Recent zbMATH articles in MSC 34https://www.zbmath.org/atom/cc/342021-04-16T16:22:00+00:00WerkzeugAn averaging principle for stochastic switched systems with Lévy noise.https://www.zbmath.org/1456.370522021-04-16T16:22:00+00:00"Ma, Shuo"https://www.zbmath.org/authors/?q=ai:ma.shuo"Kang, Yanmei"https://www.zbmath.org/authors/?q=ai:kang.yanmeiSummary: In this paper, we present an averaging method for stochastic switched systems with Lévy noise under non-Lipschitz condition. With the help of successive approximation method and Bihari's inequality, the existence and uniqueness of the solutions of original and averaged systems are proved. Then, under suitable assumptions, we show that the solution of stochastic switched system with Lévy noise strongly converges to the solution of the corresponding averaged equation.Kink solitary solutions to a hepatitis C evolution model.https://www.zbmath.org/1456.340562021-04-16T16:22:00+00:00"Telksnys, Tadas"https://www.zbmath.org/authors/?q=ai:telksnys.tadas"Navickas, Zenonas"https://www.zbmath.org/authors/?q=ai:navickas.zenonas"Sanjuán, Miguel A. F."https://www.zbmath.org/authors/?q=ai:sanjuan.miguel-a-f"Marcinkevicius, Romas"https://www.zbmath.org/authors/?q=ai:marcinkevicius.romas"Ragulskis, Minvydas"https://www.zbmath.org/authors/?q=ai:ragulskis.minvydasSummary: The standard nonlinear hepatitis C evolution model described in (Reluga et al. 2009) is considered in this paper. The generalized differential operator technique is used to construct analytical kink solitary solutions to the governing equations coupled with multiplicative and diffusive terms. Conditions for the existence of kink solitary solutions are derived. It appears that kink solitary solutions are either in a linear or in a hyperbolic relationship. Thus, a large perturbation in the population of hepatitis infected cells does not necessarily lead to a large change in uninfected cells. Computational experiments are used to illustrate the evolution of transient solitary solutions in the hepatitis C model.Existence of homoclinic orbits for a singular differential equation involving \(p\)-Laplacian.https://www.zbmath.org/1456.340452021-04-16T16:22:00+00:00"Yin, Honghui"https://www.zbmath.org/authors/?q=ai:yin.honghui"Du, Bo"https://www.zbmath.org/authors/?q=ai:du.bo"Yang, Qing"https://www.zbmath.org/authors/?q=ai:yang.qing"Duan, Feng"https://www.zbmath.org/authors/?q=ai:duan.fengIn the present manuscript, the authors are concerned with the existence of {homoclinic} solutions for the following singular ODE
\[
\Big(\Phi_p\big(x'(t)\big)\Big)'+f\big(x'(t)\big) + g\big(x(t)\big) + \frac{h(t)}{1-x(t)} = e(t), \tag{1}
\]
where $\Phi_p(s) = |s|^{p-2}s$ (for some $p > 1$), $f,g,h,e\in C(\mathbb{R};\mathbb{R})$ and, moreover, $h$ is a strictly positive $T$-periodic function.
As usual, a \textit{homoclinic solution} of (1) is a solution $x\in C(\mathbb{R};\mathbb{R})$ satisfying
\[
\text{$x(t)\to\infty$ as $|t|\to\infty$}.
\]
Due to their relevance in several contexts, homoclinic solutions for general differential systems have been studied by many authors and with different techniques (variational methods, critical-point theory, method of lower/upper solutions and fixed-point theorems, etc.); however, since equation (1) is strongly nonlinear, these traditional techniques are no-longer applicable.
Using a new continuation theorem due to Manásevich and Mawhin, the authors obtain the following theorem, which is the main result of the paper.
Theorem 1.
Assume that the following assumptions are satisfied:
\begin{itemize}
\item[{(H.1)}] $f:\mathbb{R}\to\mathbb{R}$ is continuous, bounded and non-negative;
\item[{(H.2)}] $g:\mathbb{R}\to\mathbb{R}$ is strictly monotone increasing and there are positive constants $\sigma$ and $n$ such that
\[
xg(x)\geq \sigma|x|^{n+1}\quad\text{for all $x\in\mathbb{R}$};
\]
\item[{(H.3)}] $\rho_1 := \sup_{t\in\mathbb{R}}|e(t)| < \infty$ and
\[
\rho_2 := \int_{\mathbb{R}}|e(t)|^{1+1/n}\,\mathrm{d} t < \infty.
\]
\end{itemize}
Then, if $\rho_1 > f(0)$ and $h_l/\rho_1 - f(0) < 1$ (with $h_l := \min_{t\in\mathbb{R}}h(t)$), there exists at least one positive homoclinic solution $\omega_0$, further satisfying
\[
|\omega_0'(t)|\to 0\quad\text{as $|t|\to\infty$}.
\]
Thought it is based on the continuation theorem by Manásevich and Mawhin, the proof of Theorem 1 is sophisticated and it requires some preliminary lemmas of independent interest. On the other hand, a couple of examples at the end of the paper show the wide range of applicability of this result.
Reviewer: Stefano Biagi (Milano)Classification of bifurcation diagrams in coupled phase-oscillator models with asymmetric natural frequency distributions.https://www.zbmath.org/1456.340362021-04-16T16:22:00+00:00"Yoneda, Ryosuke"https://www.zbmath.org/authors/?q=ai:yoneda.ryosuke"Yamaguchi, Yoshiyuki Y."https://www.zbmath.org/authors/?q=ai:yamaguchi.yoshiyuki-yEnergy dissipation in Hamiltonian chains of rotators.https://www.zbmath.org/1456.370652021-04-16T16:22:00+00:00"Cuneo, Noé"https://www.zbmath.org/authors/?q=ai:cuneo.noe"Eckmann, Jean-Pierre"https://www.zbmath.org/authors/?q=ai:eckmann.jean-pierre"Wayne, C. Eugene"https://www.zbmath.org/authors/?q=ai:wayne.c-eugeneStochastic resonance in a fractional oscillator with random damping strength and random spring stiffness.https://www.zbmath.org/1456.700412021-04-16T16:22:00+00:00"He, Guitian"https://www.zbmath.org/authors/?q=ai:he.guitian"Tian, Yan"https://www.zbmath.org/authors/?q=ai:tian.yan"Wang, Yan"https://www.zbmath.org/authors/?q=ai:wang.yan.6|wang.yan|wang.yan.2|wang.yan.1|wang.yan.5|wang.yan.4|wang.yan.3Multi-condition of stability for nonlinear stochastic non-autonomous delay differential equation.https://www.zbmath.org/1456.340782021-04-16T16:22:00+00:00"Shaikhet, Leonid"https://www.zbmath.org/authors/?q=ai:shaikhet.leonid-eSummary: A nonlinear stochastic differential equation with the order of nonlinearity higher than one, with several discrete and distributed delays and time varying coefficients is considered. It is shown that the sufficient conditions for exponential mean square stability of the linear part of the considered nonlinear equation also are sufficient conditions for stability in probability of the initial nonlinear equation. Some new sufficient condition of stability in probability for the zero solution of the considered nonlinear non-autonomous stochastic differential equation is obtained which can be considered as a multi-condition of stability because it allows to get for one considered equation at once several different complementary of each other sufficient stability conditions. The obtained results are illustrated with numerical simulations and figures.Increased order generalized combination synchronization of non-identical dimensional fractional-order systems by introducing different observable variable functions.https://www.zbmath.org/1456.340622021-04-16T16:22:00+00:00"Kaouache, S."https://www.zbmath.org/authors/?q=ai:kaouache.smail"Hamri, N. E."https://www.zbmath.org/authors/?q=ai:hamri.n-e"Hacinliyan, A. S."https://www.zbmath.org/authors/?q=ai:hacinliyan.avadis-simon"Kandiran, E."https://www.zbmath.org/authors/?q=ai:kandiran.e"Deruni, B."https://www.zbmath.org/authors/?q=ai:deruni.b"Keles, A. C."https://www.zbmath.org/authors/?q=ai:keles.a-cSummary: An increased order generalized combination synchronization (IOGCS) of non-identical dimensional fractional-order systems with suitable different observable variable functions is proposed and analyzed in this paper. This synchronization scheme is applied for the combination of two fractional-order unified drive systems and the fractional-order Liu response system. In view of the stability property of linear fractional-order systems, an effective nonlinear control scheme is designed to achieve the desired synchronization. Theoretical analysis and numerical simulations are shown to demonstrate the effectiveness of the proposed method.Quantum entanglement in coupled harmonic oscillator systems: from micro to macro.https://www.zbmath.org/1456.810732021-04-16T16:22:00+00:00"Kao, Jhih-Yuan"https://www.zbmath.org/authors/?q=ai:kao.jhih-yuan"Chou, Chung-Hsien"https://www.zbmath.org/authors/?q=ai:chou.chung-hsienStability of antiperiodic recurrent neural networks with multiproportional delays.https://www.zbmath.org/1456.340712021-04-16T16:22:00+00:00"Huang, Chuangxia"https://www.zbmath.org/authors/?q=ai:huang.chuangxia"Long, Xin"https://www.zbmath.org/authors/?q=ai:long.xin"Cao, Jinde"https://www.zbmath.org/authors/?q=ai:cao.jindeSummary: In general, a proportional function is obviously not antiperiodic, yet a very interesting fact in this paper shows that it is possible there is an antiperiodic solution for some proportional delayed dynamical systems. We deal with the issue of antiperiodic solutions for RNNs (recurrent neural networks) incorporating multiproportional delays. Employing Lyapunov method, inequality techniques and concise mathematical analysis proof, sufficient criteria on the existence of antiperiodic solutions including its uniqueness and exponential stability are built up. The obtained results provide us some lights for designing a stable RNNs and complement some earlier publications. In addition, simulations show that the theoretical antiperiodic dynamics are in excellent agreement with the numerically observed behavior.Boundedness of oscillation and variation of semigroups associated with Bessel Schrödinger operators.https://www.zbmath.org/1456.420272021-04-16T16:22:00+00:00"Betancor, Jorge J."https://www.zbmath.org/authors/?q=ai:betancor.jorge-j"Hu, Wenting"https://www.zbmath.org/authors/?q=ai:hu.wenting"Wu, Huoxiong"https://www.zbmath.org/authors/?q=ai:wu.huoxiong"Yang, Dongyong"https://www.zbmath.org/authors/?q=ai:yang.dongyongSummary: Let \(\lambda \in (-\frac{1}{2}, \infty)\) and \(S_\lambda := - \frac{d^2}{dx^2} + \frac{\lambda^2 - \lambda}{x^2}\) be the Bessel Schrödinger operator on \(\mathbb{R}_+ := (0, \infty)\). The authors obtain the sharp power-weighted \(L^p\), weak type and restricted weak type inequalities for the oscillation operator \(\mathcal{O}_{\{ t_i \}_{i \in \mathbb{N}}} (\{ t^m \partial_t^m \mathcal{W}_t^\lambda \}_{t > 0}, \cdot)\) and the variation operator \(\mathcal{V}_\rho (\{ t^m \partial_t^m \mathcal{W}_t^\lambda \}_{t > 0}, \cdot)\) of the heat semigroup \(\{\mathcal{W}_t^\lambda \}_{t > 0}\) associated with \(S_\lambda\), where \(\rho \in (2, \infty)\) and \(m \in \mathbb{Z}_+ := \mathbb{N} \cup \{ 0 \}\). Moreover, for \(\lambda \in (0, \infty)\), the boundedness of \(\mathcal{O}_{\{ t_i \}_{i \in \mathbb{N}}} ( \{ t^m \partial_t^m \mathcal{W}_t^\lambda \}_{t > 0} , \cdot )\) and \(\mathcal{V}_\rho ( \{ t^m \partial_t^m \mathcal{W}_t^\lambda \}_{t > 0}, \cdot)\) from the Hardy space \(H^p ( \mathbb{R}_+ )\) into \(L^p ( \mathbb{R}_+ )\) with \(p \in ( \frac{1}{2}, 1]\) and on the Campanato type spaces BMO\({}^\alpha (\mathbb{R}_+)\) with \(\alpha \in [0, 1) \cap (0, \lambda)\) are obtained.Logarithmic norm-based analysis of robust asymptotic stability of nonlinear dynamical systems.https://www.zbmath.org/1456.340652021-04-16T16:22:00+00:00"Vrabel, Robert"https://www.zbmath.org/authors/?q=ai:vrabel.robertThe article provides sufficient conditions for the robust (local and global) asymptotic stability of a semilinear differential equation in \(\mathbb{R}^n\). The conditions are given in terms of some integral estimates involving both the time-varying linear and nonlinear parts of the unperturbed equation and the time-varying nonlinear perturbation.
The contribution of the linear part is expressed in terms of the integral of the logarithmic norm of the time-varying matrix representing the linear dynamics. The construction of the logarithmic norm is recalled, and a discussion is provided on its dependence on the choice of a norm in \(\mathbb{R}^n\) (and its induced norm in the space of matrices). The combination of the different contributions is estimated using the variation of constants formula.
Several examples illustrate the effectiveness of the proposed sufficient conditions for robust asymptotic stability.
Reviewer: Mario Sigalotti (Paris)Collective behavior and stochastic resonance in a linear underdamped coupled system with multiplicative dichotomous noise and periodical driving.https://www.zbmath.org/1456.340672021-04-16T16:22:00+00:00"Li, Pengfei"https://www.zbmath.org/authors/?q=ai:li.pengfei"Ren, Ruibin"https://www.zbmath.org/authors/?q=ai:ren.ruibin"Fan, Zening"https://www.zbmath.org/authors/?q=ai:fan.zening"Luo, Maokang"https://www.zbmath.org/authors/?q=ai:luo.maokang"Deng, Ke"https://www.zbmath.org/authors/?q=ai:deng.keHomoclinic solutions for a class of nonlinear fourth order \(p\)-Laplacian differential equations.https://www.zbmath.org/1456.340442021-04-16T16:22:00+00:00"Dimitrov, Nikolay D."https://www.zbmath.org/authors/?q=ai:dimitrov.nikolay-d"Tersian, Stepan A."https://www.zbmath.org/authors/?q=ai:tersian.stepan-agopIn this paper, the authors deal with a class of fourth-order differential equation involving \(p\)-Laplacian
\[
|u''(x)|^{p-2}u''(x))''+\omega (|u'(x)|^{p-2}u'(x))'+\lambda V(x) |u(x)|^{p-2} u(x)=f(x,u(x)
\]
where \(\omega\) is a constant, \(\lambda\) is a parameter and \(f\in C (\mathbb{R},\mathbb{R})\). with the aid of critical point theory and variational methods, they prove that under suitable growth conditions, the above equation possesses at least one nontrivial homoclinic solution, i.e., a nontrivial solution satisfying \(u(x)\longrightarrow 0\) as \(x\longrightarrow \mp\infty\).
Reviewer: Mohsen Timoumi (Monastir)Generalized Yang's conjecture on the periodicity of entire functions.https://www.zbmath.org/1456.300522021-04-16T16:22:00+00:00"Liu, Kai"https://www.zbmath.org/authors/?q=ai:liu.kai.1|liu.kai.2|liu.kai|liu.kai.5|liu.kai.4|liu.kai.3"Wei, Yuming"https://www.zbmath.org/authors/?q=ai:wei.yuming"Yu, Peiyong"https://www.zbmath.org/authors/?q=ai:yu.peiyongSummary: On the periodicity of transcendental entire functions, Yang's Conjecture is proposed in [\textit{P. Li} et al., Houston J. Math. 45, No. 2, 431--437 (2019; Zbl 1428.30030); \textit{Q. Wang} and \textit{P. Hu}, Acta Math. Sci., Ser. A, Chin. Ed. 38, No. 2, 209--214 (2018; Zbl 1413.30118)]. In the paper, we mainly consider and obtain partial results on a general version of Yang's Conjecture, namely, if \(f(z)^nf^{(k)}(z)\) is a periodic function, then \(f(z)\) is also a periodic function. We also prove that if \(f(z)^n+f^{(k)}(z)\) is a periodic function with additional assumptions, then \(f(z)\) is also a periodic function, where \(n,k\) are positive integers.An introduction to the mathematical theory of inverse problems. 3rd updated edition.https://www.zbmath.org/1456.350012021-04-16T16:22:00+00:00"Kirsch, Andreas"https://www.zbmath.org/authors/?q=ai:kirsch.andreasPublisher's description: This graduate-level textbook introduces the reader to the area of inverse problems, vital to many fields including geophysical exploration, system identification, nondestructive testing, and ultrasonic tomography. It aims to expose the basic notions and difficulties encountered with ill-posed problems, analyzing basic properties of regularization methods for ill-posed problems via several simple analytical and numerical examples. The book also presents three special nonlinear inverse problems in detail: the inverse spectral problem, the inverse problem of electrical impedance tomography (EIT), and the inverse scattering problem. The corresponding direct problems are studied with respect to existence, uniqueness, and continuous dependence on parameters. Ultimately, the text discusses theoretical results as well as numerical procedures for the inverse problems, including many exercises and illustrations to complement coursework in mathematics and engineering.
