolutionary traits becomes smaller gradually. Next, by employing the adaptive dynamic theory, a long-term evolutionary model is formulated to evaluate the effects of long-term evolution on the population dynamics and the effects of pollution on the body size of prey. The invasion fitness function is given, which reflects whether the mutant can invade successfully or not. Considering the trade-off between the intrinsic growth rate and the evolutionary trait, the critical function analysis method is used to investigate the dynamics of such slow evolutionary system. The results of theoretical analysis and numerical simulations conclude that pollution affects the evolutionary traits and evolutionary dynamics. The worsening of the pollution leads to a smaller body size of prey due to natural selection, while the opposite is more likely to generate evolutionary branching.A mathematical model of population dynamics for the internet gaming addictionhttps://www.zbmath.org/1483.921122022-05-16T20:40:13.078697Z"Seno, Hiromi"https://www.zbmath.org/authors/?q=ai:seno.hiromiSummary: As the number of internet users appears to steadily increase each year, Internet Gaming Disorder (IGD) is bound to increase as well. The question how this increase will take place, and what factors have the largest impact on this increase, naturally arises. We consider a system of ordinary differential equations as a simple mathematical model of the population dynamics about the internet gaming. We assume three stages about the internet gamer's state: moderate, addictive, and under treatment. The transition of the gamer's state between the moderate and the addictive stages is significantly affected by the social nature of internet gaming. As the activity of social interaction gets higher, the gamer would be more likely to become addictive. With the inherent social reinforcement of internet game, the addictive gamer would hardly recontrol his/herself to recover to the moderate gamer. Our result on the model demonstrates the importance of earlier initiation of a system to check the IGD and lead to some medical/therapeutic treatment. Otherwise, the number of addictive gamers would become larger beyond the socially controllable level.Vaccination and vector control effect on dengue virus transmission dynamics: modelling and simulationhttps://www.zbmath.org/1483.921182022-05-16T20:40:13.078697Z"Abidemi, A."https://www.zbmath.org/authors/?q=ai:abidemi.afeez"Abd Aziz, M. I."https://www.zbmath.org/authors/?q=ai:abd-aziz.m-i"Ahmad, R."https://www.zbmath.org/authors/?q=ai:ahmad.reyaz|ahmad.riyaz|ahmad.robiah|ahmad.rauf|ahmad.riaz|ahmad.r-badlishah|ahmad.rehan|ahmad.rana-zeeshan|ahmad.rodina|ahmad.rubi|ahmad.rashdi-shah|ahmad.rashid.1|ahmad.rokiah-rozita|ahmad.rana-tariq-mehmood|ahmad.rohanin|ahmad.rais|ahmad.raheelSummary: This paper presents a two-strain compartmental dengue model with variable humans and mosquitoes populations sizes. The model incorporates two control measures: \textit{Dengvaxia} vaccine and insecticide (adulticide) to forecast the transmission and effective control strategy for dengue in Madeira Island if there is a new outbreak with a different virus serotype after the first outbreak in 2012. The basic reproduction number, \(\mathcal{R}_0=\max\{\sqrt{\mathcal{R}_{01}}, \sqrt{\mathcal{R}_{0j}}\}\), associated with the model is computed using the next generation matrix operator. The disease-free equilibrium is found to be locally asymptotically stable when both \(\mathcal{R}_{01},\mathcal{R}_{0j}<1\), but unstable otherwise. The global asymptotic stability of the model is derived using the comparison theorem. Sensitivity analysis is carried out on the model parameters. The results of the analysis show that mosquito biting and death rates are the most sensitive parameters. Three strategies: the use of \textit{Dengvaxia} vaccine only, the use of adulticide only, and the combination of \textit{Dengvaxia} vaccine and adulticide, are considered for the control implementation under two scenarios (less and more aggressive cases). The numerical results show that a strategy which is based on \textit{Dengvaxia} vaccine and adulticide is the most effective strategy for controlling dengue disease transmission in both scenarios among the considered strategies.Effects of anti-infection behavior on the equilibrium states of an infectious diseasehttps://www.zbmath.org/1483.921192022-05-16T20:40:13.078697Z"Báez Sánchez, Andrés David"https://www.zbmath.org/authors/?q=ai:sanchez.andres-david-baez"Bobko, Nara"https://www.zbmath.org/authors/?q=ai:bobko.naraSummary: We propose a mathematical model to analyze the effects of anti-infection behavior on the equilibrium states of an infectious disease. The anti-infection behavior is incorporated into a classical epidemiological SIR model, by considering the behavior adoption rate across the population as an additional variable. We consider also the effects on the adoption rate produced by the disease evolution, using a dynamic payoff function and an additional differential equation. The equilibrium states of the proposed model have remarkable characteristics: possible coexistence of two locally stable endemic equilibria, the coexistence of locally stable endemic and disease-free equilibria, and even the possibility of a stable continuum of endemic equilibrium points. We show how some of the results obtained may be used to support strategic planning leading to effective control of the disease in the long-term.A new zoonotic visceral leishmaniasis dynamic transmission