Recent zbMATH articles in MSC 37https://www.zbmath.org/atom/cc/372022-05-16T20:40:13.078697ZUnknown authorWerkzeugOne-sided fattening of the graph in the real projective planehttps://www.zbmath.org/1483.050452022-05-16T20:40:13.078697Z"Choy, Jaeyoo"https://www.zbmath.org/authors/?q=ai:choy.jaeyoo"Chu, Hahng-Yun"https://www.zbmath.org/authors/?q=ai:chu.hahng-yunSummary: The one-sided fattenings (called semi-ribbon graph in this paper) of the graph embedded in the real projective plane \(\mathbb{RP}^2\) are completely classified up to topological equivalence. A planar graph (i.e., embedded in the plane), admitting the one-sided fattening, is known to be a cactus boundary. For the graphs embedded in \(\mathbb{RP}^2\) admitting the one-sided fattening, unlike the planar graphs, a new building block appears: a bracelet along the Möbius band, which is not a connected summand of the oriented surfaces.Ihara zeta function and cycle fluctuations of random Ramanujan graph generated by \(p\)-adic pseudorandom numbershttps://www.zbmath.org/1483.051632022-05-16T20:40:13.078697Z"Naito, Koichiro"https://www.zbmath.org/authors/?q=ai:naito.koichiroSummary: Using the sequences of the pseudorandom numbers generated by the \(p\)-adic logistic map, we construct some pseudorandom adjacency matrices and their various nonregular graphs, which are characterized as weak Ramanujan graphs. For these various types of random graphs, which have some clustering properties, we statistically calculate the fitting curves to the distributions of these cycle numbers by using their cycle basis and estimating the numbers of each cycle in these graphs. We can estimate their fluctuation exponents and we call this value the cycle-fluctuation exponent (CFE). Furthermore, using the distributions of poles of the Ihara zeta function, we introduce the Ramanujan radius ratio (RRr) and the Ramanujan radius standard deviation (RRd). Comparing the value CFE to the other parameters, the small-worldness coefficient (SWC), RRr and RRd and clarifying the relations among these parameters numerically, we characterize the clustering and random properties of our random Ramanujan graphs.Problems in number theory related to aperiodic orderhttps://www.zbmath.org/1483.110042022-05-16T20:40:13.078697Z"Lagarias, Jeffrey C."https://www.zbmath.org/authors/?q=ai:lagarias.jeffrey-cSummary: This talk concerns properties of dilated floor functions \(f_\alpha(x) = [ \alpha x]\), where \(\alpha\) takes a fixed real value.
For the entire collection see [Zbl 1459.37002].A dynamical proof of the prime number theoremhttps://www.zbmath.org/1483.110112022-05-16T20:40:13.078697Z"McNamara, Redmond"https://www.zbmath.org/authors/?q=ai:mcnamara.redmondSummary: We present a new, elementary, dynamical proof of the prime number theorem.A generalization of the Gauss-Kuzmin-Wirsing constanthttps://www.zbmath.org/1483.111422022-05-16T20:40:13.078697Z"Sun, Peng"https://www.zbmath.org/authors/?q=ai:sun.peng.1|sun.pengLet \(p\) be a positive integer. The present paper deals with the generalized Gauss transformation \(T_p(x)\) on \( [0, 1]\). That is, \(T_p(x)=\left\{\frac p x\right\}\) whenever \(x\ne 0\) and \(T_p(x)=0\) whenever \(x= 0\), where \(\{a\}\) is the fractional part of a number \(a\). It is noted that such transformations are related to \(p\)-continued fractions (\(N\)-continued fractions).
A brief survey of this paper is devoted to these continued fractions, to the Gauss-Kuzmin-Wirsing operator, and to the Gauss-Kuzmin problem.
Special attention is given to the Gauss-Kuzmin-Wirsing constant \(\lambda\) and to a certain Wirsing's result ([\textit{E. Wirsing}, Acta Arith. 24, 507--528 (1974; Zbl 0283.10032)]) which is generalized in the present paper. A near-optimal estimate for the generalized Gauss-Kuzmin-Wirsing constant \(\lambda_p\) is obtained.