This updated text includes a new chapter on the theory of nonlinear inverse problems in response to the field's growing popularity, as well as a new section on the interior transmission eigenvalue problem which complements the Sturm-Liouville problem and which has received great attention since the previous edition was published.
See the review of the first edition in [Zbl 0865.35004]. For the second edition see [Zbl 1213.35004].Uniform convergence of Fourier series expansions for a fourth-order spectral problem with boundary conditions depending on the eigenparameter.https://www.zbmath.org/1456.340182021-04-16T16:22:00+00:00"Namazov, Faiq Mirzali"https://www.zbmath.org/authors/?q=ai:namazov.faiq-mirzaliSummary: In this paper, we consider the eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter contained in two of boundary conditions. We obtain refined asymptotic formulas for eigenvalues and eigenfunctions, and study uniform convergence of Fourier series expansions of continuous functions in the system of eigenfunctions of this problem.Evolution of autocatalytic sets in a competitive percolation model.https://www.zbmath.org/1456.921592021-04-16T16:22:00+00:00"Zhang, Renquan"https://www.zbmath.org/authors/?q=ai:zhang.renquan"Pei, Sen"https://www.zbmath.org/authors/?q=ai:pei.sen"Wei, Wei"https://www.zbmath.org/authors/?q=ai:wei.wei.5|wei.wei.7|wei.wei.2|wei.wei.3|wei.wei.6|wei.wei.4"Zheng, Zhiming"https://www.zbmath.org/authors/?q=ai:zheng.zhimingA reliable algorithm to check the accuracy of iterative schemes for solving nonlinear equations: an application of the CESTAC method.https://www.zbmath.org/1456.650322021-04-16T16:22:00+00:00"Fariborzi Araghi, Mohammad Ali"https://www.zbmath.org/authors/?q=ai:fariborzi-araghi.mohammad-aliSummary: The aim of this study is to apply the discrete stochastic arithmetic (DSA) to validate the class of muli-step iterative methods and find the optimal numerical solution of nonlinear equations. To this end, the Controle et Estimation Stochastique des Arrondis de Calculs (CESTAC) method and the Control of Accuracy and Debugging for Numerical Applications (CADNA) library are applied. By using this approach, the optimal number of iteration and the optimal solution with its accuracy are found. In this case, the usual stopping termination in the iterative procedure is replaced by a new criterion which is independent of the given tolerance \((\epsilon)\) such that the optimal results are evaluated computationally. A main theorem is proved which shows the accuracy of the iterative schemes by means of the concept of common significant digits. The numerical results are presented to illustrate the efficiency and importance of using the DSA in place of the floating-point arithmetic (FPA).On traveling waves in compressible Euler equations with thermal conductivity.https://www.zbmath.org/1456.350702021-04-16T16:22:00+00:00"Thanh, Mai Duc"https://www.zbmath.org/authors/?q=ai:mai-duc-thanh."Vinh, Duong Xuan"https://www.zbmath.org/authors/?q=ai:vinh.duong-xuanSummary: Heat conduction plays an important role in fluid dynamics. However, the modeling of thermal conductivity involves higher order derivatives which causes a tough obstacle for the study of traveling waves. In this work, we propose a modified term for the thermal conductivity coefficient in viscous-capillary compressible Euler equations. By approximation, which is crucial in any mathematical modeling, the heat conduction may be assumed to depend only on the specific volume. Then, we can derive a \(2\times 2\) system of first-order differential equations for traveling waves of the given model, whose equilibria can be shown to admit a stable-saddle connection for 1-shocks and a saddle-stable connection for 3-shocks. This establishes the existence of a traveling wave of the viscous-capillary Euler equations with the presence of a modified thermal conductivity effect.Efficient reduction for diagnosing Hopf bifurcation in delay differential systems: applications to cloud-rain models.https://www.zbmath.org/1456.340722021-04-16T16:22:00+00:00"Chekroun, Mickaël D."https://www.zbmath.org/authors/?q=ai:chekroun.mickael-d"Koren, Ilan"https://www.zbmath.org/authors/?q=ai:koren.ilan"Liu, Honghu"https://www.zbmath.org/authors/?q=ai:liu.honghuThe paper is concerned with the analysis of Hopf bifurcations and their characterisations (as super-critical or sub-critical) for systems of delay differential equations.
The authors present the Galerkin-Koornwinder (GK) approximation which leads to simple analytical conditions which enables the classification of Hopf bifurcations in solutions of delay differential systems. Essentially, the approach proposes the application of the centre manifold theory to the ODE resulting from the application of the GK approximation to the DDE. A Lyapunov coefficient can then be used to determine whether the bifurcation is super-critical or sub-critical. Applications to cloud-rain models (delay models of Koren and Feingold type) are discussed and new dynamical behaviors are shown for these models, such as double-Hopf bifurcations.
Reviewer: Neville Ford (Chester)Admissible Banach function spaces and nonuniform stabilities.https://www.zbmath.org/1456.340692021-04-16T16:22:00+00:00"Lupa, Nicolae"https://www.zbmath.org/authors/?q=ai:lupa.nicolae"Popescu, Liviu Horia"https://www.zbmath.org/authors/?q=ai:popescu.liviu-horiaThe paper is concerned with the nonuniform stability and nonuniform exponential stability for a nonuniform exponentially bounded evolution family \(\mathcal{U}\) of bounded linear operators on a Banach space. Equivalent conditions for nonuniform stability of \(\mathcal{U}\) and a sufficient condition for nonuniform exponential stability of \(\mathcal{U}\) are given.
Reviewer: Jin Liang (Shanghai)Sustained rotation in a vibrated disk with asymmetric supports.https://www.zbmath.org/1456.700312021-04-16T16:22:00+00:00"Peraza-Mues, Gonzalo G."https://www.zbmath.org/authors/?q=ai:peraza-mues.gonzalo-g"Moukarzel, Cristian F."https://www.zbmath.org/authors/?q=ai:moukarzel.cristian-fFractional differential equations involving Hadamard fractional derivatives with nonlocal multi-point boundary conditions.https://www.zbmath.org/1456.340072021-04-16T16:22:00+00:00"Subramanian, Muthaiah"https://www.zbmath.org/authors/?q=ai:subramanian.muthaiah"Manigandan, Murugesan"https://www.zbmath.org/authors/?q=ai:manigandan.murugesan"Gopal, Thangaraj Nandha"https://www.zbmath.org/authors/?q=ai:gopal.thangaraj-nandhaSummary: In this paper, we investigate the existence and uniqueness of solutions for the Hadamard fractional boundary value problems with nonlocal multipoint boundary conditions. By using Leray-Schauder nonlinear alternative, Leray Schauder degree theory, Krasnoselskii fixed point theorem, Schaefer fixed point theorem, Banach fixed point theorem, Nonlinear Contractions, the existence and uniqueness of solutions are obtained. As an application, two examples are given to demonstrate our results.Variable order fractional permanent magnet synchronous motor: dynamical analysis and numerical simulation.https://www.zbmath.org/1456.340582021-04-16T16:22:00+00:00"Zahra, Waheed K."https://www.zbmath.org/authors/?q=ai:zahra.waheed-k"Hikal, M. M."https://www.zbmath.org/authors/?q=ai:hikal.m-mSummary: In this paper, the variable order fractional permanent magnet synchronous motor (VOFPMSM) is investigated. Conditions for existence and uniqueness of the solution of the VOFPMSM are proposed. The stability behavior of the system's equilibrium points along with the variation of the motor parameters and the order of differentiation is discussed. Sufficient conditions that guarantee the asymptotic stability of each of the equilibrium points of the system are established. Also, the required conditions that give the effect of Hopf bifurcation of the system are established in terms of the system parameters and the order of differentiation and consequently the appearance of the chaotic behavior of the VOFPMSM. New numerical techniques based on the modified backward Euler's schemes for continuous and discontinuous variable order fractional model are presented. The obtained numerical results demonstrate the merits of the proposed method and the variable order fractional permanent magnet synchronous motor over the fractional permanent magnet synchronous motor.Solutions for a singular Hadamard-type fractional differential equation by the spectral construct analysis.https://www.zbmath.org/1456.340082021-04-16T16:22:00+00:00"Zhang, Xinguang"https://www.zbmath.org/authors/?q=ai:zhang.xinguang"Yu, Lixin"https://www.zbmath.org/authors/?q=ai:yu.lixin"Jiang, Jiqiang"https://www.zbmath.org/authors/?q=ai:jiang.jiqiang"Wu, Yonghong"https://www.zbmath.org/authors/?q=ai:wu.yonghong.1"Cui, Yujun"https://www.zbmath.org/authors/?q=ai:cui.yujunSummary: In this paper, we consider the existence of positive solutions for a Hadamard-type fractional differential equation with singular nonlinearity. By using the spectral construct analysis for the corresponding linear operator and calculating the fixed point index of the nonlinear operator, the criteria of the existence of positive solutions for equation considered are established. The interesting point is that the nonlinear term possesses singularity at the time and space variables.Basins of attraction for chimera states.https://www.zbmath.org/1456.340342021-04-16T16:22:00+00:00"Martens, Erik A."https://www.zbmath.org/authors/?q=ai:martens.erik-a"Panaggio, Mark J."https://www.zbmath.org/authors/?q=ai:panaggio.mark-j"Abrams, Daniel M."https://www.zbmath.org/authors/?q=ai:abrams.daniel-mMathematical modeling of the immune system response to pathogens.https://www.zbmath.org/1456.340542021-04-16T16:22:00+00:00"Solis, Francisco J."https://www.zbmath.org/authors/?q=ai:solis.francisco-javier"Azofeifa, Danalie"https://www.zbmath.org/authors/?q=ai:azofeifa.danalieSummary: The detection and elimination of pathogens in an organism are the main tasks of its immune system. The most important cells involved in these processes are neutrophils and macrophages. These processes might have two resolutions: The first is the possibility of pathogen elimination, and the other the possibility of the inflammation resolution. In this work, we present several mathematical models involving immune cell densities and inflammation levels. Our general goal is to exhibit the possible pathogen eradication or the inflammation resolution. We use bifurcation techniques in order to analyze how parameter variations may change the system evolution. Our results indicate that the elimination of apoptotic neutrophils by macrophages has a dichotomy effect: It contributes to the decrease of the inflammation level, but it may hinder the pathogen elimination. Also, an increment of the average neutrophil life can improve healthy outcomes. Moreover, we find scenarios when pathogens cannot be eliminated, as well as conditions for their successful eradication.Noise-induced transitions in optomechanical synchronization.https://www.zbmath.org/1456.920192021-04-16T16:22:00+00:00"Weiss, Talitha"https://www.zbmath.org/authors/?q=ai:weiss.talitha"Kronwald, Andreas"https://www.zbmath.org/authors/?q=ai:kronwald.andreas"Marquardt, Florian"https://www.zbmath.org/authors/?q=ai:marquardt.florianSome theorems of existence of solutions for fractional hybrid \(q\)-difference inclusion.https://www.zbmath.org/1456.340062021-04-16T16:22:00+00:00"Samei, Mohammad Esmael"https://www.zbmath.org/authors/?q=ai:samei.mohammad-esmael"Ranjbar, Ghorban Khalilzadeh"https://www.zbmath.org/authors/?q=ai:ranjbar.ghorban-khalilzadehSummary: The purpose of this study is to obtain the existence of solutions for the fractional hybrid \(q\)-differential inclusions with the boundary conditions. Besides, we give the solution set of these \(q\)-differential inclusions with boundary values. Also, we investigate dimension of the results set for second fractional \(q\)-differential inclusions. Lastly, we present an example to elaborate our results and present the obtained results of close to mathematical calculations.Implicit differential inclusions with acyclic right-hand sides: an essential fixed points approach.https://www.zbmath.org/1456.340152021-04-16T16:22:00+00:00"Andres, Jan"https://www.zbmath.org/authors/?q=ai:andres.jan"Górniewicz, Lech"https://www.zbmath.org/authors/?q=ai:gorniewicz.lechSummary: Effective criteria are given for the solvability of initial as well as boundary valueproblems to implicit ordinary differential inclusions whose right-hand sides are governed by compactacyclic maps. Cauchy and periodic implicit problems are also considered on proximate retracts. Ournew approach is based on the application of the topological essential fixed point theory. Implicitproblems for partial differential inclusions are only indicated.A computationally efficient iterative scheme for solving fourth-order boundary value problems.https://www.zbmath.org/1456.650532021-04-16T16:22:00+00:00"Tomar, Saurabh"https://www.zbmath.org/authors/?q=ai:tomar.saurabhSummary: A new rapid-converging analytical scheme is introduced to obtain the approximate analytical solutions of nonlinear fourth-order two-point boundary value problems, which appear in various physical phenomena. The idea of the method to obtain the solution of such problems is essentially based on reducing the solution of the main problem to the solution of an integral problem. The introduced technique consists of two steps. First, construct an integral operator by introducing Green's function, and then, the Normal-S iterative scheme is applied to this integral operator to construct the iterative approach for such problems, which yields a simple way to improve the convergence of the iterative solutions to the problem. We also discuss the convergence of the introduced iterative method. To exhibit the performance of the method, we consider some numerical test examples. The obtained results are compared with the existing analytical and numerical approaches to reveal the superiority and computational efficiency of the proposed approach. In fact, it is a direct recursive and computationally cost-effective method for dealing with strong nonlinearity. The numerical simulations signify the applicability and effectiveness of the present work.Improvement of random coefficient differential models of growth of anaerobic photosynthetic bacteria by combining Bayesian inference and gPC.https://www.zbmath.org/1456.340462021-04-16T16:22:00+00:00"Calatayud, Julia"https://www.zbmath.org/authors/?q=ai:calatayud.julia"Cortés, Juan Carlos"https://www.zbmath.org/authors/?q=ai:cortes.juan-carlos"Jornet, Marc"https://www.zbmath.org/authors/?q=ai:jornet.marcSummary: The time evolution of microorganisms, such as bacteria, is of great interest in biology. In the article by \textit{D. Stanescu} et al. [ETNA, Electron. Trans. Numer. Anal. 34, 44--58 (2009; Zbl 1173.60333)], a logistic model was proposed to model the growth of anaerobic photosynthetic bacteria. In the laboratory experiment, actual data for two species of bacteria were considered: \textit{Rhodobacter capsulatus} and \textit{Chlorobium vibrioforme}. In this paper, we suggest a new nonlinear model by assuming that the population growth rate is not proportional to the size of the bacteria population, but to the number of interactions between the microorganisms, and by taking into account the beginning of the death phase in the kinetic curve. Stanescu et al. evaluated the effect of randomness into the model coefficients by using generalized polynomial chaos (gPC) expansions, by setting arbitrary distributions without taking into account the likelihood of the data. By contrast, we utilize a Bayesian inverse approach for parameter estimation to obtain reliable posterior distributions for the random input coefficients in both the logistic and our new model. Since our new model does not possess an explicit solution, we use gPC expansions to construct the Bayesian model and to accelerate the Markov chain Monte Carlo algorithm for the Bayesian inference.Periodic solutions for the Lorentz force equation with singular potentials.https://www.zbmath.org/1456.340412021-04-16T16:22:00+00:00"Garzón, Manuel"https://www.zbmath.org/authors/?q=ai:garzon.manuel"Torres, Pedro J."https://www.zbmath.org/authors/?q=ai:torres.pedro-joseThe authors consider the Lorentz force equation
\[
(\phi(q'))'=E(t,q)+q'\times B(t,q)\,,
\]
where \(E\) and \(B\) are the electric and magnetic field, respectively, and
\[
\phi(v)=\frac{v}{\sqrt{1-|v|^2}}\,.
\]
They take \(E(t,q)=-\nabla V(q)+h(t)\), with \(h\in L^1([0,T],\mathbb{R}^3)\), and they assume
(H1) \(V\in C^1(\mathbb{R}^3\setminus\{0\},R)\) and
\[\lim_{|q|\to\infty}\nabla V(q)=0;\]
(H2) \(q\cdot \nabla V(q)<0\) for every \(q\in \mathbb{R}^3\) and there exist \(c_0,\varepsilon_0>0\) and \(\gamma\ge1\) such that \(q\cdot\nabla V(q)\le-c_0|q|^{-\gamma}\) when \(|q|<\varepsilon_0\);
(H3) \(B\in C([0,T]\times \mathbb{R}^3\setminus\{0\},\mathbb{R}^3)\) and there exists \(C_B>0\) such that \[\limsup_{|q|\to\infty}|B(t,q)|<C_B<\frac{1}{T}\Big|\int_0^Th(t)\,dt\Big|;\]
(H4) there exist \(c_1,\varepsilon_1>0\) and \(\beta\in(0,\gamma)\) such that \(|B(t,q)|\le c_1|q|^{-\beta-1}\) for all \(t\in[0,T]\) when \(|q|<\varepsilon_1\).
Under the above assumptions, by the use of topological degree methods they prove that the Lorenz force equation has at least one solution \(q:[0,T]\to \mathbb{R}^3\) such that \(|q'(t)|<1\) for every \(t\in[0,T]\) and \(q(0)-q(T)=0=q'(0)-q'(T)\).