model with age-structurehttps://www.zbmath.org/1483.921222022-05-16T20:40:13.078697Z"Bi, Kaiming"https://www.zbmath.org/authors/?q=ai:bi.kaiming"Chen, Yuyang"https://www.zbmath.org/authors/?q=ai:chen.yuyang"Zhao, Songnian"https://www.zbmath.org/authors/?q=ai:zhao.songnian"Ben-Arieh, David"https://www.zbmath.org/authors/?q=ai:ben-arieh.david"Wu, Chih-Hang (John)"https://www.zbmath.org/authors/?q=ai:wu.chih-hangSummary: Visceral leishmaniasis (VL) is a fatal, neglected tropical disease primarily caused by \textit{Leishmania donovani} (\textit{L. donovani}) and \textit{Leishmania infantum} (\textit{L. infantum}). According to VL infectious data reports from severely affected countries, children and teenagers (ages 0--20) have a significantly higher vulnerability to VL infection than other populations. This paper utilizes an infected function (by age) established from epidemic prevalence data to propose a new partial differential equation (PDE) model for infection transmission patterns for various age groups. This new PDE model can be used to study VL epidemics in time and age dimensions. Disease-free and endemic equilibriums are discussed in relation to theoretical stability of the PDE system. This paper also proposes system output adjustment using historical VL data from the World Health Organization. Statistical methods such as the moving average and the autoregressive methods are used to calibrate estimated prevalence trends, potentially minimizing differences between stochastic stimulation results and reported real-world data. Results from simulation experiments using the PDE model were used to predict the worldwide VL severity of the epidemic in the next four years (from 2017 to 2020).Analysis of a stochastic distributed delay epidemic model with relapse and Gamma distribution kernelhttps://www.zbmath.org/1483.921252022-05-16T20:40:13.078697Z"Caraballo, Tomás"https://www.zbmath.org/authors/?q=ai:caraballo.tomas"El Fatini, Mohamed"https://www.zbmath.org/authors/?q=ai:el-fatini.mohamed"El Khalifi, Mohamed"https://www.zbmath.org/authors/?q=ai:el-khalifi.mohamed"Gerlach, Richard"https://www.zbmath.org/authors/?q=ai:gerlach.richard-h"Pettersson, Roger"https://www.zbmath.org/authors/?q=ai:pettersson.rogerSummary: In this work, we investigate a stochastic epidemic model with relapse and distributed delay. First, we prove that our model possesses and unique global positive solution. Next, by means of the Lyapunov method, we determine some sufficient criteria for the extinction of the disease and its persistence. In addition, we establish the existence of a unique stationary distribution to our model. Finally, we provide some numerical simulations for the stochastic model to assist and show the applicability and efficiency of our results.Understanding dynamics of \textit{Plasmodium falciparum} gametocytes production: insights from an age-structured modelhttps://www.zbmath.org/1483.921282022-05-16T20:40:13.078697Z"Djidjou-Demasse, Ramsès"https://www.zbmath.org/authors/?q=ai:demasse.ramses-djidjou"Ducrot, Arnaud"https://www.zbmath.org/authors/?q=ai:ducrot.arnaud"Mideo, Nicole"https://www.zbmath.org/authors/?q=ai:mideo.nicole"Texier, Gaëtan"https://www.zbmath.org/authors/?q=ai:texier.gaetanSummary: Many models of within-host malaria infection dynamics have been formulated since the pioneering work of \textit{R. M. Anderson} et al. in [``Non-linear phenomena in host-parasite interactions'', Parasitology 99, Suppl S1, S59--S79 (1989; \url{doi:10.1017/s0031182000083426.})]. Biologically, the goal of these models is to understand what governs the severity of infections, the patterns of infectiousness, and the variation thereof across individual hosts. Mathematically, these models are based on dynamical systems, with standard approaches ranging from \(K\)-compartments ordinary differential equations (ODEs) to delay differential equations (DDEs), to capture the relatively constant duration of replication and bursting once a parasite infects a host red blood cell. Using malaria therapy data, which offers fine-scale resolution on the dynamics of infection across a number of individual hosts, we compare the fit and robustness of one of these standard approaches \((K\)-compartments ODE) with a partial differential equations (PDEs) model, which explicitly tracks the ``age'' of an infected cell. While both models perform quite similarly in terms of goodness-of-fit for suitably chosen \(K\), the \(K\)-compartments ODE model particularly overestimates parasite densities early on in infections when the number of repeated compartments is not large enough. Finally, the \(K\)-compartments ODE model (for suitably chosen \(K)\) and the PDE model highlight a strong qualitative connection between the density of transmissible parasite stages (\textit{i.e.}, gametocytes) and the density of host-damaging (and asexually-replicating) parasite stages. This finding provides a simple tool for predicting which hosts are most infectious to mosquitoes -- vectors of \textit{Plasmodium} parasites -- which is a crucial component of global efforts to control and eliminate malaria.A fractional order model for Ebola virus with the new Caputo fractional derivative without singular kernelhttps://www.zbmath.org/1483.921292022-05-16T20:40:13.078697Z"Dokuyucu, Mustafa Ali"https://www.zbmath.org/authors/?q=ai:dokuyucu.mustafa-ali"Dutta, Hemen"https://www.zbmath.org/authors/?q=ai:dutta.hemenSummary: In this study, the model of the Ebola virus, which has been rapidly spreading in certain parts of Africa, was rearranged using the fractional derivative operator without a singular kernel proposed by Caputo and Fabrizio. It is aimed to obtain better results from the model using this approach of the model. In the first stage, the Ebola virus model was extended to the Caputo-Fabrizio fractional derivative operator. After, existence and uniqueness solutions were obtained for the fractional Ebola virus model via fixed-point theorem. Then, numerical solutions were obtained for the extended model by using Atangana and Owolabi new numerical approach via Adam-Basford method for the Caputo-Fabrizio fractional derivative. Finally, some numerical simulations were presented for different values of fractional order.Transmission dynamic and backward bifurcation of Middle Eastern respiratory syndrome coronavirushttps://www.zbmath.org/1483.921312022-05-16T20:40:13.078697Z"Fatima, Bibi"https://www.zbmath.org/authors/?q=ai:fatima.bibi"Zaman, Gul"https://www.zbmath.org/authors/?q=ai:zaman.gul"Jarad, Fahd"https://www.zbmath.org/authors/?q=ai:jarad.fahdSummary: Middle East respiratory syndrome coronavirus (MERS-CoV) remains an emerging disease threat with regular human cases on the Arabian Peninsula driven by recurring camels to human transmission events. In this paper, we present a new deterministic model for the transmission dynamics of (MERS-CoV). In order to do this, we develop a model formulation and analyze the stability of the proposed model. The stability conditions are obtained in term of \(R_0\), we find those conditions for which the model become stable. We discuss basic reproductive number \(R_0\) along with sensitivity analysis to show the impact of every epidemic parameter. We show that the proposed model exhibits the phenomena of backward bifurcation. Finally, we show the numerical simulation of our proposed model for supporting our analytical work. The aim of this work is to show via mathematical model the transmission of MERS-CoV between humans and camels, which are suspected to be the primary source of infection.Impact of fear on an eco-epidemiological modelhttps://www.zbmath.org/1483.921332022-05-16T20:40:13.078697Z"Hossain, Mainul"https://www.zbmath.org/authors/?q=ai:hossain.mainul"Pal, Nikhil"https://www.zbmath.org/authors/?q=ai:pal.nikhil-ranjan"Samanta, Sudip"https://www.zbmath.org/authors/?q=ai:samanta.sudip-kSummary: In the present paper, we investigate the impact of fear in a predator-prey model with disease in the prey species. The logistically growing prey population is divided into two groups: susceptible and infected. We take the fear of predators among prey population into consideration, which costs lowering of prey's growth rate and slims down the interactions among prey individuals. In our model, it is reflected by two decreasing factors of the fear parameter and the predator population: one in the growth term of the susceptibles and the other in the disease transmission term. We choose a general disease transmission function for which mass action, standard incidence, and saturation laws are particular cases. The predator-prey interactions are described by generalized Holling type-II functional response. We explore the effect of fear for three subcases of our model, all of which happen to be identical to three published works (without fear effect). Apart from the preliminary mathematical analysis of our model (e.g. positivity, boundedness, etc.), we find the conditions for existence and local stability of the equilibrium points and study Hopf-bifurcation around the endemic equilibrium point w.r.t. the fear parameters. We observe that fear can eliminate the chaotic oscillations of the system, produced in the absence of fear, by either making the endemic equilibrium regular (stable limit cycle or stable equilibrium point) or moving it towards the disease-free state. We also observe the presence of multiple attractors in the phase-space and different types of bistabilities. We perform extensive numerical simulations to explore the rich dynamics of our model.Analysis of linear and nonlinear mathematical models for monitoring diabetic population with minor and major complicationshttps://www.zbmath.org/1483.921452022-05-16T20:40:13.078697Z"Modu, Goni Umar"https://www.zbmath.org/authors/?q=ai:modu.goni-umar"Hadejia, Yunusa Aliyu"https://www.zbmath.org/authors/?q=ai:hadejia.yunusa-aliyu"Ahmed, Idris"https://www.zbmath.org/authors/?q=ai:ahmed.idris"Kumam, Wiyada"https://www.zbmath.org/authors/?q=ai:kumam.wiyada"Thounthong, Phatiphat"https://www.zbmath.org/authors/?q=ai:thounthong.phatiphatSummary: A mathematical analysis of linear and nonlinear models for monitoring diabetic populations with minor and major complications are considered in this work. The equilibrium point of the linear system is shown to be globally asymptotically stable (GAS) using direct Lyapunov method. For the nonlinear model, three positive equilibrium points were obtained and analyzed and only one of the equilibrium points is globally asymptotically stable (GAS), shown using the direct Lyapunov method. Some numerical simulations are carried out to demonstrate the analytical results. It is found that the prevalence/incidence of diabetes is on the rise. Our results are effective in monitoring diabetic populations with minor and major complications and the mathematical methods used in the analysis can be applied in different work. The models can be used to monitor global diabetic populations over time.Dynamic analysis and optimal control of a class of SISP respiratory