Reviewer: Alessandro Fonda (Trieste)Global solution curves for first order periodic problems, with applications.https://www.zbmath.org/1456.340432021-04-16T16:22:00+00:00"Korman, Philip"https://www.zbmath.org/authors/?q=ai:korman.philip-l"Schmidt, Dieter S."https://www.zbmath.org/authors/?q=ai:schmidt.dieter-sThe authors consider a scalar first order differential equation of the type
\[
u'+g(t,u)=0,
\]
where the function \(g(t,u)\) belongs to \(C^{0,1}\) and is \(T\)-periodic in \(t\). They study the multiplicity of \(T\)-periodic solutions by continuation and bifurcation methods. An example of application is provided by a population model with fishing. Some numerical simulations are also provided.
Reviewer: Alessandro Fonda (Trieste)Degenerate band edges in periodic quantum graphs.https://www.zbmath.org/1456.811942021-04-16T16:22:00+00:00"Berkolaiko, Gregory"https://www.zbmath.org/authors/?q=ai:berkolaiko.gregory"Kha, Minh"https://www.zbmath.org/authors/?q=ai:kha.minhSummary: Edges of bands of continuous spectrum of periodic structures arise as maxima and minima of the dispersion relation of their Floquet-Bloch transform. It is often assumed that the extrema generating the band edges are non-degenerate. This paper constructs a family of examples of \({\mathbb{Z}}^3\)-periodic quantum graphs where the non-degeneracy assumption fails: the maximum of the first band is achieved along an algebraic curve of co-dimension 2. The example is robust with respect to perturbations of edge lengths, vertex conditions and edge potentials. The simple idea behind the construction allows generalizations to more complicated graphs and lattice dimensions. The curves along which extrema are achieved have a natural interpretation as moduli spaces of planar polygons.A prey predator model with disease in prey and recovery.https://www.zbmath.org/1456.340502021-04-16T16:22:00+00:00"Pal, Samares"https://www.zbmath.org/authors/?q=ai:pal.samaresh"Sarkar, Ikbal Hossein"https://www.zbmath.org/authors/?q=ai:sarkar.ikbal-hosseinThe authors propose and study the model
\begin{align*}
& \frac{dS}{dt}=aS\left( 1-\frac{S+I}{k} \right)-\frac{{{m}_{1}}SX}{{{a}_{1}}+S+{{b}_{1}}I}+\gamma I-\beta SI, \\
& \frac{dI}{dt}=\beta SI-\frac{{{m}_{2}}IX}{{{a}_{2}}+S+{{b}_{2}}I}-\gamma I-bI, \\
& \frac{dX}{dt}=\left( \frac{{{e}_{1}}{{m}_{1}}S}{{{a}_{1}}+S+{{b}_{1}}I}+\frac{{{e}_{2}}{{m}_{2}}I}{{{a}_{2}}+S+{{b}_{2}}I}-l \right)X,
\end{align*}
where \(S=S(t)\) is the population of susceptible prey, \(I=I(t)\) is the population of infected prey and \(X=X(t)\) is the predator's population; \(a,{{a}_{i}},b,{{b}_{i}},{{e}_{i}},k,l,{{m}_{i}},\gamma ,\beta \) are the parameters of the model.
The authors prove existence and boundedness of solutions of the model. Stability of the equilibrium points of the model and the mechanism of the Hopf bifurcation are investigated. Theoretical results illustrated by numerical examples.
Reviewer: Eduard Musafirov (Grodno)On the behavior of solutions with positive initial data to higher-order differential equations with general power-law nonlinearity.https://www.zbmath.org/1456.340102021-04-16T16:22:00+00:00"Korchemkina, Tatiana"https://www.zbmath.org/authors/?q=ai:korchemkina.tatianaSummary: Higher-order differential equation with general power-law nonlinearity are considered. In particular, solutions with positive initial data are studied depending on the values of nonlinearity exponents. It is proven that if the sum of nonlinearity exponents is greater than one, then any considered solution has a finite right domain boundary. In the case of a constant potential solutions with power-law behavior are found.
For the entire collection see [Zbl 1445.34003].The Kuramoto model revisited.https://www.zbmath.org/1456.340312021-04-16T16:22:00+00:00"da Fonseca, J. D."https://www.zbmath.org/authors/?q=ai:da-fonseca.j-d"Abud, C. V."https://www.zbmath.org/authors/?q=ai:abud.c-vA Green's function iterative approach for the solution of a class of fractional BVPs arising in physical models.https://www.zbmath.org/1456.650522021-04-16T16:22:00+00:00"Hadid, S."https://www.zbmath.org/authors/?q=ai:hadid.samir-b"Khuri, S. A."https://www.zbmath.org/authors/?q=ai:khuri.suheil-a"Sayfy, A."https://www.zbmath.org/authors/?q=ai:sayfy.ali-m-sSummary: The aim of this study is to present an alternative approach for the numerical solution of a wide class of fractional boundary value problems (FBVPs) that arise in various physical applications. Examples of such FBVP include but not limited to Bagley-Torvik, Riccati, Bratu, and Troesch's problems that appear in applied mathematics and mechanics. The method is based on first constructing an integral operator that is given in terms of the Green's function associated with the linear differential term of the fractional differential equation. Fixed point iterative procedures, such as Picard's and Mann's, are then applied to the operator to generate an iterative scheme that yields a convergent semi-analytical solution. Numerical examples are reported to confirm the efficiency, reliability, accuracy and fast convergence of the scheme.Existence of solutions of fuzzy fractional panto-graph equations.https://www.zbmath.org/1456.340772021-04-16T16:22:00+00:00"Agilan, K."https://www.zbmath.org/authors/?q=ai:agilan.k"Parthiban, V."https://www.zbmath.org/authors/?q=ai:parthiban.vijayaAuthors consider nonlinear fuzzy fractional panto-graph equation with the Caputo gH-derivative
\[(_{gH}D^q_{0+})u(x)=f(x,u(x),u(\lambda x)), \quad u(0)=u_0,\]
where \(0 < q \leq 1\) is a real number and the operator \( (_{gH}D^q_{0+})\) indicate the Caputo fractional generalized derivative of order \(q,\) \(f: J\times R_f\times R_f\to R_f\) is the continuous function. Conditions for the existence of a solution are obtained. Finally, the authors give an example to support the results.
Reviewer: Tatyana Komleva (Odessa)New shock-wave and periodic-wave solutions for some physical and engineering models: Vakhnenko-Parkes, GEWB, GRLW and some integrable equations.https://www.zbmath.org/1456.340012021-04-16T16:22:00+00:00"Alquran, Marwan"https://www.zbmath.org/authors/?q=ai:alquran.marwan-taiseer"Jaradat, Imad"https://www.zbmath.org/authors/?q=ai:jaradat.imad"Sivasundaram, Seenith"https://www.zbmath.org/authors/?q=ai:sivasundaram.seenith"Al Shraiedeh, Laila"https://www.zbmath.org/authors/?q=ai:al-shraiedeh.lailaSummary: In this work, the modified unified expansion and the Bernoulli sub-equation methods are implemented to extract new shock-wave and periodic-wave solutions for important physical and engineering models. We study four models; the Vakhnenko-Parkes (VP) equation, the generalized equal width-Burgers (GEWB) equation with \(p \in \{1, 2\}\) and the generalized regularized-long-wave (GRLW) equation by the modified unified method, whereas the first-second fourth-order integrable equations by the Bernoulli sub-equation method. Shock-wave and periodic wave solutions are obtained for these models. All obtained solutions are verified and categorized regarding its physical structures.Generalized stochastic resonance in a linear fractional system with a random delay.https://www.zbmath.org/1456.827672021-04-16T16:22:00+00:00"Gao, Shi-Long"https://www.zbmath.org/authors/?q=ai:gao.shilongOn existence theorems for generalized abstract measure integrodifferential equations.https://www.zbmath.org/1456.450022021-04-16T16:22:00+00:00"Dhage, Bapurao Chandrabhan"https://www.zbmath.org/authors/?q=ai:dhage.bapurao-chandrabhanSummary: In this paper, an existence and uniqueness results for a nonlinear abstract measure integrodifferential equation are proved via classical fixed point theorems of Schauder (see [\textit{A. Granas} and \textit{J. Dugundji}, Fixed point theory. New York, NY: Springer (2003; Zbl 1025.47002)]) and the author [Electron. J. Qual. Theory Differ. Equ. 2002, Paper No. 6, 9 p. (2002; Zbl 1029.47034)] under weaker Carathéodory condition. The existence for extremal solutions is also proved under certain Chandrabhan condition and using a hybrid fixed point theorem of the author [loc. cit.] in an ordered Banach space. Our existence results presented in this paper include the existence results of \textit{R. R. Sharma} [Proc. Am. Math. Soc. 32, 503--510 (1972; Zbl 0213.36201)], \textit{S. R. Joshi} [J. Math. Phys. Sci. 13, 497--506 (1979; Zbl 0435.34053)], \textit{G. R. Shendge} and \textit{S. R. Joshi} [Acta Math. Hung. 41, 53--59 (1983; Zbl 0536.34040)] and the author [J. Math. Phys. Sci. 20, 367--380 (1986; Zbl 0619.45005); with \textit{P. R. M. Reddy}, Jñānābha 49, No. 2, 82--93 (2019; Zbl 07273233)] on nonlinear abstract measure and abstract measure integrodifferential equations as special cases under weaker continuity condition.Review on the behaviour of a many predator-one prey system.https://www.zbmath.org/1456.340532021-04-16T16:22:00+00:00"Söderbacka, G. J."https://www.zbmath.org/authors/?q=ai:soderbacka.g-j"Petrov, A. S."https://www.zbmath.org/authors/?q=ai:petrov.a-sSummary: We consider a known predator-prey system, where more than one predator compete for the same prey. Mainly the case with two predators is considered. A review of general results is given, among them conditions for the extinction of one predator and an investigation of the different types of coexistence of predators. In non-degenerate cases the predators in this model cannot coexist at an equilibrium, but there can be a cyclic or more complicated coexistence. Many numerical results are presented. A model map for a Poincaré map is given under some conditions. But the most interesting case where there can arise ``spiral-like'' attractors is not well known here, and we pose open questions. We discuss some bifurcations and the existence of systems with several attractors.Bendixson criterion in impulsive systems.https://www.zbmath.org/1456.340112021-04-16T16:22:00+00:00"Ding, Changming"https://www.zbmath.org/authors/?q=ai:ding.changming"Duan, Zhipeng"https://www.zbmath.org/authors/?q=ai:duan.zhipeng"Pan, Shiyao"https://www.zbmath.org/authors/?q=ai:pan.shiyaoConsider the sufficiently smooth planar vector field \(F(x,y)\). Under the assumption that the divergence of \(F\) has constant sign in some simply connected region \(U\subset\mathbb{R}^2\) and is not identically zero in any subregion of \(U\), then the criterion of Bendixson says that there is no closed orbit of \(F\) entirely located in \(U\). The authors recall basic definitions and properties of planar impulsive systems. They present a simple example showing that the Bendixson criterion does not hold for planar impulsive systems in its original form. Introducing the concept of a periodic orbit of order \(k\), where \(k\) is related to the number of jumps, they prove an extended Bendixson criterion which excludes the existence of periodic orbits of order \(k \leq 2\).
Reviewer: Klaus R. Schneider (Berlin)Dynamical models of interrelation in a class of artificial networks.https://www.zbmath.org/1456.340522021-04-16T16:22:00+00:00"Sadyrbaev, Felix"https://www.zbmath.org/authors/?q=ai:sadyrbaev.felix-zh"Atslega, Svetlana"https://www.zbmath.org/authors/?q=ai:atslega.svetlana"Brokan, Eduard"https://www.zbmath.org/authors/?q=ai:brokan.eduardSummary: The system of ordinary differential equations that models a type of artificial networks is considered. The system consists of a sigmoidal function that depends on linear combinations of the arguments minus the linear part. The linear combinations of the arguments are described by the regulatory matrix W. For the three-dimensional cases, several types of matrices W are considered and the behavior of solutions of the system is analyzed. The attractive sets are constructed for most cases. The illustrative examples are provided. The list of references consists of 12 items.
For the entire collection see [Zbl 1445.34003].Multiple periodic solutions for a Duffing type equation with one-sided sublinear nonlinearity: beyond the Poincaré-Birkhoff twist theorem.https://www.zbmath.org/1456.340402021-04-16T16:22:00+00:00"Dondè, Tobia"https://www.zbmath.org/authors/?q=ai:donde.tobia"Zanolin, Fabio"https://www.zbmath.org/authors/?q=ai:zanolin.fabioSummary: We prove the existence of multiple periodic solutions for a planar Hamiltonian system generated from the second order scalar ODE of Duffing type \(x'' + q(t)g(x) = 0\) with \(g\) satisfying a one-sided condition of sublinear type. We consider the classical approach based on the Poincaré-Birkhoff fixed point theorem as well as some refinements on the side of the theory of bend-twist maps and topological horseshoes. We focus our analysis to the case of a stepwise weight function, in order to highlight the underlying geometrical structure.
For the entire collection see [Zbl 1445.34003].Truncated Euler-Maruyama method for classical and time-changed non-autonomous stochastic differential equations.https://www.zbmath.org/1456.650072021-04-16T16:22:00+00:00"Liu, Wei"https://www.zbmath.org/authors/?q=ai:liu.wei"Mao, Xuerong"https://www.zbmath.org/authors/?q=ai:mao.xuerong"Tang, Jingwen"https://www.zbmath.org/authors/?q=ai:tang.jingwen"Wu, Yue"https://www.zbmath.org/authors/?q=ai:wu.yueSummary: The truncated Euler-Maruyama (EM) method is proposed to approximate a class of non-autonomous stochastic differential equations (SDEs) with the Hölder continuity in the temporal variable and the super-linear growth in the state variable. The strong convergence with the convergence rate is proved. Moreover, the strong convergence of the truncated EM method for a class of highly non-linear time-changed SDEs is studied.Various spectral problems with the same characteristic determinant.https://www.zbmath.org/1456.340202021-04-16T16:22:00+00:00"Akhtyamov, A. M."https://www.zbmath.org/authors/?q=ai:akhtyamov.azamat-moukhtarovich|akhtyamov.azamat-mukhtarovichSummary: We show that there exist whole classes of various boundary value problems having the same characteristic determinant, with the respective problems allowed to have differing orders of the differential equations and to be defined both on intervals and on geometric graphs.On oscillation conditions for solutions to delay differential equations.https://www.zbmath.org/1456.340702021-04-16T16:22:00+00:00"Malygina, V. V."https://www.zbmath.org/authors/?q=ai:malygina.v-v"Chudinov, K. M."https://www.zbmath.org/authors/?q=ai:chudinov.k-mSummary: The possibilities of obtaining effective oscillation conditions for solutions to linear non-autonomous first-order differential equations with aftereffect are investigated. We consider estimates, on the functional of the equation parameters, guaranteeing the oscillation of all solutions. These estimates are results of the development of known sufficient oscillation conditions for a linear equation with one concentrated delay, in the form of an estimate of the integral of the coefficient over the delay length. For an equation with several concentrated delays, several results achieved in recent years are compared and new proofs and examples are given that justify the effectiveness of some of the developed methods and their advantages over earlier approaches. A new approach is applied to equations with distributed delay, for which new effective conditions for the oscillation of solutions are obtained.Hereditary circularity for energy minimal diffeomorphisms.https://www.zbmath.org/1456.300432021-04-16T16:22:00+00:00"Koh, Ngin-Tee"https://www.zbmath.org/authors/?q=ai:koh.ngin-teeSummary: We reveal some geometric properties of energy minimal diffeomorphisms defined on an annulus, whose existence was established in works by
\textit{T. Iwaniec} et al. [Invent. Math. 186, No. 3, 667--707 (2011; Zbl 1255.30031); J. Am. Math. Soc. 24, No. 2, 345--373 (2011; Zbl 1214.31001)]
and \textit{D. Kalaj} [Calc. Var. Partial Differ. Equ. 51, No. 1--2, 465--494 (2014; Zbl 1296.30052)].A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation.https://www.zbmath.org/1456.340192021-04-16T16:22:00+00:00"Dzhumabaev, Dulat"https://www.zbmath.org/authors/?q=ai:dzhumabaev.dulat-syzdykbekovich"Bakirova, Elmira"https://www.zbmath.org/authors/?q=ai:bakirova.elmira-a"Mynbayeva, Sandugash"https://www.zbmath.org/authors/?q=ai:mynbayeva.sandugash-tabyldyevnaIn this paper, the authors study a nonlinear loaded differential equation with a parameter on a finite interval
\[
\frac{dx}{dt}=f_0(t,x,\mu)+f_1(t,x(\theta_0),x(\theta_1),\dots,x(\theta_{m-1}),x(\theta_m),\mu),\ t\in(0,T),x\in \mathbb{R}^n,\mu\in \mathbb{R}^l
\]
subjected to the boundary value conditions
\[
g[x(0),x(T),\mu]=0,
\]
where \(f_0:[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^l\to \mathbb{R}^n\), \(f_1:[0,T]\times\mathbb{R}^{n(m+1)}\times\mathbb{R}^l\to \mathbb{R}^n\), and \(g:\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}^l\to \mathbb{R}^{n+l}\) are continuous functions, \(0=\theta_0<\theta_1<\cdots<\theta_{m-1}<\theta_m=T\), \(\|x\|=\max_{i=1,\dots,n}|x_i|\).