diseaseshttps://www.zbmath.org/1483.921472022-05-16T20:40:13.078697Z"Shi, Lei"https://www.zbmath.org/authors/?q=ai:shi.lei.3|shi.lei|shi.lei.1|shi.lei.2|shi.lei.4"Qi, Longxing"https://www.zbmath.org/authors/?q=ai:qi.longxingSummary: In this paper, the actual background of the susceptible population being directly patients after inhaling a certain amount of \(\mathrm{PM_{2.5}}\) is taken into account. The concentration response function of \(\mathrm{PM_{2.5}}\) is introduced, and the SISP respiratory disease model is proposed. Qualitative theoretical analysis proves that the existence, local stability and global stability of the equilibria are all related to the daily emission \(P_0\) of \(\mathrm{PM_{2.5}}\) and \(\mathrm{PM_{2.5}}\) pathogenic threshold \(K\). Based on the sensitivity factor analysis and time-varying sensitivity analysis of parameters on the number of patients, it is found that the conversion rate \(\beta\) and the inhalation rate \(\eta\) has the largest positive correlation. The cure rate \(\gamma\) of infected persons has the greatest negative correlation on the number of patients. The control strategy formulated by the analysis results of optimal control theory is as follows: The first step is to improve the clearance rate of \(\mathrm{PM_{2.5}}\) by reducing the \(\mathrm{PM_{2.5}}\) emissions and increasing the intensity of dust removal. Moreover, such removal work must be maintained for a long time. The second step is to improve the cure rate of patients by being treated in time. After that, people should be reminded to wear masks and go out less so as to reduce the conversion rate of susceptible people becoming patients.Numerical bifurcation analysis and pattern formation in a minimal reaction-diffusion model for vegetationhttps://www.zbmath.org/1483.921552022-05-16T20:40:13.078697Z"Kabir, M. Humayun"https://www.zbmath.org/authors/?q=ai:kabir.md-humayun"Gani, M. Osman"https://www.zbmath.org/authors/?q=ai:gani.m-osmanSummary: Model-aided understanding of the mechanism of vegetation patterns and desertification is one of the burning issues in the management of sustainable ecosystems. A pioneering model of vegetation patterns was proposed by \textit{C. A. Klausmeier} [``Regular and irregular patterns in semiarid vegetation'', Science 284, No. 5421, 1826--1828 (1999; \url{doi:10.1126/science.284.5421.1826})] that involves a downhill flow of water. In this paper, we study the diffusive Klausmeier model that can describe the flow of water in flat terrain incorporating a diffusive flow of water. It consists of a two-component reaction-diffusion system for water and plant biomass. The paper presents a numerical bifurcation analysis of stationary solutions of the diffusive Klausmeier model extensively. We numerically investigate the occurrence of diffusion-driven instability and how this depends on the parameters of the model. Finally, the model predicts some field observed vegetation patterns in a semiarid environment, e.g. spot, stripe (labyrinth), and gap patterns in the transitions from bare soil at low precipitation to homogeneous vegetation at high precipitation. Furthermore, we introduce a two-component reaction-diffusion model considering a bilinear interaction of plant and water instead of their cubic interaction. It is inspected that no diffusion-driven instability occurs as if vegetation patterns can be generated. This confirms that the diffusive Klausmeier model is the minimal reaction-diffusion model for the occurrence of vegetation patterns from the viewpoint of a two-component reaction-diffusion system.Dynamical study of fractional order mutualism parasitism food web modulehttps://www.zbmath.org/1483.921562022-05-16T20:40:13.078697Z"Khan, Aziz"https://www.zbmath.org/authors/?q=ai:khan.aziz"Abdeljawad, Thabet"https://www.zbmath.org/authors/?q=ai:abdeljawad.thabet"Gómez-Aguilar, J. F."https://www.zbmath.org/authors/?q=ai:gomez-aguilar.jose-francisco"Khan, Hasib"https://www.zbmath.org/authors/?q=ai:khan.hasibSummary: In literature, many researchers have examined food web modules for different aspects and kinds such as exploitative competition, energy flow web, apparent competition, source web, trophic cascades of food chains, functional web, paleoecological web and intraguild predation. These food webs have been analyzed for competition and predation, where as the module connected with mutualism and parasitism have attracted the attention of researchers. In this article, we study mutualism parasitism food web module (MPFWM) by replacing the ordinary derivative by Atangana-Baleanu (AB) fractional order (FO) derivative, which is a generalization of classical derivative. This new type of operators enables us to use the essential information of the variables in the nonlocal systems. Existence and uniqueness (EU) of solutions have been proved by employing fixed point theorem. Picard's stability approach is used for the stability analysis. Finally, numerical solutions of the ABC fractional order MPFWM were obtained for the particular parameter values.Modeling the effects of insecticides and external efforts on crop productionhttps://www.zbmath.org/1483.921642022-05-16T20:40:13.078697Z"Misra, A. K."https://www.zbmath.org/authors/?q=ai:misra.arun-k|misra.aruna-kumari|misra.abhishek-kumar|misra.arvind-kumar|misra.amit-kumar"Patel, Rahul"https://www.zbmath.org/authors/?q=ai:patel.rahul"Jha, Navnit"https://www.zbmath.org/authors/?q=ai:jha.navnitSummary: In this paper a nonlinear mathematical model is proposed and analyzed to understand the effects of insects, insecticides