The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, the authors obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, they compose a system of nonlinear algebraic equations in parameters. They develop a method of solving the boundary value problem with a parameter for the above loaded differential equation which is based on finding the solution to the system of nonlinear algebraic equations by an iterative process.
Reviewer: Yanqiong Lu (Lanzhou)Stochastic resonance in a fractional oscillator driven by multiplicative quadratic noise.https://www.zbmath.org/1456.700362021-04-16T16:22:00+00:00"Ren, Ruibin"https://www.zbmath.org/authors/?q=ai:ren.ruibin"Luo, Maokang"https://www.zbmath.org/authors/?q=ai:luo.maokang"Deng, Ke"https://www.zbmath.org/authors/?q=ai:deng.keSchur-\(m\) power convexity of Cauchy means and its application.https://www.zbmath.org/1456.260132021-04-16T16:22:00+00:00"Wang, Dong-Sheng"https://www.zbmath.org/authors/?q=ai:wang.dongsheng"Fu, Chunru"https://www.zbmath.org/authors/?q=ai:fu.chunru"Shi, Huannan"https://www.zbmath.org/authors/?q=ai:shi.huannanThe Cauchy mean generalizes many well-known classical means of two variables. Schur convexity, Schur geometric convexity, Schur harmonic convexity and Schur power convexity are important tools in their study. In recent years, the application of majorization theory to the Schur convexity of various means has been very active and many results have been obtained. The authors of this paper obtain necessary and sufficient conditions for the Cauchy mean to be Schur-\(m\) power convex or concave. As applications, Schur-\(m\) power convexity of the exponential mean is discussed and a comparative inequality between the Gini mean and the Stolarsky mean is given.
Reviewer: Ioan Raşa (Cluj-Napoca)Error analysis of the Wiener-Askey polynomial chaos with hyperbolic cross approximation and its application to differential equations with random input.https://www.zbmath.org/1456.651732021-04-16T16:22:00+00:00"Luo, Xue"https://www.zbmath.org/authors/?q=ai:luo.xueSummary: It is well-known that sparse grid algorithm has been widely accepted as an efficient tool to overcome the ``curse of dimensionality'' in some degree. In this note, we give the error estimate of hyperbolic cross (HC) approximations with all sorts of Askey polynomials. These polynomials are useful in generalized polynomial chaos (gPC) in the field of uncertainty quantification. The exponential convergences in both regular and optimized HC approximations have been shown under the condition that the random variable depends on the random inputs smoothly in some degree. Moreover, we apply gPC to numerically solve the ordinary differential equations with slightly higher dimensional random inputs. Both regular and optimized HC have been investigated with Laguerre-chaos, Charlier-chaos and Hermite-chaos in the numerical experiment. The discussion of the connection between the standard ANOVA approximation and Galerkin approximation is in the appendix.The KW equations and the Nahm pole boundary condition with knots.https://www.zbmath.org/1456.813112021-04-16T16:22:00+00:00"Mazzeo, Rafe"https://www.zbmath.org/authors/?q=ai:mazzeo.rafe-r"Witten, Edward"https://www.zbmath.org/authors/?q=ai:witten.edwardIn this detailed technical paper the authors extend further their previous analysis of the Kapustin-Witten (KW) equations with Nahm pole boundary condition now adapted to general 4-manifolds-with-boundary such that the boundary-3-manifold contains a knot or more generally a link.
The motivation is a conjecture of the second author that the coefficients of the Laurent expansion of the Jones polynomial of a link \(L\subset {\mathbb R}^3\) arise by counting solutions of the KW equations on the half-space \({\mathbb R}^4_+\) obeying a generalized Nahm pole boundary condition on \(\partial {\mathbb R}^4_+={\mathbb R}^3\supset L\) i.e. the Nahm pole boundary condition generalized to be compatible with the extra information of containing a link on the boundary. Roughly this means to prescribe further singularities in the Higgs field part of the KW pair along each link component while the connection part is continuous up to the boundary as before. The conjecture is important because it is well-known that computing the Jones polynomial of a link is an exponentially difficult problem in terms of e.g. the crossing number of any plane diagram of the link.
Reviewer: Gabor Etesi (Budapest)Oscillation criteria of singular initial-value problem for second order nonlinear dynamic equation on time scales.https://www.zbmath.org/1456.340852021-04-16T16:22:00+00:00"Negi, Shekhar Singh"https://www.zbmath.org/authors/?q=ai:negi.shekhar-singh"Abbas, Syed"https://www.zbmath.org/authors/?q=ai:abbas.syed"Malik, Muslim"https://www.zbmath.org/authors/?q=ai:malik.muslimSummary: By using of generalized Opial's type inequality on time scales, a new oscillation criterion is given for a singular initial-value problem of second-order dynamic equation on time scales. Some oscillatory results of its generalizations are also presented. Example with various time scales is given to illustrate the analytical findings.On stability of small periodic solutions.https://www.zbmath.org/1456.340392021-04-16T16:22:00+00:00"Abramov, V. V."https://www.zbmath.org/authors/?q=ai:abramov.vladimir-viktorovichThe author considers normal time-periodic systems of ordinary differential equations whose right-hand smoothly depend on phase variables and small parameters. Branching conditions for small periodic solution of the system was found and the Lyapunov stability test for small solution with respect to parameters or variables are established. The Lyapunov stability test iss used for the analysis of the branching of small periodic solution. The conditions that guarantee asymptotic stability of a small periodic solution of the considered system is established in the work.
Reviewer: Babatunde Ogundare (Ile-Ife)Implicit parametrizations and applications in optimization and control.https://www.zbmath.org/1456.490082021-04-16T16:22:00+00:00"Tiba, Dan"https://www.zbmath.org/authors/?q=ai:tiba.danThe subject is the characterization (with numerical applications in mind) of the manifold \(V\) of solutions of the nonlinear system
\[
F_j(x_1, x_2, \dots, x_d) = 0 \quad (1 \le j \le l )\quad l \le d - 1 \tag{1}
\]
in the vicinity of \(x^0 = (x^0_1, x^0_2 , \dots ,x^0_d) \in V,\) under the Jacobian assumption
\[
\frac{\partial (F_1, F_2, \dots, F_l)}{\partial (x_1, x_2, \dots, x_l)} \ne 0
\quad \hbox{in} \ x^0 = (x^0_1, x^0_2 , \dots ,x^0_d) \, .
\]
The first step involves the underdetermined linear system
\[
v(x) \cdot \nabla F_j(x) = 0 \quad (1 \le j \le l)
\]
which is used to obtain bases \((v_1(x), v_2(x), \dots, v_{d - l}(x))\) for the tangent spaces of \(V.\) Next, the chain of differential equations
\begin{align*}
\frac{\partial y_1(t_1)}{\partial t_1} &= v_1(y_1(t_1)), y_1(0) = x^0
\cr
\frac{\partial y_2(t_1, t_2)}{\partial t_2} &= v_2(y_2(t_1, t_2)),\quad y_2(t_1, 0) = y(t_1)
\cr
& \hskip 2em \dots \dots \dots \dots
\cr
\frac{\partial y_{d - l}(t_1, t_2, \dots, t_{d - l})}{ \partial t_{d - l}}
&= v_{d - l}(y_{d - l}(t_1, t_2, \dots, t_{d - l})) \, ,
\cr
& \hskip 2.7em y_{d - l}(t_1, \dots , t_{d - l - 1}, 0) = y_{d - l - 1}(t_1, t_2, \dots , t_{d - l - 1})
\end{align*}
is set up, thus constructing a parametrization of \(V\) which may be considered as an explicit form of the implicit function theorem. The result is used to construct an algorithm for the solution of the problem of minimizing a function
\(g(x_0, x_2, \dots , x_d)\) subject to (1). Some generalizations are covered, such as the case where (1) includes inequality constraints and/or regularity is relaxed. In the last section the results are applied to the control problem
of minimizing \(l(x(0), x(1))\) among the trajectories of the system
\(x'(t) = f(t, x(t), u(t))\) subject to the state-control constraint \(h(x(t), u(t)) = 0.\) There are several numerical implementation of the algorithms and the author notes that computations can be carried out using standard Matlab routines.
Reviewer: Hector O. Fattorini (Los Angeles)Global dynamics of neoclassical growth model with multiple pairs of variable delays.https://www.zbmath.org/1456.340792021-04-16T16:22:00+00:00"Huang, Chuangxia"https://www.zbmath.org/authors/?q=ai:huang.chuangxia"Zhao, Xian"https://www.zbmath.org/authors/?q=ai:zhao.xian"Cao, Jinde"https://www.zbmath.org/authors/?q=ai:cao.jinde"Alsaadi, Fuad E."https://www.zbmath.org/authors/?q=ai:alsaadi.fuad-eid-sA Lorentz-covariant interacting electron-photon system in one space dimension.https://www.zbmath.org/1456.812602021-04-16T16:22:00+00:00"Kiessling, Michael K.-H."https://www.zbmath.org/authors/?q=ai:kiessling.michael-karl-heinz"Lienert, Matthias"https://www.zbmath.org/authors/?q=ai:lienert.matthias"Tahvildar-Zadeh, A. Shadi"https://www.zbmath.org/authors/?q=ai:tahvildar-zadeh.a-shadiSummary: A Lorentz-covariant system of wave equations is formulated for a quantum-mechanical two-body system in one space dimension, comprised of one electron and one photon. Manifest Lorentz covariance is achieved using Dirac's formalism of multi-time wave functions, i.e., wave functions \(\Psi^{(2)}(\mathbf{x}_{\mathrm{ph}},\mathbf{x}_{\mathrm{el}})\) where \(\mathbf{x}_{\mathrm{el}},\mathbf{x}_{\mathrm{ph}}\) are the generic spacetime events of the electron and photon, respectively. Their interaction is implemented via a Lorentz-invariant no-crossing-of-paths boundary condition at the coincidence submanifold \(\{\mathbf{x}_{\mathrm{el}}=\mathbf{x}_{\mathrm{ph}}\} \), compatible with particle current conservation. The corresponding initial-boundary-value problem is proved to be well-posed. Electron and photon trajectories are shown to exist globally in a hypersurface Bohm-Dirac theory, for typical particle initial conditions. Also presented are the results of some numerical experiments which illustrate Compton scattering as well as a new phenomenon: photon capture and release by the electron.Hybrid projective combination-combination synchronization in non-identical hyperchaotic systems using adaptive control.https://www.zbmath.org/1456.340632021-04-16T16:22:00+00:00"Khan, Ayub"https://www.zbmath.org/authors/?q=ai:khan.ayub"Chaudhary, Harindri"https://www.zbmath.org/authors/?q=ai:chaudhary.harindriSummary: In this paper, we investigate a hybrid projective combination-combination synchronization scheme among four non-identical hyperchaotic systems via adaptive control method. Based on Lyapunov stability theory, the considered approach identifies the unknown parameters and determines the asymptotic stability globally. It is observed that various synchronization techniques, for instance, chaos control problem, combination synchronization, projective synchronization, etc. turn into particular cases of combination-combination synchronization. The proposed scheme is applicable to secure communication and information processing. Finally, numerical simulations are performed to demonstrate the effectivity and correctness of the considered technique by using MATLAB.Study of a boundary value problem for fractional order \(\psi\)-Hilfer fractional derivative.https://www.zbmath.org/1456.340052021-04-16T16:22:00+00:00"Harikrishnan, S."https://www.zbmath.org/authors/?q=ai:harikrishnan.sugumaran"Shah, Kamal"https://www.zbmath.org/authors/?q=ai:shah.kamal"Kanagarajan, K."https://www.zbmath.org/authors/?q=ai:kanagarajan.kana|kanagarajan.kuppusamySummary: This manuscript is devoted to the existence theory of a class of random fractional differential equations (RFDEs) involving boundary condition (BCs). Here we take the corresponding derivative of arbitrary order in \(\psi\)-Hilfer sense. By utilizing classical fixed point theory and nonlinear analysis we establish some basic results of the qualitative theory such as existence, uniqueness and stability of solutions to the considered boundary value problem of RFDEs. Further, for the justification of our analysis we provide two examples.Boundary value problems for Caputo fractional differential equations with nonlocal and fractional integral boundary conditions.https://www.zbmath.org/1456.340042021-04-16T16:22:00+00:00"Derbazi, Choukri"https://www.zbmath.org/authors/?q=ai:derbazi.choukri"Hammouche, Hadda"https://www.zbmath.org/authors/?q=ai:hammouche.haddaSummary: In this paper, we study the existence and uniqueness of solutions for fractional differential equations with nonlocal and fractional integral boundary conditions. New existence and uniqueness results are established using the Banach contraction principle. Other existence results are obtained using O'Regan fixed point theorem and Burton and Kirk fixed point. In addition, an example is given to demonstrate the application of our main results.A new stable collocation method for solving a class of nonlinear fractional delay differential equations.https://www.zbmath.org/1456.650552021-04-16T16:22:00+00:00"Shi, Lei"https://www.zbmath.org/authors/?q=ai:shi.lei.2|shi.lei.4|shi.lei.1|shi.lei.3|shi.lei"Chen, Zhong"https://www.zbmath.org/authors/?q=ai:chen.zhong"Ding, Xiaohua"https://www.zbmath.org/authors/?q=ai:ding.xiaohua"Ma, Qiang"https://www.zbmath.org/authors/?q=ai:ma.qiangSummary: In this paper, a stable collocation method for solving the nonlinear fractional delay differential equations is proposed by constructing a new set of multiscale orthonormal bases of \(W^1_{2,0}\). Error estimations of approximate solutions are given and the highest convergence order can reach four in the sense of the norm of \(W_{2,0}^1\). To overcome the nonlinear condition, we make use of Newton's method to transform the nonlinear equation into a sequence of linear equations. For the linear equations, a rigorous theory is given for obtaining their \(\varepsilon \)-approximate solutions by solving a system of equations or searching the minimum value. Stability analysis is also obtained. Some examples are discussed to illustrate the efficiency of the proposed method.Revealing the role of the effector-regulatory T cell loop on autoimmune disease symptoms via nonlinear analysis.https://www.zbmath.org/1456.340592021-04-16T16:22:00+00:00"Zhang, Wenjing"https://www.zbmath.org/authors/?q=ai:zhang.wenling"Yu, Pei"https://www.zbmath.org/authors/?q=ai:yu.peiSummary: In this paper, we investigate the influence of the effector-regulatory (Teff-Treg) T cell interaction on the T-cell-mediated autoimmune disease dynamics. The simple 3-dimensional Teff-Treg model is derived from the two-step model reduction of an established 5-dimensional model. The reduced 4- and 3-dimensional models preserve the dynamical behaviors in the original 5-dimensional model, which represents the chronic and relapse-remitting autoimmune symptoms. Moreover, we find three co-existing limit cycles in the reduced 3-dimensional model, in which two stable periodic solutions enclose an unstable one. The existence of multiple limit cycles provides a new mechanism to explain varying oscillating amplitudes of lesion grade in multiple sclerosis. The complex multiphase symptom could be caused by a noise-driven Teff population traveling between two coexisting stable periodic solutions. The simulated phase portrait and time history of coexisting limit cycles are given correspondingly.Edge connectivity and the spectral gap of combinatorial and quantum graphs.https://www.zbmath.org/1456.811932021-04-16T16:22:00+00:00"Berkolaiko, Gregory"https://www.zbmath.org/authors/?q=ai:berkolaiko.gregory"Kennedy, James B."https://www.zbmath.org/authors/?q=ai:kennedy.james-b"Kurasov, Pavel"https://www.zbmath.org/authors/?q=ai:kurasov.pavel-b"Mugnolo, Delio"https://www.zbmath.org/authors/?q=ai:mugnolo.delioModeling of invasive process with spontaneous transition from critical \(K\)-capacity of the environment to an alternative asymptotic states of population.https://www.zbmath.org/1456.340802021-04-16T16:22:00+00:00"Perevaryukha, A. Yu."https://www.zbmath.org/authors/?q=ai:perevaryukha.andrei-yurevichSummary: In the article we discuss the simulation of critical development of the invasive process of a population with a high reproductive \(p\)-parameter. Situations of invasion of alien species can variably cause the phenomenon of outbreaks, depending on the regulation, \(N(0)\) and resistance of the environment. We propose to describe variability by new equations with delay. We have developed a model where, depending on the ratio of the parameters \(r,\tau,\gamma\), two practically significant scenarios are realized after a monotonic attainment of the maximum of the invaded species \(N(t)\to\max N(t_m)\). Destruction of the invasive population after depletion of the environment's exhaustion after a time \(t_\infty>t_m\) is realized with the output of the trajectory to unlimited growth \(N(t)\to\infty\). With the installation of slightly lower values of reproductive activity, a monotonic decrease to a small equilibrium level is realized: \(\lim_{t\to\infty}N(t)=L,L<\max N(t_m)\). Similarly, depending on minor changes in \(N(0)\), an acute infection can develop -- along the lethal pathway, or with the transformation into a chronic one, for example hepatitis C. In the second part, we considered the model of the scenario of the population passing through the critical minimum of the population \(\min N(t)<L<K\) after an unstable equilibrium \(N(t_k)=K\). The scenario is observed with a sharp increase in the resistance of the biotic environment or the development of a delayed immune response, then the situation occurs \(N(t)\to L\ll K\). The appearance of relaxation cycles, as in the previously proposed equation \(\dot N=rf(N^2(t-\tau))\), we do not observe. The equation in a scenario with a spontaneous transition from the \(K\)-capacity of saturation of the environment to the minimum \(\mathbf{H}\)-equilibrium describes the chronicization of viral hepatitis. Model scenarios consider the form of an immune response.Exact solutions and conservation laws of multi Kaup-Boussinesq system with fractional order.https://www.zbmath.org/1456.352212021-04-16T16:22:00+00:00"Singla, Komal"https://www.zbmath.org/authors/?q=ai:singla.komal"Rana, M."https://www.zbmath.org/authors/?q=ai:rana.mehwish|rana.meenakshiSummary: The purpose of the present work is to investigate exact solutions of the fractional order multi Kaup-Boussinesq system with \(l=2\) by using the group invariance approach and power series expansion method. Due to the significance of conserved vectors in terms of integrability and behaviour of nonlinear systems, the conservation laws are also derived by testing the nonlinear self-adjointness.A proof of unlimited multistability for phosphorylation cycles.https://www.zbmath.org/1456.340472021-04-16T16:22:00+00:00"Feliu, Elisenda"https://www.zbmath.org/authors/?q=ai:feliu.elisenda"Rendall, Alan D."https://www.zbmath.org/authors/?q=ai:rendall.alan-d"Wiuf, Carsten"https://www.zbmath.org/authors/?q=ai:wiuf.carstenFurther results on Ulam stability for a system of first-order nonsingular delay differential equations.https://www.zbmath.org/1456.340742021-04-16T16:22:00+00:00"Zada, Akbar"https://www.zbmath.org/authors/?q=ai:zada.akbar"Pervaiz, Bakhtawar"https://www.zbmath.org/authors/?q=ai:pervaiz.bakhtawar"Alzabut, Jehad"https://www.zbmath.org/authors/?q=ai:alzabut.jehad-o"Shah, Syed Omar"https://www.zbmath.org/authors/?q=ai:shah.syed-omarSummary: This paper is concerned with a system governed by nonsingular delay differential equations. We study the \(\beta\)-Ulam-type stability of the mentioned system. The investigations are carried out over compact and unbounded intervals. Before proceeding to the main results, we convert the system into an equivalent integral equation and then establish an existence theorem for the addressed system. To justify the application of the reported results, an example along with graphical representation is illustrated at the end of the paper.Effective construction of Poincaré-Bendixson regions.https://www.zbmath.org/1456.340262021-04-16T16:22:00+00:00"Gasull, Armengol"https://www.zbmath.org/authors/?q=ai:gasull.armengol"Giacomini, Hector"https://www.zbmath.org/authors/?q=ai:giacomini.hector-j"Grau, Maite"https://www.zbmath.org/authors/?q=ai:grau.maiteConsider planar autonomous polynomial systems. A traditional approach to establish the existence of at least one limit cycle \(\Gamma\) is to construct an annulus \(\mathbb{A}\) bounded by transversal ovals and which contains no equilibrium point. Then, the Poincaré-Bendixson theorem provides the existence of at least one limit cycle in \(\mathbb{A}\).
In this paper, the focus of the authors is on the possibility to use the annulus \(\mathbb{A}\) for the location of the limit cycle \(\Gamma\). They prove that if \(\Gamma\) is hyperbolic, then there is a neighborhood \(\mathbb{A}_\Gamma\) of \(\Gamma\) bounded by transversal ovals. Additionally, they present a method to construct these ovals. The basic idea of the authors is as follows: Suppose \(\tilde{\Gamma}\) is an approximation of a hyperbolic limit cycle \(\Gamma\), where \(\tilde{\Gamma}\) is located in \(\mathbb{A}_\Gamma\). Then \(\tilde{\Gamma}\) can be used to construct a Poincaré-Bendixson annulus \(\tilde{\mathbb{A}}\) containing \(\Gamma\) and which provides an excellent inclusion of \(\Gamma\). In applications, \(\tilde{\Gamma}\) is a numerical approximation of \(\Gamma\). The presented approach is applied to several
well-known systems.
Reviewer: Klaus R. Schneider (Berlin)Centers for the Kukles homogeneous systems with even degree.https://www.zbmath.org/1456.340272021-04-16T16:22:00+00:00"Gine, Jaume"https://www.zbmath.org/authors/?q=ai:gine.jaume"Llibre, Jaume"https://www.zbmath.org/authors/?q=ai:llibre.jaume"Valls, Claudia"https://www.zbmath.org/authors/?q=ai:valls.claudiaThe polynomial autonomous differential system
\[
\dot{x}=-y,\ \dot{y}=x+Q_n(x,y),\tag{1}
\]
where \(Q_n(x,y)\) is a homogeneous polynomial of even degree \(n\geqslant 6\) is considered. Necessary and sufficient conditions are obtained, when: 1) the system (1) has a center at the origin; 2) has an isochronous center at the origin.
Reviewer: Valentine Tyshchenko (Grodno)On increasing solutions of half-linear delay differential equations.https://www.zbmath.org/1456.340732021-04-16T16:22:00+00:00"Matucci, Serena"https://www.zbmath.org/authors/?q=ai:matucci.serena"Řehák, Pavel"https://www.zbmath.org/authors/?q=ai:rehak.pavelSummary: We establish conditions guaranteeing that all eventually positive increasing solutions of a half-linear delay differential equation are regularly varying and derive precise asymptotic formulae for them. The results presented here are new also in the linear case and some of the observations are original also for non-functional equations. A substantial difference is pointed out between the delayed and nondelayed case for eventually positive decreasing solutions.Limit cycle bifurcation for a nilpotent system in \(Z_3\)-equivariant vector field.https://www.zbmath.org/1456.340372021-04-16T16:22:00+00:00"Du, Chaoxiong"https://www.zbmath.org/authors/?q=ai:du.chaoxiong"Wang, Qinlong"https://www.zbmath.org/authors/?q=ai:wang.qinlong"Liu, Yirong"https://www.zbmath.org/authors/?q=ai:liu.yirong"Zhang, Qi"https://www.zbmath.org/authors/?q=ai:zhang.qi.1Authors' abstract: Our work is concerned with the problem on limit cycle bifurcation for a class of Z3-equivariant Lyapunov system of five degrees with three third-order nilpotent critical points which lie in a Z3-equivariant vector field. With the help of computer algebra system-MATHEMATICA, the first 5 quasi-Lyapunov constants are deduced. The fact of existing 12 small amplitude limit cycles created from the three third-order nilpotent critical points is also proved. Our proof is algebraic and symbolic, obtained result is new and interesting in terms of nilpotent critical points' Hilbert number in equivariant vector field.
Reviewer: Tao Li (Chengdu)Positive solutions of the \(p\)-Laplacian dynamic equations on time scales with sign changing nonlinearity.https://www.zbmath.org/1456.340842021-04-16T16:22:00+00:00"Dogan, Abdulkadir"https://www.zbmath.org/authors/?q=ai:dogan.abdulkadirSummary: This article concerns the existence of positive solutions for \(p\)-Laplacian boundary value problem on time scales. By applying fixed point index we obtain the existence of solutions. Emphasis is put on the fact that the nonlinear term is allowed to change sign. An example illustrates our results.Oscillations in enzymatic reaction with periodic input.https://www.zbmath.org/1456.340482021-04-16T16:22:00+00:00"Lara-Aguilar, Brenda"https://www.zbmath.org/authors/?q=ai:lara-aguilar.brenda"Osuna, Osvaldo"https://www.zbmath.org/authors/?q=ai:osuna.osvaldo"Wencer, Giovanni"https://www.zbmath.org/authors/?q=ai:wencer.giovanniSummary: In this work, we prove the existence of periodic solutions for some enzyme catalysed reaction models subject to periodic substrate input. We also obtain uniqueness and asymptotic stability of the periodic solution of some classes of reaction equations. Numerical simulations are performed using specific substrate functions to illustrate our analytical findings.Oscillatory behavior of a fractional partial differential equation.https://www.zbmath.org/1456.352262021-04-16T16:22:00+00:00"Wang, Jiangfeng"https://www.zbmath.org/authors/?q=ai:wang.jiangfeng"Meng, Fanwei"https://www.zbmath.org/authors/?q=ai:meng.fanweiSummary: In this paper, a fractional partial differential equation subject to the Robin boundary condition is considered. Based on the properties of Riemann-Liouville fractional derivative and a generalized Riccati technique, we obtained sufficient conditions for oscillation of the solutions of such equation. Examples are given to illustrate the main results.Remarks on the slow relaxation for the fractional Kuramoto model for synchronization.https://www.zbmath.org/1456.340612021-04-16T16:22:00+00:00"Ha, Seung-Yeal"https://www.zbmath.org/authors/?q=ai:ha.seung-yeal"Jung, Jinwook"https://www.zbmath.org/authors/?q=ai:jung.jinwookSummary: The collective behavior of an oscillatory system is ubiquitous in our nature, and one interesting issue in the dynamics of many-body oscillatory systems is the relaxation dynamics toward relative equilibria such as phase-locked states. For the Kuramoto model, relaxation dynamics occurs exponentially fast for generic initial data. However, some synchronization phenomena observed in our nature exhibit a slow subexponential relaxation. Thus, as one of the possible attempts for such slow relaxation, a second-order inertia term was added to the Kuramoto model in the previous literature so that the resulting second-order model can exhibit a slow relaxation dynamics for some range of inertia and coupling strength. In this paper, we present another Kuramoto type model exhibiting a slow algebraic relaxation. More precisely, our proposed model replaces the classical derivative by the Caputo fractional derivative in the original Kuramoto model. For this new model, we present several sufficient frameworks for fractional complete synchronization and practical synchronization.{
\copyright 2018 American Institute of Physics}Hidden attractors and multistability in a modified Chua's circuit.https://www.zbmath.org/1456.340572021-04-16T16:22:00+00:00"Wang, Ning"https://www.zbmath.org/authors/?q=ai:wang.ning"Zhang, Guoshan"https://www.zbmath.org/authors/?q=ai:zhang.guoshan"Kuznetsov, N. V."https://www.zbmath.org/authors/?q=ai:kuznetsov.nikolay-v"Bao, Han"https://www.zbmath.org/authors/?q=ai:bao.hanSummary: The first hidden chaotic attractor was discovered in a dimensionless piecewise-linear Chua's system with a special Chua's diode. But designing such physical Chua's circuit is a challenging task due to the distinct slopes of Chua's diode. In this paper, a modified Chua's circuit is implemented using a 5-segment piecewise-linear Chua's diode. In particular, the coexisting phenomena of hidden attractors and three point attractors are noticed in the entire period-doubling bifurcation route. Attraction basins of different coexisting attractors are explored. It is demonstrated that the hidden attractors have very small basins of attraction not being connected with any fixed point. The PSIM circuit simulations and DSP-assisted experiments are presented to illustrate the existence of hidden attractors and coexisting attractors.Quenching and restoration of oscillations under environmental interactions.https://www.zbmath.org/1456.340302021-04-16T16:22:00+00:00"Bera, Bidesh K."https://www.zbmath.org/authors/?q=ai:bera.bidesh-kSummary: Different types of collective dynamical behavior of the coupled oscillator networks are investigated under heterogeneous environmental coupling. This type of interaction pattern mainly occurs indirectly between two or more dynamical units. By interplaying the diffusive and the environmental coupling, the transition scenarios among several collective states, such as complete synchronization, amplitude, and oscillation death are explored in the coupled dynamical network. Here we consider a heterogeneous environmental coupling scheme, meaning that two or more dynamical units are not only connected via one medium but they can rely on their information through more than one medium. Another type of heterogeneity is introduced in terms of the coupling asymmetry in the interacting network structure and it is observed that the proper tuning of the coupling heterogeneity parameter is capable of restoring the dynamic rhythm from the oscillation suppressed state. Using detailed bifurcation analysis it is shown that the asymmetry parameter plays a key role in the transition among the several collective dynamical states and we map them in the different parameter space. We analytically derived the stability conditions for the existence of different dynamical states. The analytical findings are confirmed by numerical results. We performed the numerical simulation on networks of Stuart-Landau oscillators. Finally, we extend this investigation to large network sizes. In this case, we observe the novel transitions from amplitude death to multicluster oscillation death states and correspondingly, the revival of oscillations from the different suppressed states is also articulated.Measure-preserving symmetries and reversibilities of ordinary differential systems.https://www.zbmath.org/1456.370312021-04-16T16:22:00+00:00"Sabatini, Marco"https://www.zbmath.org/authors/?q=ai:sabatini.marcoThe author studies the dynamics induced by a class of \(n\)-dimensional ordinary autonomous differential systems with measure-preserving symmetries. It is proved that for such systems both divergence and the divergence derivatives along the solutions are preserved. The results obtained are applied to planar Lotka-Volterra and Liénard systems.
Reviewer: Valentine Tyshchenko (Grodno)Persistence time of solutions of the three-dimensional Navier-Stokes equations in Sobolev-Gevrey classes.https://www.zbmath.org/1456.351512021-04-16T16:22:00+00:00"Biswas, Animikh"https://www.zbmath.org/authors/?q=ai:biswas.animikh"Hudson, Joshua"https://www.zbmath.org/authors/?q=ai:hudson.joshua"Tian, Jing"https://www.zbmath.org/authors/?q=ai:tian.jingSummary: In this paper, we study existence times of strong solutions of the three-dimensional Navier-Stokes equations in time-varying analytic Gevrey classes based on Sobolev spaces \(H^s, s > \frac{1}{2}\). This complements the seminal work of \textit{C. Foias} and \textit{R. Temam} on \(H^1\) based Gevrey classes [J. Funct. Anal. 87, No. 2, 359--369 (1989; Zbl 0702.35203)], thus enabling us to improve estimates of the analyticity radius of solutions for certain classes of initial data. The main thrust of the paper consists in showing that the existence times in the much stronger Gevrey norms (i.e. the norms defining the analytic Gevrey classes which quantify the radius of real-analyticity of solutions) match the best known persistence times in Sobolev classes. Additionally, as in the case of persistence times in the corresponding Sobolev classes, our existence times in Gevrey norms are optimal for \(\frac{1}{2} < s < \frac{5}{2}\).Ultimate tumor dynamics and eradication using oncolytic virotherapy.https://www.zbmath.org/1456.340552021-04-16T16:22:00+00:00"Starkov, Konstantin E."https://www.zbmath.org/authors/?q=ai:starkov.konstantin-e"Kanatnikov, Anatoly N."https://www.zbmath.org/authors/?q=ai:kanatnikov.anatoly-n"Andres, Giovana"https://www.zbmath.org/authors/?q=ai:andres.giovanaSummary: In this paper we study ultimate dynamics of one three-dimensional model for tumor growth under oncolytic virotherapy which describes interactions between cytotoxic T-cells, uninfected tumor cells and infected tumor cells. Using the localization theorem of compact invariant sets we derive ultimate upper bounds for all cell populations and establish the property of the existence of the attracting set. Next, we find several conditions under which our system demonstrates convergence dynamics to equilibrium points located in invariant planes corresponding cases of the absence of uninfected or infected tumor cells. These assertions mean global eradication of uninfected or infected tumor cell populations and are presented as algebraic inequalities respecting virus replication rate \(\theta\). In particular, we find in Theorems 4 and 5 the following curious phenomenon. Namely, when we vary \(\theta\) from the instability range of the infected tumor free equilibrium point to the stability range we obtain in the latter range convergence dynamics to one of tumor free equilibrium points; this means that the local eradication of infected tumor cells implies their global eradication. Besides, we give conditions under which the infected tumor cell population persists. Our theoretical studies are supplied by results of numerical simulation.Stable and non-symmetric pitchfork bifurcations.https://www.zbmath.org/1456.340382021-04-16T16:22:00+00:00"Pujals, Enrique"https://www.zbmath.org/authors/?q=ai:pujals.enrique-ramiro"Shub, Michael"https://www.zbmath.org/authors/?q=ai:shub.michael"Yang, Yun"https://www.zbmath.org/authors/?q=ai:yang.yunAuthors' abstract: In this paper, we present a criterion for pitchfork bifurcations of smooth vector fields based on a topological argument. Our result expands Rajapakse and Smale's result significantly. Based on our criterion, we present a class of families of non-symmetric vector fields undergoing a pitchfork bifurcation.