and external efforts on the agricultural crop productions. In the modeling process, we have assumed that crops grow logistically and decrease due to insects, which are wholly dependent on crops. Insecticides and external efforts are applied to control the insect population and enhance the crop production, respectively. The external efforts affect the intrinsic growth rate and carrying capacity of crop production. The feasibility of equilibria and their stability properties are discussed. We have identified the key parameters for the formulation of effective control strategies necessary to combat the insect population and increase the crop production using the approach of global sensitivity analysis. Numerical simulation is performed, which supports the analytical findings. It is shown that periodic oscillations arise through Hopf bifurcation as spraying rate of insecticides decreases. Our findings suggest that to gain the desired crop production, the rate of spraying and the quality of insecticides with proper use of external efforts are much important.New insight kinetic modeling: models above classical chemical mechanichttps://www.zbmath.org/1483.921682022-05-16T20:40:13.078697Z"Atangana, Ernestine"https://www.zbmath.org/authors/?q=ai:atangana.ernestineSummary: New trends of differential operators have been suggest very recently and have been proven to be accurate in modeling real world problems in many fields of science. We present in this paper some kinetic reaction model where the process does not follow the classical law of chemical mechanic. We present kinetic reactions where the processes follow the power law, exponential decay law and crossover behavior from exponential to power law. Analytical technique were used where the model is linear and new numerical where the model is non-linear. Results obtained here are very closer to reality than those models designed with classical chemical mechanic.Controllability of semilinear impulsive Atangana-Baleanu fractional differential equations with delayhttps://www.zbmath.org/1483.930352022-05-16T20:40:13.078697Z"Aimene, D."https://www.zbmath.org/authors/?q=ai:aimene.djihad"Baleanu, D."https://www.zbmath.org/authors/?q=ai:baleanu.dimitru|baleanu.dumitru-i"Seba, D."https://www.zbmath.org/authors/?q=ai:seba.djamilaSummary: We discuss the controllability of semilinear differential equations of fractional order with impulses and delay. We make use of the Atangana-Baleanu derivative. Our main tools are semigroup theory, the fixed point theorem due to Darbo and their combination with the properties of measures of noncompactness. Our abstract results are well supported by an illustrative example.Predictor-based super-twisting sliding mode observer for synchronisation of nonlinear chaotic systems with delayed measurementshttps://www.zbmath.org/1483.930572022-05-16T20:40:13.078697Z"Hamoudi, Ahcene"https://www.zbmath.org/authors/?q=ai:hamoudi.ahcene"Djeghali, Nadia"https://www.zbmath.org/authors/?q=ai:djeghali.nadia"Bettayeb, Maamar"https://www.zbmath.org/authors/?q=ai:bettayeb.maamarSummary: The most used configuration for chaos synchronisation is the drive-response or master-slave pattern, where the response of chaotic systems at the receiver side must track the drive chaotic trajectory at the emitter side. The synchronisation is achieved by sending, through the public channel, a suitable control signal delivered by the emitter on its output to the receiver. One of the major problems encountered in this configuration is transmission delay which can degrade the synchronisation. In this paper, we focus on the synchronisation problem of nonlinear chaotic systems in the presence of output transmission delay. In the new proposed method, the slave system is made up of a super-twisting sliding mode observer and a predictor arranged in cascade in order to compensate for the delayed transmission signal from the transmitter to the receiver. The observer estimates the delayed states and the predictor provides the estimated states at the current time. The convergence conditions of the proposed method are established. Numerical examples are given. The computer simulation results are provided to demonstrate the effectiveness of the proposed synchronisation approach.Design of functional interval observers for nonlinear fractional-order interconnected systemshttps://www.zbmath.org/1483.932182022-05-16T20:40:13.078697Z"Huong, Dinh Cong"https://www.zbmath.org/authors/?q=ai:huong.dinh-congThe systems are
\[ \begin{aligned}
D^{\alpha_j} x_i(t) &= A_{ii} x_i(t) + B_i u_i(t) + \sum_{j = 1, j \ne i}^N A_{ij}x_j(t) + f_i(t, x_i(t)) \\ y_i(t) &= C_i x_i(t) \qquad i = 1,2,\dots,N
\end{aligned} \tag{1} \]
where fractional derivatives are understood in the sense of Caputo,
\[
D^\alpha f(t) = \frac{1}{ \Gamma (m - \alpha)} \int_0^t \frac{f^{(m)}(\tau)}{(t - \tau)^{\alpha + 1 - m}} d\tau
\]
\(m\) an integer and \(m - 1 \le \alpha < m\). The \(x_j \in \mathbb{R}^{n_j}\) are the \textit{states,} \(u_j(t) \in\mathbb{R}^{m_j}\) the \textit{control inputs} and \(y_j(t) \in \mathbb{R}^{p_j}\) the \textit{outputs.} If we fix \(i\) then \(x_i(t), u_i(t), y_i(t)\) are the \textit{local} state, control input and output; for \(j \ne i\) \(x_j(t), u_j(t), y_j(t)\) are the \textit{remote} states. control inputs and outputs. A control system like (1) is \textit{observable} if the states \(x_j(t)\) can be reconstructed (or approximated) from the controls \(u_j(t)\) and outputs \(y_j(t)\). An \textit{observer} is a system that does the reconstruction.