Reviewer: Tao Li (Chengdu)Automorphic Schwarzian equations.https://www.zbmath.org/1456.110462021-04-16T16:22:00+00:00"Sebbar, Abdellah"https://www.zbmath.org/authors/?q=ai:sebbar.abdellah"Saber, Hicham"https://www.zbmath.org/authors/?q=ai:saber.hichamSummary: This paper concerns the study of the Schwartz differential equation \(\{h,\tau\}=s\operatorname{E}_4(\tau)\), where \(\operatorname{E}_4\) is the weight 4 Eisenstein series and \(s\) is a complex parameter. In particular, we determine all the values of \(s\) for which the solutions \(h\) are modular functions for a finite index subgroup of \(\operatorname{SL}_2({\mathbb{Z}})\). We do so using the theory of equivariant functions on the complex upper-half plane as well as an analysis of the representation theory of \(\operatorname{SL}_2({\mathbb{Z}})\). This also leads to the solutions to the Fuchsian differential equation \(y^{\prime\prime}+s\operatorname{E}_4y=0\).Limit cycles for a class of \(Z_p\)-equivariant differential systems.https://www.zbmath.org/1456.340252021-04-16T16:22:00+00:00"Gao, Jing"https://www.zbmath.org/authors/?q=ai:gao.jing"Zhao, Yulin"https://www.zbmath.org/authors/?q=ai:zhao.yulinThe authors consider a class of analytic planar autonomous systems which is \(\mathbb{Z}_p\)-equivariant (invariant under a special rotation) and depends on four parameters. In complex notation this class can be rewritten as
\[
\frac{{dz}}{{dt}} = (a+i)z+(b+i)z|z|^{2(p-2)}+cz|z|^2-\frac{{5i}}{{2}}\overline{z}^{p-1} \tag{1}
\]
with \(a,b,c \in \mathbb{R}, p \in \mathbb{N}\). The main result of the paper reads:
Theorem. For any \(p\) with \(4\le p \le \mathbb{N}\) there is a system of type (1) having at least \(2p\) limit cycles.
The proof is based on the bifurcation of limit cycles of a special Hamiltonian system by using the Melnikov function.
Reviewer: Klaus R. Schneider (Berlin)Probabilistic characteristics of noisy Van der Pol type oscillator with nonlinear damping.https://www.zbmath.org/1456.340322021-04-16T16:22:00+00:00"Dubkov, A. A."https://www.zbmath.org/authors/?q=ai:dubkov.alexander-a"Litovsky, I. A."https://www.zbmath.org/authors/?q=ai:litovsky.i-aResponse to: ``Comment on `The asymptotic iteration method revisited'''.https://www.zbmath.org/1456.340132021-04-16T16:22:00+00:00"Ismail, Mourad E. H."https://www.zbmath.org/authors/?q=ai:ismail.mourad-el-houssieny"Saad, Nasser"https://www.zbmath.org/authors/?q=ai:saad.nasserSummary: This response concers the Comment by \textit{F. M. Fernández} [J. Math. Phys. 61, No. 6, 064101, 2 p. (2020; Zbl 1456.34012)].
{\copyright 2020 American Institute of Physics}Comment on: ``The asymptotic iteration method revisited'''.https://www.zbmath.org/1456.340122021-04-16T16:22:00+00:00"Fernández, Francisco M."https://www.zbmath.org/authors/?q=ai:fernandez.francisco-mSummary: This comment concerns [\textit{M. E. H. Ismail} and \textit{N. Saad}, ibid. 61, No. 3, 033501, 12 p. (2020; Zbl 1443.34021)]. In this comment, we show that the eigenvalues of a quartic anharmonic oscillator obtained recently by means of the asymptotic iteration method may not be as accurate as the authors claim them to be.
{\copyright 2020 American Institute of Physics}Riesz basis of exponential family for a hyperbolic system.https://www.zbmath.org/1456.340682021-04-16T16:22:00+00:00"Intissar, Abdelkader"https://www.zbmath.org/authors/?q=ai:intissar.abdelkader"Jeribi, Aref"https://www.zbmath.org/authors/?q=ai:jeribi.aref"Walha, Ines"https://www.zbmath.org/authors/?q=ai:walha.inesOn a Hilbert space \(H\) the authors consider an unbounded linear operator \(A\). Their main comcern is on conditions for the eigenvectors of \(A\) to form a Riesz basis in \(H\). The following result is obtained: If \(A\) generates a \(C_0\) semigroup, the resolvent of \(A\) is compact, its eigenvaluse \(\{\lambda_n\}_{n\ge 1}\) are simple, the family \(\{ e^{\lambda_n}\}_{n\ge 1}\) is a Riesz basis for \(L^2(0,T), T>0,\) and the system of eigenvectors of \(A\) is complete in \(H\), then the eigenvectors of \(A\) form a Riesz basis in \(H\). Applications of this result are given.
Reviewer: Minh Van Nguyen (Little Rock)Extremal problems of the density for vibrating string equations with applications to gap and ratio of eigenvalues.https://www.zbmath.org/1456.340222021-04-16T16:22:00+00:00"Qi, Jiangang"https://www.zbmath.org/authors/?q=ai:qi.jiangang"Li, Jing"https://www.zbmath.org/authors/?q=ai:li.jing.13"Xie, Bing"https://www.zbmath.org/authors/?q=ai:xie.bingIn this paper, the authors obtain the infimum of the densities for vibrating string equations
\[
-y''=\lambda w y,\, y=y(x),\, x\in (0,1)
\]
together with Dirichlet-type boundary conditions \(y(0)=y(1)=0\) in terms of the gap and ratio of the first two eigenvalues. Here, the density \(w\) is a nonnegative integrable function on \([0, 1]\). As a main result of this investigation the authors prove a generalized version of the Lyapunov inequality involving the first two eigenvalues. Furthermore, they find some new estimates of the gap and ratio for the first and second eigenvalues of the above-mentioned boundary-value problem.
Reviewer: Erdogan Sen (Tekirdağ)Spectral expansion for singular conformable fractional Sturm-Liouville problem.https://www.zbmath.org/1456.340032021-04-16T16:22:00+00:00"Allahverdiev, Bilender P."https://www.zbmath.org/authors/?q=ai:allahverdiev.bilender-pasaoglu"Tuna, Hüseyin"https://www.zbmath.org/authors/?q=ai:tuna.huseyin"Yalçinkaya, Yüksel"https://www.zbmath.org/authors/?q=ai:yalcinkaya.yukselSummary: With this study, the spectral function for singular conformable fractional Sturm-Lioville problem is demonstrated. Further, we establish a Parseval equality and spectral expansion formula by terms of the spectral function.Approximate controllability for semilinear second-order stochastic evolution systems with infinite delay.https://www.zbmath.org/1456.340762021-04-16T16:22:00+00:00"Su, Xiaofeng"https://www.zbmath.org/authors/?q=ai:su.xiaofeng"Fu, Xianlong"https://www.zbmath.org/authors/?q=ai:fu.xianlongSummary: In this work, we study the approximate controllability for a class of semilinear second-order stochastic evolution systems with infinite delay. The main technique is the fundamental solution theory constructed through Laplace transformation. Some sufficient conditions for the approximate controllability result is obtained via the so-called resolvent condition and cosine family of linear operators. Due to the fundamental solution theory applied, the nonlinear terms are only required to be partly uniformly bounded. Finally, an example is provided to illustrate the obtained results.Bessel inequality and the basis property for a \(2m\times 2m\) Dirac type system with an integrable potential.https://www.zbmath.org/1456.340822021-04-16T16:22:00+00:00"Kurbanov, V. M."https://www.zbmath.org/authors/?q=ai:kurbanov.vali-m"Gadzhieva, G. R."https://www.zbmath.org/authors/?q=ai:gadzhieva.g-rSummary: We study the Bessel and basis properties of root vector functions of a Dirac type operator with an integrable potential. Criteria for the Bessel and basis properties are established, and a theorem on the equivalent basis property in \(L_2^{2m}(0,2\pi )\) is proved.Calculating the Lyapunov exponents of a piecewise-smooth soft impacting system with a time-delayed feedback controller.https://www.zbmath.org/1456.370972021-04-16T16:22:00+00:00"Zhang, Zhi"https://www.zbmath.org/authors/?q=ai:zhang.zhi"Liu, Yang"https://www.zbmath.org/authors/?q=ai:liu.yang.8|liu.yang.21|liu.yang.23|liu.yang.3|liu.yang.9|liu.yang.19|liu.yang.20|liu.yang.6|liu.yang.12|liu.yang.11|liu.yang.2|liu.yang.17|liu.yang.4|liu.yang.15|liu.yang.1|liu.yang.5|liu.yang|liu.yang.16|liu.yang.10|liu.yang.22|liu.yang.18|liu.yang.14|liu.yang.13"Sieber, Jan"https://www.zbmath.org/authors/?q=ai:sieber.janSummary: Lyapunov exponent is a widely used tool for studying dynamical systems. When calculating Lyapunov exponents for piecewise-smooth systems with time-delayed arguments one faces a lack of continuity in the variational problem. This paper studies how to build a variational equation for the efficient construction of Jacobians along trajectories of a delayed nonsmooth system. Trajectories of a piecewise-smooth system may encounter the so-called grazing event where the trajectory approaches a discontinuity surface in the state space in a non-transversal manner. For this event we develop a grazing point estimation algorithm to ensure the accuracy of trajectories for the nonlinear and the variational equations. We show that the eigenvalues of the Jacobian matrix computed by the algorithm converge with an order consistent with the order of the numerical integration method, therefore guaranteeing the reliability of the proposed numerical method. Finally, the method is demonstrated on a periodically forced impacting oscillator under the time-delayed feedback control.Coupled systems of linear differential-algebraic and kinetic equations with application to the mathematical modelling of muscle tissue.https://www.zbmath.org/1456.340512021-04-16T16:22:00+00:00"Plunder, Steffen"https://www.zbmath.org/authors/?q=ai:plunder.steffen"Simeon, Bernd"https://www.zbmath.org/authors/?q=ai:simeon.berndSummary: We consider a coupled system composed of a linear differential-algebraic equation (DAE) and a linear large-scale system of ordinary differential equations where the latter stands for the dynamics of numerous identical particles. Replacing the discrete particles by a kinetic equation for a particle density, we obtain in the mean-field limit the new class of partially kinetic systems.
We investigate the influence of constraints on the kinetic theory of those systems and present necessary adjustments. We adapt the mean-field limit to the DAE model and show that index reduction and the mean-field limit commute. As a main result, we prove Dobrushin's stability estimate for linear systems. The estimate implies convergence of the mean-field limit and provides a rigorous link between the particle dynamics and their kinetic description.
Our research is inspired by mathematical models for muscle tissue where the macroscopic behaviour is governed by the equations of continuum mechanics, often discretised by the finite element method, and the microscopic muscle contraction process is described by Huxley's sliding filament theory. The latter represents a kinetic equation that characterises the state of the actin-myosin bindings in the muscle filaments. Linear partially kinetic systems are a simplified version of such models, with focus on the constraints.
For the entire collection see [Zbl 1445.34004].Controlling subharmonic generation by vibrational and stochastic resonance in a bistable system.https://www.zbmath.org/1456.700352021-04-16T16:22:00+00:00"Sarkar, Prasun"https://www.zbmath.org/authors/?q=ai:sarkar.prasun"Paul, Shibashis"https://www.zbmath.org/authors/?q=ai:paul.shibashis"Ray, Deb Shankar"https://www.zbmath.org/authors/?q=ai:ray.deb-shankarA characterization of generalized exponential dichotomy.https://www.zbmath.org/1456.340092021-04-16T16:22:00+00:00"Wang, Liugen"https://www.zbmath.org/authors/?q=ai:wang.liugen"Xia, Yonghui"https://www.zbmath.org/authors/?q=ai:xia.yonghui"Zhao, Ninghong"https://www.zbmath.org/authors/?q=ai:zhao.ninghongGiven the linear system
\[
x'(t) = A(t)x(t) \tag{1}
\]
\((x(t)\) a \(n\)-vector, \(A(t)\) a continuous \(n \times n\) matrix) let \(X(t)\) be a fundamental matrix. The system has an \textit{exponential dichotomy} if there exist a projection \(P\) and two positive constants \(K, \alpha\) with
\begin{align*}
\|X(t)PX^{-1}(s)\| &\le K \exp(-\alpha(t - s)) \quad (t \ge s) \\
\|X(t)(I - P)X^{-1}(s)\| &\le K \exp(\alpha(t - s)) \hskip 0.8em \quad (t \le s)\tag{2}
\end{align*}
(for \(A(t) = A\) existence of an exponential dichotomy means absence of purely imaginary eigenvalues). This definition is generalized by the authors to that of \textit{generalized exponential dichotomy}, where (2) becomes
\[
\begin{aligned}
\|X(t)PX^{-1}(s)\| &\le K \exp\bigg(- \int_s^t \alpha(\tau) d\tau\bigg) \quad (t \ge s) \\
\|X(t)(I - P)X^{-1}(s)\| &\le K \exp \bigg( \int_s^t \alpha(\tau) d\tau \bigg) \hskip 0.8em \quad (t \le s)
\end{aligned}
\]
with \(\alpha(t)\) a nonnegative continuous function satisfying
\[
\int_{-\infty}^0 \alpha(\tau) d\tau = \int_0^\infty \alpha(\tau) d\tau = + \infty \, . \tag{3}
\]
For instance, if \(\alpha(t)\) satisfies (3) the system \(x'(t) = -\alpha(t), y'(t) = \alpha(t)\) has a generalized exponential dichotomy but not necessarily an exponential dichotomy. The system (1) is of \textit{generalized bounded growth} if
\[
\|X(t) X^{-1}(s)\| \le \mu \exp \bigg( \int_s^t \rho(\tau) d\tau \bigg) \quad (t \ge s)
\]
where \(\mu \ge 1\) and the function \(\rho(t)\) is continuous, nonnegative and nonincreasing. The authors prove several results on existence of a generalized exponential dichotomy for a system (1) having generalized bounded growth. The last theorem relates existence of a generalized exponential dichotomy for (1) and existence of nontrivial bounded solutions.
Reviewer: Hector O. Fattorini (Los Angeles)Inferring topologies via driving-based generalized synchronization of two-layer networks.https://www.zbmath.org/1456.340642021-04-16T16:22:00+00:00"Wang, Yingfei"https://www.zbmath.org/authors/?q=ai:wang.yingfei"Wu, Xiaoqun"https://www.zbmath.org/authors/?q=ai:wu.xiaoqun"Feng, Hui"https://www.zbmath.org/authors/?q=ai:feng.hui"Lu, Jun-An"https://www.zbmath.org/authors/?q=ai:lu.junan"Xu, Yuhua"https://www.zbmath.org/authors/?q=ai:xu.yuhuaEffects of time delay on the synchronized states of globally coupled network.https://www.zbmath.org/1456.370872021-04-16T16:22:00+00:00"Nag, Mayurakshi"https://www.zbmath.org/authors/?q=ai:nag.mayurakshi"Poria, Swarup"https://www.zbmath.org/authors/?q=ai:poria.swarupSummary: The effects of the time delay on the stability of different synchronized states of a globally coupled network are investigated. Conditions for the stability of the synchronized fixed points, synchronized periodic orbits, or synchronized chaos in a network of globally coupled chaotic smooth maps over a ring lattice with a homogeneous delay are derived analytically. Our analysis reveals that the stability properties of the synchronized dynamics are significantly different for odd and even time delays. The conditions for the stability of a synchronized fixed point and synchronized period-2 orbits for both odd and even delays are determined analytically. The range of parameter values for the stability of synchronized chaos has been calculated for a unit delay. All theoretical results are illustrated with the help of numerical examples.
{\copyright 2020 American Institute of Physics}Rigid local systems on \(\mathbb{A}^{1}\) with finite monodromy.https://www.zbmath.org/1456.112322021-04-16T16:22:00+00:00"Katz, Nicholas M."https://www.zbmath.org/authors/?q=ai:katz.nicholas-mSummary: We formulate some conjectures about the precise determination of the monodromy groups of certain rigid local systems on \(\mathbb{A}^{1}\) whose monodromy groups are known, by results of Kubert, to be finite. We prove some of them.Synchronization and spatial patterns in a light-dependent neural network.https://www.zbmath.org/1456.340492021-04-16T16:22:00+00:00"Liu, Yong"https://www.zbmath.org/authors/?q=ai:liu.yong.2|liu.yong.1|liu.yong.3|liu.yong.5|liu.yong.4"Xu, Ying"https://www.zbmath.org/authors/?q=ai:xu.ying"Ma, Jun"https://www.zbmath.org/authors/?q=ai:ma.junSummary: Nonlinear oscillators and networks can be synchronized by channel coupling for signal exchange, while non-coupling synchronization between chaotic oscillators can be obtained by applying the same stochastic disturbance for inducing resonance. For most of realistic dynamical systems, physical energy and biophysical energy are pumped along the coupling channels and then the variables are regulated to present different modes in oscillation. In this paper, a new photosensitive neuron is proposed to detect the dynamics in isolated neuron and synchronization stability by changing the illumination, which can adjust the photocurrent across the branch circuit, even no direct synapse coupling is applied. The generation of photocurrents with diversity is explained from physical viewpoint. Furthermore, the collective responses of these photosensitive neurons in network are detected by calculating the synchronization stability and pattern formation. It is found that the spatial patterns in the network are dependent on the illumination. Uniform illumination can induce complete synchronization while non-uniform illumination can develop rich spatial patterns. Furthermore, uniform and stochastic photocurrents are imposed on all neurons to realize complete synchronization even synapse connection are removed from the network. These results can give potential guidance for designing functional neural circuits with potential application to identify optical signals as electronic eyes.Kuramoto model of synchronization: equilibrium and nonequilibrium aspects.https://www.zbmath.org/1456.340602021-04-16T16:22:00+00:00"Gupta, Shamik"https://www.zbmath.org/authors/?q=ai:gupta.shamik"Campa, Alessandro"https://www.zbmath.org/authors/?q=ai:campa.alessandro"Ruffo, Stefano"https://www.zbmath.org/authors/?q=ai:ruffo.stefanoForced oscillation of sublinear impulsive differential equations via nonprincipal solution.https://www.zbmath.org/1456.340292021-04-16T16:22:00+00:00"Mostepha, Naceri"https://www.zbmath.org/authors/?q=ai:mostepha.naceri"Özbekler, Abdullah"https://www.zbmath.org/authors/?q=ai:ozbekler.abdullahSummary: In this paper, we give new oscillation criteria for forced sublinear impulsive differential equations of the form
\[
\begin{cases}
(r(t)x^\prime)^\prime + q(t)|x|^{\gamma - 1} x = f(t), &t \neq \theta_i;\\
\Delta r(t) x^\prime + q_i|x|^{\gamma-1}x = f_i, &t= \theta_i,
\end{cases}
\]
where \(\gamma \in (0,1)\), under the assumption that associated homogenous linear equation
\[
\begin{cases}
(r(t)z^\prime)^\prime + q(t)z = 0, &t \neq \theta_i;\\
\Delta r (t)z^\prime + q_i z = 0, &t = \theta_i
\end{cases}
\]
is nonoscillatory.Hopf bifurcation of KdV-Burgers-Kuramoto system with delay feedback.https://www.zbmath.org/1456.371072021-04-16T16:22:00+00:00"Guan, Junbiao"https://www.zbmath.org/authors/?q=ai:guan.junbiao"Liu, Jie"https://www.zbmath.org/authors/?q=ai:liu.jie|liu.jie.3|liu.jie.4|liu.jie.5|liu.jie.2|liu.jie.1|liu.jie.7"Feng, Zhaosheng"https://www.zbmath.org/authors/?q=ai:feng.zhaoshengDynamics and optimal control of a Monod-Haldane predator-prey system with mixed harvesting.https://www.zbmath.org/1456.921222021-04-16T16:22:00+00:00"Liu, Xinxin"https://www.zbmath.org/authors/?q=ai:liu.xinxin"Huang, Qingdao"https://www.zbmath.org/authors/?q=ai:huang.qingdaoRelaxation for a class of control systems with unilateral constraints.https://www.zbmath.org/1456.340172021-04-16T16:22:00+00:00"Papageorgiou, Nikolaos S."https://www.zbmath.org/authors/?q=ai:papageorgiou.nikolaos-s"Vetro, Calogero"https://www.zbmath.org/authors/?q=ai:vetro.calogero"Vetro, Francesca"https://www.zbmath.org/authors/?q=ai:vetro.francescaThe paper is concerned with a nonlinear feedback control system of type
\[
\left\{\begin{array}{ll}-x' \in A(x(t))+f(t,x(t))u(t)\\
x(0)=x_0, \quad u(t) \in U(t,x(t))\end{array}\right.
\]
in a time interval \(T=[0,b]\). Here, \(A:D(A)\subset \mathbb{R}^N \to 2^{\mathbb{R}^N}\) is a maximal monotone mapping and the control constraint multifunction \(U:T\times \mathbb{R}^N \to 2^{\mathbb{R}^N} \setminus \{ \emptyset \}\) has nonconvex values. It is assumed that \(U(t, \cdot )\) is lower semicontinuous for a.a. \(t\in T\). The authors introduce a control relaxed system by \(Q\)-regularization (in the sense of Cesari). Then, they show that every original state is a relaxed state and the set of the original states is dense in the set of the relaxed states, which is closed in \(C(T, \mathbb{R}^N)\).
Reviewer: Petru Jebelean (Timişoara)Stochastic transitions between in-phase and anti-phase synchronization in coupled map-based neural oscillators.https://www.zbmath.org/1456.370542021-04-16T16:22:00+00:00"Bashkirtseva, Irina"https://www.zbmath.org/authors/?q=ai:bashkirtseva.irina-adolfovna"Ryashko, Lev"https://www.zbmath.org/authors/?q=ai:ryashko.lev-borisovich"Pisarchik, Alexander N."https://www.zbmath.org/authors/?q=ai:pisarchik.alexander-nSummary: A problem of mathematical modeling and analysis of complex oscillatory behavior in coupled nonlinear stochastic systems is considered. We study stochastic bifurcations and transitions between in-phase and anti-phase dynamics in two coupled map-based neural oscillators with regular and chaotic attractors. Interesting dynamical regimes of isolated and coupled oscillators are considered in a wide range of both deterministic and stochastic modes associated with in-phase and anti-phase synchronization. The comprehensive nonlinear and stochastic analyses using a stochastic sensitivity approach and confidence ellipses allowed us to reveal the geometry of multiple basins of attraction of coexisting states and confidence areas in both synchronous and asynchronous regimes. Coupling-induced and noise-induced transitions between order and chaos are also discussed.Stochastic resonance in a non-Poissonian dichotomous process: a new analytical approach.https://www.zbmath.org/1456.600912021-04-16T16:22:00+00:00"Bologna, Mauro"https://www.zbmath.org/authors/?q=ai:bologna.mauro"Chandía, Kristopher J."https://www.zbmath.org/authors/?q=ai:chandia.kristopher-j"Tellini, Bernardo"https://www.zbmath.org/authors/?q=ai:tellini.bernardoOn the existence of three solutions for some classes of two-point semi-linear and quasi-linear differential equations.https://www.zbmath.org/1456.340232021-04-16T16:22:00+00:00"Saiedinezhad, Somayeh"https://www.zbmath.org/authors/?q=ai:saiedinezhad.somayehThe paper under review deals with the semi-linear boundary-value problem which consists the differential equation
\[
u''(t)+\lambda f(u)=0\, t\in(0,1),
\]
and Dirichlet type boundary conditions \(u(0)=u(1)=0\).
The author prove the existence of three solutions for this two-point boundary value problem in an appropriate Sobolev space. Furthermore, some existence results for a quasi-linear Dirichlet problem is obtained.
The results in the paper can be considered as an extension and generalizations of results of the paper [\textit{G. Bonannao}, Appl. Math. Lett. 13, No. 5, 53--57 (2000; Zbl 1009.34019)].
Reviewer: Erdogan Sen (Tekirdağ)A generalized fractional-order Chebyshev wavelet method for two-dimensional distributed-order fractional differential equations.https://www.zbmath.org/1456.651302021-04-16T16:22:00+00:00"Do, Quan H."https://www.zbmath.org/authors/?q=ai:do.quan-h"Ngo, Hoa T. B."https://www.zbmath.org/authors/?q=ai:ngo.hoa-t-b"Razzaghi, Mohsen"https://www.zbmath.org/authors/?q=ai:razzaghi.mohsenSummary: We provide a new effective method for the two-dimensional distributed-order fractional differential equations (DOFDEs). The technique is based on fractional-order Chebyshev wavelets. An exact formula involving regularized beta functions for determining the Riemann-Liouville fractional integral operator of these wavelets is given. The given wavelets and this formula are utilized to find the solutions of the given two-dimensional DOFDEs. The method gives very accurate results. The given numerical examples support this claim.On the Riesz basisness of root functions of a Sturm-Liouville operator with conjugate conditions.https://www.zbmath.org/1456.340242021-04-16T16:22:00+00:00"Cabri, O."https://www.zbmath.org/authors/?q=ai:cabri.olgun"Mamedov, K. R."https://www.zbmath.org/authors/?q=ai:mamedov.khanlar-rIn this study, authors consider the discontinuous Sturm-Liouville operator
\[
l(y) =
\begin{cases}
l_1 (y_1), \quad & x \in (-1,0) \\
l_2 (y_2), \quad & x \in (0,1)
\end{cases}
\]
where
\[
l_1(y_1)= y_1''+q_1(x)y_1,\, l_2(y_2)=y_2''+q_2(x)y_2,
\]
\(q_1(x) \in C^1[-1,0)\) and \(q_1(x) \in C^1(0,1]\) are complex-valued functions and have finite limits \(\lim_{x \to \mp 0}q_k(x)\) for \(k=1, 2\).
The authors deal with the problem of the operator \(l(y)\) with the periodic boundary conditions and with conjugate conditions. Both conjugate conditions have different finite one-sided limits at the point zero. By using the fundamental solution of problem, the asymptotic expression of eigenvalues and eigenfunctions are obtained. By means of asymptotic formulas of eigenfunctions and Bessel properties of eigenfunctions, Riesz basisness of the root functions the boundary value problem is proved. Similar spectral properties are studied for same operator with anti-periodic boundary conditions and with the same conjugate conditions.
Reviewer: Rakib Efendiev (Baku)Lyapunov exponents of stochastic systems -- from micro to macro.https://www.zbmath.org/1456.826732021-04-16T16:22:00+00:00"Laffargue, Tanguy"https://www.zbmath.org/authors/?q=ai:laffargue.tanguy"Tailleur, Julien"https://www.zbmath.org/authors/?q=ai:tailleur.julien"van Wijland, Frédéric"https://www.zbmath.org/authors/?q=ai:van-wijland.fredericThe unique periodic solution of Abel's differential equation.https://www.zbmath.org/1456.340422021-04-16T16:22:00+00:00"Hua, Ni"https://www.zbmath.org/authors/?q=ai:hua.niConsider the scalar differential equation
\[
\frac{{dx}}{{dt}} = a(t) x^3 + b(t) x^2 + c(t) x+ d(t) \tag{1}
\]
under the assumption that \(a,b,c,d\) are continuous \(\omega\)-periodic functions. The authors prove that under
the additional conditions
(\(i\)). \( a(t) <0 \quad (a(t)>0)\)
(\(ii\)). \(b^2(t)-3a(t)<0\)
equation (1) has a unique \(\omega\)-periodic solution which is asymptotically stable (unstable).
Reviewer: Klaus R. Schneider (Berlin)Numerical solution of backward fuzzy SDEs with time delayed coefficients.https://www.zbmath.org/1456.650092021-04-16T16:22:00+00:00"Zabiba, Mohammed S."https://www.zbmath.org/authors/?q=ai:zabiba.mohammed-s"Falah, Sarhan"https://www.zbmath.org/authors/?q=ai:falah.sarhanSummary: In this work, we consider the fuzzy stochastic differential delay equation and study the numerical solution of backward fuzzy stochastic differential delay equations (FSDDEs). Finally, we examine numerical convergence for FSDDEs.Existence of metastable, hyperchaos, line of equilibria and self-excited attractors in a new hyperjerk oscillator.https://www.zbmath.org/1456.370502021-04-16T16:22:00+00:00"Rajagopal, Karthikeyan"https://www.zbmath.org/authors/?q=ai:rajagopal.karthikeyan"Singh, Jay Prakash"https://www.zbmath.org/authors/?q=ai:singh.jay-prakash"Karthikeyan, Anitha"https://www.zbmath.org/authors/?q=ai:karthikeyan.anitha"Roy, Binoy Krishna"https://www.zbmath.org/authors/?q=ai:roy.binoy-krishnaSpectral inclusion and pollution for a class of dissipative perturbations.https://www.zbmath.org/1456.811822021-04-16T16:22:00+00:00"Stepanenko, Alexei"https://www.zbmath.org/authors/?q=ai:stepanenko.alexeiSummary: Spectral inclusion and spectral pollution results are proved for sequences of linear operators of the form \(T_0 + i \gamma s_n\) on a Hilbert space, where \(s_n\) is strongly convergent to the identity operator and \(\gamma > 0\). We work in both an abstract setting and a more concrete Sturm-Liouville framework. The results provide rigorous justification for a method of computing eigenvalues in spectral gaps.
{\copyright 2021 American Institute of Physics}Pure point spectrum for the Maryland model: a constructive proof.https://www.zbmath.org/1456.370532021-04-16T16:22:00+00:00"Jitomirskaya, Svetlana"https://www.zbmath.org/authors/?q=ai:jitomirskaya.svetlana-ya"Yang, Fan"https://www.zbmath.org/authors/?q=ai:yang.fan.6Summary: We develop a constructive method to prove and study pure point spectrum for the Maryland model with Diophantine frequencies.Expressions of meromorphic solutions of a certain type of nonlinear complex differential equations.https://www.zbmath.org/1456.300572021-04-16T16:22:00+00:00"Chen, Jun-Fan"https://www.zbmath.org/authors/?q=ai:chen.junfan"Lian, Gui"https://www.zbmath.org/authors/?q=ai:lian.guiSummary: In this paper, the expressions of meromorphic solutions of the following nonlinear complex differential equation of the form \[f^n+Q_d(z,f)=\sum_{i=1}^3p_i(z)e^{\alpha_i(z)}\] are studied by using Nevanlinna theory, where \(n\geq5\) is an integer, \(Q_d(z,f)\) is a differential polynomial in \(f\) of degree \(d\leq n-4\)~with rational functions as its coefficients, \(p_1(z), p_2(z), p_3(z)\)~are non-vanishing rational functions, and \(\alpha_1(z), \alpha_2(z), \alpha_3(z)\) are nonconstant polynomials such that \(\alpha_1'(z), \alpha_2'(z), \alpha_3'(z)\) are distinct each other. Moreover, examples are given to illustrate the accuracy of the condition.Positive solutions for nonlinear problems involving the one-dimensional \(\phi\)-Laplacian.https://www.zbmath.org/1456.340212021-04-16T16:22:00+00:00"Kaufmann, Uriel"https://www.zbmath.org/authors/?q=ai:kaufmann.uriel"Milne, Leandro"https://www.zbmath.org/authors/?q=ai:milne.leandroSummary: Let \({\Omega} : = (a, b) \subset \mathbb{R}\), \(m \in L^1(\Omega)\) and \(\lambda > 0\) be a real parameter. Let \(\mathcal{L}\) be the differential operator given by \(\mathcal{L} u : = - \phi(u^\prime)^\prime + r(x) \phi(u)\), where \(\phi : \mathbb{R} \rightarrow \mathbb{R}\) is an odd increasing homeomorphism and \(0 \leq r \in L^1(\Omega)\). We study the existence of \textit{positive} solutions for problems of the form
\[
\begin{cases} \mathcal{L} u = \lambda m(x) f(u) & \text{in } \Omega, \\ u = 0 & \text{on } \partial \Omega, \end{cases}
\]
where \(f : [0, \infty) \rightarrow [0, \infty)\) is a continuous function which is, roughly speaking, sublinear with respect to \(\phi\). Our approach combines the sub and supersolution method with some estimates on related nonlinear problems. We point out that our results are new even in the cases \(r \equiv 0\) and/or \(m \geq 0\).Assessing observability of chaotic systems using delay differential analysis.https://www.zbmath.org/1456.370902021-04-16T16:22:00+00:00"Gonzalez, Christopher E."https://www.zbmath.org/authors/?q=ai:gonzalez.christopher-e"Lainscsek, Claudia"https://www.zbmath.org/authors/?q=ai:lainscsek.claudia-s-m"Sejnowski, Terrence J."https://www.zbmath.org/authors/?q=ai:sejnowski.terrence-j"Letellier, Christophe"https://www.zbmath.org/authors/?q=ai:letellier.christopheSummary: Observability can determine which recorded variables of a given system are optimal for discriminating its different states. Quantifying observability requires knowledge of the equations governing the dynamics. These equations are often unknown when experimental data are considered. Consequently, we propose an approach for numerically assessing observability using Delay Differential Analysis (DDA). Given a time series, DDA uses a delay differential equation for approximating the measured data. The lower the least squares error between the predicted and recorded data, the higher the observability. We thus rank the variables of several chaotic systems according to their corresponding least square error to assess observability. The performance of our approach is evaluated by comparison with the ranking provided by the symbolic observability coefficients as well as with two other data-based approaches using reservoir computing and singular value decomposition of the reconstructed space. We investigate the robustness of our approach against noise contamination.