The object of this paper is the design of separate observers for each of the states \(x_i(t)\). Each observer is \textit{stand alone} if it doesn't need information on the approximations to the remote states \(x_j(t)\), although it may use the remote inputs and outputs \(u_j(t)\) and \(y_i(t)\). In observer design, the basic requirement is that a bound for the approximation be given, that is a bound for \(\|x_i(t) - \hat x_i(t) \|\) where \(\hat x_i(t)\) is the approximate state constructed by the observer. A different way of bounding the error is to use \textit{vector interval bounds} of he form \(z^- \le z \le z^+\) where the inequalities are understood elementwise; this allows to bring into play order theorems for fractional ODE. The main result is the construction of an observer that provides an interval estimation
\[
z^-_i(t) \le z_i(t) \le z^+_i(t)
\]
for \(z_i(t) = Fx_i(t)\). The author verifies numerically the results in various examples.
Reviewer: Hector O. Fattorini (Los Angeles)Observer for differential inclusion systems with incremental quadratic constraintshttps://www.zbmath.org/1483.932492022-05-16T20:40:13.078697Z"Yang, Lin"https://www.zbmath.org/authors/?q=ai:yang.lin"Huang, Jun"https://www.zbmath.org/authors/?q=ai:huang.jun"Zhang, Min"https://www.zbmath.org/authors/?q=ai:zhang.min.5|zhang.min.1|zhang.min.3|zhang.min.4|zhang.min|zhang.min.2|zhang.min.6|zhang.min.7"Yang, Ming"https://www.zbmath.org/authors/?q=ai:yang.ming.2|yang.ming.1|yang.mingThe system is
\[
\begin{aligned} x'(t) &= Ax(t) + B\varphi(g(t)) - G\omega(t) \\
g(t) &= Fx(t) \qquad \omega(t) \in \psi(Hx(t)) \\
y(t) &= Cx(t) \end{aligned}\tag{1}
\]
where \(x(t) \in \mathbb{R}^n\) is the \textit{state}, \(y(t) \in\mathbb{R}^s\) is the \textit{output}, and \(\omega(t) \in\mathbb{R}^m\) is the \textit{control}. The function \(\varphi(g)\) is nonlinear and \(\psi(u)\) is a set function. Finally, \(A, B, C, F, G, H\) are matrices of appropriate dimensions. A control system like (1) is \textit{observable} if the state \(x(t)\) can be reconstructed (or approximated) from the control \(\omega(t)\) and output \(y(t),\) which in applications corresponds to information from sensors. The device doing the approximate reconstruction \(\hat x(t)\) of \(x(t)\) is called the \textit{observer}, in this case
\[
\begin{aligned}\hat x'(t) &= A \hat x(t) + B\varphi(\hat g) - G \hat \omega(t) + L(y(t) - C \hat x(t)) \\
\hat g(t) &= F \hat x(t) + J(y(t) - C \hat x(t)) \qquad \hat \omega(t) \in \psi( H \hat x(t) + K(y(t) - C \hat x(t)) \\
\hat y(t) &= C \hat x(t).\end{aligned} \tag{2}
\]
The observer is \textit{asymptotically accurate} if \(\| \hat x(t) - x(t)\| \to 0\) as \(t \to \infty.\) Under conditions that include quadratic growth of \(\varphi(g)\) and monotonicity of \(\psi(z)\) the authors derive the estimate
\[
\| \hat x(t) - x(t)\| \le \sigma \| \hat x(0) - x(0)\| e^{- \alpha t} \tag{3}
\]
where \(\sigma, \alpha > 0\) are explicit (not just generic) constants.Then they introduce an algorithm that leads to the exact computation of \(x(t)\) and \(\hat x(t)\) in some cases and test (3) in several applications: a rotor model, a chaotic system, and a model for VTOL aircraft. The paper includes a large list of references and the results are compared with existing theorems.