{\copyright 2020 American Institute of Physics}Inverse problems for Sturm-Liouville differential operators with two constant delays under Robin boundary conditions.https://www.zbmath.org/1456.340752021-04-16T16:22:00+00:00"Vojvodic, Biljana"https://www.zbmath.org/authors/?q=ai:vojvodic.biljana-m|vojvodich.bilyana"Pikula, Milenko"https://www.zbmath.org/authors/?q=ai:pikula.milenko"Vladicic, Vladimir"https://www.zbmath.org/authors/?q=ai:vladicic.vladimirThe paper deals with the boundary value problems \(D_{ki}\), \(k,i=1,2\) of the form
\[
-y''(x)+q_1(x)y(x-\tau_1)+(-1)^{k-1} q_2(x)y(x-\tau_2)=\lambda y(x),\quad 0<x<\pi,
\]
\[
y'(0)-h_iy(0)=y'(\pi)+Hy(\pi)=0,
\]
where \(\tau\in [\pi/2,\pi),\) and \(q_j(x)=0\) for \(x<\tau_j\). The authors study the inverse problem of recovering the functions \(q_j\) and the coefficients \(\tau_j, h_i, H\) from the given four spectra of the boundary value problems \(D_{ki}\). The result of the paper is the uniqueness theorem along with an algorithm for constructing the solution of this inverse problem.
Reviewer: Vjacheslav Yurko (Saratov)Bistable labyrinth-like structures and chimera states in a 2D lattice of van der Pol oscillators.https://www.zbmath.org/1456.340352021-04-16T16:22:00+00:00"Shepelev, Igor A."https://www.zbmath.org/authors/?q=ai:shepelev.igor-aleksandrovich"Anishchenko, V. S."https://www.zbmath.org/authors/?q=ai:anishchenko.vadim-semenovichSummary: The present work is devoted to the numerical analysis of the dynamics of a 2D lattice of coupled van der Pol oscillators in the regime of relaxation oscillations. It is shown that the influence of coupling leads to the shift of effective values of the control parameters of individual oscillators. The strong coupling can even cause the transition to bistable dynamics which is never observed in a single van der Pol oscillator. The numerically constructed phase-parametric diagram that takes into account the shifts of parameters shows that the bistability arises through the pitchfork bifurcation when varying the coupling strength. The lattice dynamics is analyzed when the control and coupling parameters are varied within a wide range, and the regime diagrams are constructed in the planes of system parameters. A new type of spatiotemporal pattern, a so-called ``labyrinth-like structure'', is found and described in detail. We also reveal for the first time and study a spiral wave chimera with a new kind of the incoherence core in the form of ``labyrinth-like structure''.Analysis of traveling waveform of flexible waveguides containing absorbent material along flanged junctions.https://www.zbmath.org/1456.740852021-04-16T16:22:00+00:00"Afzal, Muhammad"https://www.zbmath.org/authors/?q=ai:afzal.muhammad-zeshan|afzal.muhammad-u"Shafique, Sajid"https://www.zbmath.org/authors/?q=ai:shafique.sajid"Wahab, Abdul"https://www.zbmath.org/authors/?q=ai:wahab.abdul-fatahSummary: The traveling waveform of a flexible waveguide bounded by elastic plates and with an inserted expansion chamber having flanges at two junctions and a finite elastic membrane atop is investigated through a mode-matching technique. The modeled problem is governed by Helmholtz's equation and includes Dirichlet and higher-order boundary conditions. An acoustically absorbent lining is placed along the inner sides of the flanges at the junctions while their outer sides are kept rigid. Moreover, the edge conditions are imposed to define the physical behavior of the elastic membrane and plates at finite edges. The configuration is excited with the structure as well as fluid-borne modes. The influence of the imposed edge conditions at the connections of the plates and the prescribed incident forcing on the transmission-loss along the duct is elaborated. Specifically, the effects of edge conditions on the transmission-loss of structure-borne vibrations and fluid-borne noise are specified. The performance of low-frequency approximation is compared with that of the benchmark mode-matching method and is found to be in good agreement with relative merits. Apposite numerical simulations are performed to substantiate the validity of the mode-matching technique.Finite difference approximation of a generalized time-fractional telegraph equation.https://www.zbmath.org/1456.650612021-04-16T16:22:00+00:00"Delić, Aleksandra"https://www.zbmath.org/authors/?q=ai:delic.aleksandra"Jovanović, Boško S."https://www.zbmath.org/authors/?q=ai:jovanovic.bosko-s"Živanović, Sandra"https://www.zbmath.org/authors/?q=ai:zivanovic.sandra.1Summary: We consider a class of a generalized time-fractional telegraph equations. The existence of a weak solution of the corresponding initial-boundary value problem has been proved. A finite difference scheme approximating the problem is proposed, and its stability is proved. An estimate for the rate of convergence, in special discrete energetic Sobolev's norm, is obtained. The theoretical results are confirmed by numerical examples.Asymptotics of the solution to the boundary-value problems when limited equation has singular point.https://www.zbmath.org/1456.340662021-04-16T16:22:00+00:00"Kozhobekov, K. G."https://www.zbmath.org/authors/?q=ai:kozhobekov.kudaiberdi-gaparalievich"Erkebaev, U. Z."https://www.zbmath.org/authors/?q=ai:erkebaev.ulukbek-zairbekovich"Tursunov, D. A."https://www.zbmath.org/authors/?q=ai:tursunov.dilmurat-abdillazhanovichIn this paper, the authors deal with the two-point boundary value problem for a linear second-order ordinary differential equation
\[
\varepsilon y_{\varepsilon}''(x)-x^np(x)y_{\varepsilon}'(x)-q(x)y_{\varepsilon}(x)=f(x),\ p(x),q(x)>0,\ 0\leq x\leq1,
\]
with boundary condition of one of the three following conditions types:
\[
y_{\varepsilon}(0)=a,\ y_{\varepsilon}(1)=b \text{ (Dirichlet problem)},
\]
\[
y_{\varepsilon}'(0)=a,\ y_{\varepsilon}'(1)=b \text{ (Neumann problem)},
\]
\[
y_{\varepsilon}'(0)-h_1y_{\varepsilon}'(0)=a, \ y_{\varepsilon}'(1)+h_2y_{\varepsilon}'(1)=b,\ h_1,h_2>0,\ n\geq2\ \mathrm{(Robin\ problem)},
\]
wherethe corresponding reduced problem (Eq. with \(\varepsilon=0\)) has an irregular singular point \(x=0.\)
The goal of the paper is to construct a complete asymptotic expansion of the solution \(y_{\varepsilon}(t)\) on the interval \([0,1]\) (as \(\varepsilon\to 0^+\)) of the singularly perturbed problems with irregular singular points.
Reviewer: Robert Vrabel (Trnava)Topology trivialization transition in random non-gradient autonomous ODEs on a sphere.https://www.zbmath.org/1456.829132021-04-16T16:22:00+00:00"Fyodorov, Y. V."https://www.zbmath.org/authors/?q=ai:fyodorov.yan-vApproximate solution of Bagley-Torvik equations with variable coefficients and three-point boundary-value conditions.https://www.zbmath.org/1456.651832021-04-16T16:22:00+00:00"Huang, Q. A."https://www.zbmath.org/authors/?q=ai:huang.qiongao|huang.qiu-an"Zhong, X. C."https://www.zbmath.org/authors/?q=ai:zhong.xichang|zhong.xiaochun|zhong.xian-ci"Guo, B. L."https://www.zbmath.org/authors/?q=ai:guo.baolin|guo.boling|guo.baolongSummary: The fractional Bagley-Torvik equation with variable coefficients is investigated under three-point boundary-value conditions. By using the integration method, the considered problems are transformed into Fredholm integral equations of the second kind. It is found that when the fractional order is \(1< \alpha <2\), the obtained Fredholm integral equation is with a weakly singular kernel. When the fractional order is \(0< \alpha <1\), the given Fredholm integral equation is with a continuous kernel or a weakly singular kernel depending on the applied boundary-value conditions. The uniqueness of solution for the obtained Fredholm integral equation of the second kind with weakly singular kernel is addressed in continuous function spaces. A new numerical method is further proposed to solve Fredholm integral equations of the second kind with weakly singular kernels. The approximate solution is made and its convergence and error estimate are analyzed. Several numerical examples are computed to show the effectiveness of the solution procedures.On spectral curves and complexified boundaries of the phase-lock areas in a model of Josephson junction.https://www.zbmath.org/1456.340832021-04-16T16:22:00+00:00"Glutsyuk, A. A."https://www.zbmath.org/authors/?q=ai:glutsyuk.alexey-a"Netay, I. V."https://www.zbmath.org/authors/?q=ai:netay.igor-vA three-parameters family of special double confluent Heun equations, with real parameters \(l\), \(\lambda\), \(\mu\), is considered. The study of the real part of the spectral curve leads to applications to model of Josephson junction which is a family of dynamical systems on 2-torus depending on parameters \((B, A, \omega)\), where \(\omega\) is called the frequency. The authors provide an approach to study the boundaries of the phase-lock areas in \(\mathbb R^2(B, A)\) and their solutions, as \(\omega\) decreases to 0.They prove the irreducibility of the complex spectral curve \(\Gamma_l\) for every \(l\in\mathbb N\). They calculate its genus for \(l\le 20\) and present a conjecture on general genus formula. They apply the irreducibility result to the complexified boundaries of the phase-lock areas of model of Josephson junction. They show as well that its complexification is a complex analytic subset consisting of just four two-dimensional irreducible components, and describe them. They prove that the spectral curve has no real ovals. Finally, they present a Monotonicity Conjecture on the evolution of the phase-lock area portraits, as \(\omega\) decreases, and a partial positive result towards its confirmation.
Reviewer: Bertin Zinsou (Johannesburg)Solvability and stability of the inverse Sturm-Liouville problem with analytical functions in the boundary condition.https://www.zbmath.org/1456.340142021-04-16T16:22:00+00:00"Bondarenko, Natalia P."https://www.zbmath.org/authors/?q=ai:bondarenko.natalia-pThe paper deals with the boundary value problem
\[
-y''(x)+q(x)y(x)=\lambda y(x),\; 0<x<\pi,
\]
\[
y(0)=0,\; f_1(\lambda)y'(\pi)+f_2(\lambda)y(\pi)=0,
\]
where \(f_k(\lambda)\) are entire functions in \(\lambda.\) The author studies the inverse problem of recovering the potential \(q(x)\) from a part of the spectrum. Local and global solvability are established for the solution of this nonlinear inverse problem.
Reviewer: Vjacheslav Yurko (Saratov)Consensus dynamics on weighted multiplex networks: a long-range interaction perspective.https://www.zbmath.org/1456.340332021-04-16T16:22:00+00:00"Kumar, Rajesh"https://www.zbmath.org/authors/?q=ai:kumar.rajesh-s"Singh, Anurag"https://www.zbmath.org/authors/?q=ai:singh.anurag-k|singh.anuragFinite-time stability for differential inclusions with applications to neural networks.https://www.zbmath.org/1456.340162021-04-16T16:22:00+00:00"Matusik, Radosław"https://www.zbmath.org/authors/?q=ai:matusik.radoslaw"Nowakowski, Andrzej"https://www.zbmath.org/authors/?q=ai:nowakowski.andrzej-f"Plaskacz, Sławomir"https://www.zbmath.org/authors/?q=ai:plaskacz.slawomir"Rogowski, Andrzej"https://www.zbmath.org/authors/?q=ai:rogowski.andrzejThe paper studies differential inclusions of the form
\[
x'\in F(t,x),
\]
where \(F:[0,\infty )\times\mathbb{R}^n\to \mathcal{P}(\mathbb{R}^n)\) is a set-valued map with non-empty compact convex values. It is assumed that \(F(t,.)\) is upper semicontinuous, \(F\) satisfies a certain linear growth condition and that the origin is an equilibrium point (i.e, \(0\in F(t,0)\) for almost all \(t\in [0,\infty )\)).
By using a nonsmooth Lyapunov function, sufficient conditions for weak and strong finite-time stability are obtained in terms of contingent epiderivatives and hypoderivatives of the Lyapunov function.
An application to a class of Hopfield neural networks is also provided.
Reviewer: Aurelian Cernea (Bucharest)Memory-dependent derivative versus fractional derivative. I: Difference in temporal modeling.https://www.zbmath.org/1456.260102021-04-16T16:22:00+00:00"Wang, Jin-Liang"https://www.zbmath.org/authors/?q=ai:wang.jinliang.2"Li, Hui-Feng"https://www.zbmath.org/authors/?q=ai:li.huifengSummary: Since the memory-dependent derivative (MDD) was developed in 2011, it has become a new branch of Fractional Calculus which is still in the ascendant nowadays. How to understand MDD and fractional derivative (FD)? What are the advantages and disadvantages for them? How do they behave in Modeling? These questions guide going deep into the illustration of memory effect. Though the FD is defined on an interval, it mainly reflects the local change. Relative to the FD, the physical meaning of MDD is much clearer. The time-delay reflects the duration of memory effect, and the kernel function reflects the dependent weight. The results show that the MDD is more suitable for temporal modeling. In addition, a numerical algorithm for MDD is also developed here.A class of planar differential systems with explicit expression for two limit cycles.https://www.zbmath.org/1456.340282021-04-16T16:22:00+00:00"Hamizi, Saad Eddine"https://www.zbmath.org/authors/?q=ai:hamizi.saad-eddine"Boukoucha, Rachid"https://www.zbmath.org/authors/?q=ai:boukoucha.rachidSummary: The existence of limit cycles is interesting and very important in applications. It is a key to understand the dynamic of polynomial differential systems. The aim of this paper is to investigate a class of a multi-parameter planar polynomial differential systems. Under some suitable conditions, the existence of two limit cycles, these limit cycles are explicitly given. Some examples are presented in order to illustrate the applicability of our results. algebras.On conditional stability of Inverse scattering problem on a Lasso-shaped graph.https://www.zbmath.org/1456.340812021-04-16T16:22:00+00:00"Mochizuki, Kiyoshi"https://www.zbmath.org/authors/?q=ai:mochizuki.kiyoshi"Trooshin, Igor"https://www.zbmath.org/authors/?q=ai:trooshin.igor-yuSummary: We investigate the conditional stability of the inverse scattering problem on a lasso-shaped graph using the fundamental equation of inverse scattering theory.
For the entire collection see [Zbl 1415.35004].Spectral analysis for discontinuous non-self-adjoint singular Dirac operators with eigenparameter dependent boundary condition.https://www.zbmath.org/1456.470152021-04-16T16:22:00+00:00"Li, Kun"https://www.zbmath.org/authors/?q=ai:li.kun.2"Sun, Jiong"https://www.zbmath.org/authors/?q=ai:sun.jiong"Hao, Xiaoling"https://www.zbmath.org/authors/?q=ai:hao.xiaoling"Bao, Qinglan"https://www.zbmath.org/authors/?q=ai:bao.qinglanSummary: In this paper, a discontinuous non-self-adjoint (dissipative) Dirac operator with eigenparameter dependent boundary condition, and with two singular endpoints is studied. The interface conditions are imposed on the discontinuous point. Firstly, we pass the considered problem to a maximal dissipative operator \(L_h\) by using operator theoretic formulation. The self-adjoint dilation \(\mathcal{T}_h\) of \(L_h\) in the space \(\mathcal{H}\) is constructed, furthermore, the incoming and outgoing representations of \(\mathcal{T}_h\) and functional model are also constructed, hence in light of Lax-Phillips theory, we derive the scattering matrix. Using the equivalence between scattering matrix and characteristic function, a completeness theorem on the eigenvectors and associated vectors of this dissipative operator is proved.A minimal model of self-consistent partial synchrony.https://www.zbmath.org/1456.920182021-04-16T16:22:00+00:00"Clusella, Pau"https://www.zbmath.org/authors/?q=ai:clusella.pau"Politi, Antonio"https://www.zbmath.org/authors/?q=ai:politi.antonio-z"Rosenblum, Michael"https://www.zbmath.org/authors/?q=ai:rosenblum.michael-gIntegrability analysis of the Shimizu-Morioka system.https://www.zbmath.org/1456.340022021-04-16T16:22:00+00:00"Huang, Kaiyin"https://www.zbmath.org/authors/?q=ai:huang.kaiyin"Shi, Shaoyun"https://www.zbmath.org/authors/?q=ai:shi.shaoyun"Li, Wenlei"https://www.zbmath.org/authors/?q=ai:li.wenleiSummary: The aim of this paper is to give some new insights into the Shimizu-Morioka system
\[
\dot{x}=y,\quad \dot{y}=x-\lambda y-xz,\quad \dot{z}=-\alpha z+x^2,
\]
from the integrability point of view. Firstly, we propose a linear scaling in time and coordinates which converts the Shimizu-Morioka system into a special case of the Rucklidge system when \(\alpha\neq 0\) and discuss the relationship between Shimizu-Morioka system and Rucklidge system. Based on this observation, Darboux integrability of the Shimizu-Morioka system with \(\alpha\neq 0\) is trivially derived from the corresponding results on the Rucklidge system. When \(\alpha=0\), we investigate Darboux integrability of the Shimizu-Morioka system by the Gröbner basis in algebraic geometry. Secondly, we use the stability of the singular points and periodic orbits to study the nonexistence of global \(C^1\) first integrals of the Shimizu-Morioka system. Finally, in the case \(\alpha\neq 0\), we prove it is not rationally integrable for almost all parameter values by an extended Morales-Ramis theory, and in the case \(\alpha=0\), we show that it is not algebraically integrable by quasi-homogeneous decompositions and Kowalevski exponents. Our results are in accord with the fact that this system admits chaotic behaviors for a large range of its parameters.