Reviewer: Hector O. Fattorini (Los Angeles)Synchronization patterns with strong memory adaptive control in networks of coupled neurons with chimera states dynamicshttps://www.zbmath.org/1483.933082022-05-16T20:40:13.078697Z"Vázquez-Guerrero, P."https://www.zbmath.org/authors/?q=ai:vazquez-guerrero.p"Gómez-Aguilar, J. F."https://www.zbmath.org/authors/?q=ai:gomez-aguilar.jose-francisco"Santamaria, F."https://www.zbmath.org/authors/?q=ai:santamaria.fidel"Escobar-Jiménez, R. F."https://www.zbmath.org/authors/?q=ai:escobar-jimenez.ricardo-fabricioSummary: This work presents the Hindmarsh-Rose fractional model of three-state using the Atangana-Baleanu-Caputo fractional derivative with strong memory. The model allows simulating the chimera states in a neural network. To achieve the synchronization was developed a fractional adaptive controller which is based on the uncertainty of the coupling parameters. The synchronization was studied using different fractional-orders and for 15, 40, 65 and 90 neurons. We consider fractional derivatives with nonlocal and non-singular Mittag-Leffler law. The simulations results show that the neurons synchronization is reached using the proposed method. We believe that the application of fractional operators to synchronization of chimera states open a new direction of research in the near future.Orbital stability analysis for perturbed nonlinear systems and natural entrainment via adaptive Andronov-Hopf oscillatorhttps://www.zbmath.org/1483.933292022-05-16T20:40:13.078697Z"Zhao, Jinxin"https://www.zbmath.org/authors/?q=ai:zhao.jinxin"Iwasaki, Tetsuya"https://www.zbmath.org/authors/?q=ai:iwasaki.tetsuyaEditorial remark: No review copy delivered.Stability analysis of nonlinear oscillator networks based on the mechanism of cascading failureshttps://www.zbmath.org/1483.934662022-05-16T20:40:13.078697Z"Huang, Yubo"https://www.zbmath.org/authors/?q=ai:huang.yubo"Dong, Hongli"https://www.zbmath.org/authors/?q=ai:dong.hongli"Zhang, Weidong"https://www.zbmath.org/authors/?q=ai:zhang.weidong"Lu, Junguo"https://www.zbmath.org/authors/?q=ai:lu.junguoSummary: Traditional methods for resisting cascading failures mainly focus on optimizing the topology of the network. Recent research indicates that the exceptional dynamics of the nodes, which caused by an intentional attack, can also induce cascading failures of nonlinear oscillator networks. This implies that the tolerance of the network to cascading failures can be enhanced by stabilizing the exceptional nodes after the attack. In this paper, we have constructed a framework to identify and control the exceptional nodes of the network to avoid further cascading failures. First, the node with the largest load is removed to simulate the situation that the network is intentionally attacked and that will cause the exceptional dynamics of other nodes. Then, the exceptional nodes which can induce cascading failures are identified by the small disturbance analysis method. Finally, the specific external control will be applied in the exceptional nodes respectively to stabilize them based on the phase differences analysis of their dynamic equations. Results show that the number of exceptional nodes in the Sci-Grid and the Spanish power grid can be reduced by more than 50\% via increasing the coupling strength and the unstable network can be stabilized by controlling the remaining exceptional nodes (usually no more than 10\% of the total).Stabilisation for cascade of nonlinear ODEs and counter-convecting transport dynamicshttps://www.zbmath.org/1483.934992022-05-16T20:40:13.078697Z"Cai, Xiushan"https://www.zbmath.org/authors/?q=ai:cai.xiushan"Wang, Dingchao"https://www.zbmath.org/authors/?q=ai:wang.dingchao"Liu, Yang"https://www.zbmath.org/authors/?q=ai:liu.yang.7|liu.yang.8"Zhan, Xisheng"https://www.zbmath.org/authors/?q=ai:zhan.xisheng"Yan, Huaicheng"https://www.zbmath.org/authors/?q=ai:yan.huaichengSummary: We consider stabilisation for a nonlinear ordinary differential equation (ODE) and counter-convecting transport partial differential equations (PDEs) cascaded system in which the transport coefficients depend on the ODE state. Stability analysis of the closed-loop system is based on the infinite-dimensional backstepping transformations and a Lyapunov functional. A predictor control is proposed such that the closed-loop system is globally asymptotically stable. The proposed design method is illustrated by a single-link manipulator.Exponential stabilization by delay feedback control for highly nonlinear hybrid stochastic functional differential equations with infinite delayhttps://www.zbmath.org/1483.935272022-05-16T20:40:13.078697Z"Mei, Chunhui"https://www.zbmath.org/authors/?q=ai:mei.chunhui"Fei, Chen"https://www.zbmath.org/authors/?q=ai:fei.chen"Fei, Weiyin"https://www.zbmath.org/authors/?q=ai:fei.weiyin"Mao, Xuerong"https://www.zbmath.org/authors/?q=ai:mao.xuerongAuthors' abstract: Given an unstable hybrid stochastic functional differential equation, how to design a delay feedback controller to make it stable? Some results have been obtained for hybrid systems with finite delay. However, the state of many stochastic differential equations are related to the whole history of the system, so it is necessary to discuss the feedback control of stochastic functional differential equations with infinite delay. On the other hand, in many practical stochastic models, the coefficients of these systems do not satisfy the linear growth condition, but are highly nonlinear. In this paper, the delay feedback controls are designed for a class of infinite delay stochastic systems with highly nonlinear and the influence of switching state.
Reviewer: Cristina Pignotti (L'Aquila)Fixed-time synchronization of memristor chaotic systems via a new extended high-gain observerhttps://www.zbmath.org/1483.935422022-05-16T20:40:13.078697Z"Al-Saggaf, Ubaid Mohsen"https://www.zbmath.org/authors/?q=ai:al-saggaf.ubaid-mohsen"Bettayeb, Maamar"https://www.zbmath.org/authors/?q=ai:bettayeb.maamar"Djennoune, Said"https://www.zbmath.org/authors/?q=ai:djennoune.saidSummary: Memristor based chaotic oscillators are often chosen for secure communication owing to their interesting feature. In chaos-based secure communication applications, synchronization is a central issue. Most of synchronization methods proposed in the literature are asymptotic. In practice, it is desirable that synchronization be established in a user predefined time. This paper provides new developments in the design of filtered extended high-gain observer dedicated for prescribed-time synchronization of memristor chaotic systems used in a master-slave based secure communication process subject to channel noise. The proposed prescribed-time extended filtered-high gain observer is constructed on the basis of a time-dependent coordinates transformation based on modulating functions which annihilate the effect of initial conditions on the synchronization time. Simulations performed on a numerical example illustrate the efficiency of the proposed approach.Finite-time projective synchronization of fractional-order complex-valued memristor-based neural networks with delayhttps://www.zbmath.org/1483.935672022-05-16T20:40:13.078697Z"Zhang, Yanlin"https://www.zbmath.org/authors/?q=ai:zhang.yanlin"Deng, Shengfu"https://www.zbmath.org/authors/?q=ai:deng.shengfuSummary: This paper studies the finite-time projective synchronization of fractional-order complex-valued memristor-based neural networks (FCVMNNs) with delay. By applying the set-valued map, the differential inclusion theory and Gronwall-Bellman integral inequalities, some sufficient criteria are established to achieve the finite time projective synchronization of the FCVMNNs. The upper bound of the settling time for synchronization is also estimated. Moreover, two numerical examples are designed to verify the correctness and effectiveness of the obtained theoretical results.Capturing and shunting energy in chaotic Chua circuithttps://www.zbmath.org/1483.940772022-05-16T20:40:13.078697Z"Wang, Chunni"https://www.zbmath.org/authors/?q=ai:wang.chunni"Liu, Zhilong"https://www.zbmath.org/authors/?q=ai:liu.zhilong"Hobiny, Aatef"https://www.zbmath.org/authors/?q=ai:hobiny.aatef-d"Xu, Wenkang"https://www.zbmath.org/authors/?q=ai:xu.wenkang"Ma, Jun"https://www.zbmath.org/authors/?q=ai:ma.jun.1|ma.junSummary: Nonlinear electric devices are critical for building chaotic circuits and the outputs voltage from the capacitor are often detected for further analyzing the dynamics of the nonlinear circuits. Continuous exchange between the electric field energy in the capacitor and magnetic field energy in the induction coil is effective to keep continuous oscillation in the circuit. That is, energy encoding and transmission can regulate the dynamical behaviors in chaotic circuits. In this paper, a branch circuit, which is built by using a capacitor and induction coil, is paralleled with one output end of chaotic Chua circuit, and this external branch circuit is activated to control chaos by pumping energy from Chua circuit and capturing external electromagnetic radiation. From dynamical control view, it explains the mechanism for differential control via capacitor and integral control via induction coil. While in the view of energy encoding, the control branch of circuit built by using a capacitor connected with induction coil in series can capture some external field energy and thus the nonlinear behaviors are controlled by generating equivalent current in the branch of control circuit. The circuit equation and also the dimensionless dynamical system under energy control are obtained, and numerical studies are supplied to confirm the control mechanism from physical view. In the end, it gives possible suggestions to enhance the control effectiveness in experimental way.A new approach to fuzzy sets: application to the design of nonlinear time series, symmetry-breaking patterns, and non-sinusoidal limit-cycle oscillationshttps://www.zbmath.org/1483.940782022-05-16T20:40:13.078697Z"García-Morales, Vladimir"https://www.zbmath.org/authors/?q=ai:garcia-morales.vladimirSummary: It is shown that characteristic functions of sets can be made fuzzy by means of the \(\mathcal{B}_\kappa\)-function, recently introduced by the author, where the fuzziness parameter \(\kappa\in\mathbb{R}\) controls how much a fuzzy set deviates from the crisp set obtained in the limit \(\kappa\rightarrow 0\). As applications, we present first a general expression for a switching function that may be of interest in electrical engineering and in the design of nonlinear time series. We then introduce another general expression that allows wallpaper and frieze patterns for every possible planar symmetry group (besides patterns typical of quasicrystals) to be designed. We show how the fuzziness parameter \(\kappa\) plays an analogous role to temperature in physical applications and may be used to break the symmetry of spatial patterns. As a further, important application, we establish a theorem on the shaping of limit cycle oscillations far from bifurcations in smooth deterministic nonlinear dynamical systems governed by differential equations. Following this application, we briefly discuss a generalization of the Stuart-Landau equation to non-sinusoidal oscillators.