Reviewer: Symon Serbenyuk (Kyïv)Higher-dimensional gap theorems for the maximum metrichttps://www.zbmath.org/1483.111432022-05-16T20:40:13.078697Z"Haynes, Alan"https://www.zbmath.org/authors/?q=ai:haynes.alan-k"Ramirez, Juan J."https://www.zbmath.org/authors/?q=ai:ramirez.juan-jThe celebrated three-distance theorem states that given a real number \(\alpha\) and a positive integer \(n\), there are at most three distances between consecutive points \(\{k\alpha\}_{k=0}^n\), when the points are taken on the circle viewed as the interval \([0,1)\) modulo \(1\). In recent joint work [Ann. Sci. Ec. Norm. Super. (4) 53, No. 2, 537--557 (2020; Zbl 1475.11130)], the first author extended this theorem to higher-dimensional toral rotations, providing upper bounds for any Riemannian metric. In the paper under review, the authors give upper bounds in all dimensions for the maximum metric. In particular, they show that in dimension \(d\), an upper bound for the number of distinct difference of neighbouring points is \(2^d+1\), and as well, they show that in dimensions \(2\) and \(3\) this bound is best possible.
Reviewer: Michael Coons (Callaghan)An extension of Weyl's equidistribution theorem to generalized polynomials and applicationshttps://www.zbmath.org/1483.111532022-05-16T20:40:13.078697Z"Bergelson, Vitaly"https://www.zbmath.org/authors/?q=ai:bergelson.vitaly"Håland Knutson, Inger J."https://www.zbmath.org/authors/?q=ai:haland-knutson.inger-j"Son, Younghwan"https://www.zbmath.org/authors/?q=ai:son.younghwanSummary: Generalized polynomials are mappings obtained from the conventional polynomials by the use of the operations of addition and multiplication and taking the integer part. Extending the classical theorem of Weyl on equidistribution of polynomials, we show that a generalized polynomial \(q(n)\) has the property that the sequence \((q(n) \lambda)_{n\in\mathbb{Z}}\) is well-distributed mod 1 for all but countably many \(\lambda\in\mathbb{R}\) if and only if \(\lim_{\substack{|n|\rightarrow \infty \\ n\notin J}}|q(n)|=\infty\) for some (possibly empty) set \(J\) having zero natural density in \(\mathbb{Z}\). We also prove a version of this theorem along the primes (which may be viewed as an extension of classical results of Vinogradov and Rhin). Finally, we utilize these results to obtain new examples of sets of recurrence and van der Corput sets.A fresh look at the notion of normalityhttps://www.zbmath.org/1483.111542022-05-16T20:40:13.078697Z"Bergelson, Vitaly"https://www.zbmath.org/authors/?q=ai:bergelson.vitaly"Downarowicz, Tomasz"https://www.zbmath.org/authors/?q=ai:downarowicz.tomasz"Misiurewicz, Michał"https://www.zbmath.org/authors/?q=ai:misiurewicz.michalThe point of departure of this paper is the classical notion of normality of a \(0\)-\(1\) sequence \((x_n)_{n\in\mathbb N}\in \{0,1\}^{\mathbb N}\): for any positive integer \(k\) and any word \(w\in\{0,1\}^k\), the asymptotic density of the set of indices \(n\in\mathbb N\) such that \((x_n,x_{n+1},\ldots,x_{n+k-1})=w\) has to be equal to \(2^{-k}\).
This definition is extended to Følner sequences \((F_n)_{n\in \mathbb N}\), which are sequences of finite subsets \(F_n\subset\mathbb N\) satisfying the Følner condition
\[
(\text{ for all } k\in\mathbb N) \lim_{n\rightarrow\infty}\frac{\lvert F_n\cap(F_n-k)\rvert}{\lvert F_n\rvert}=1.
\]
A sequence \(x\in\{0,1\}^{\mathbb N}\) is \((F_n)\)-normal if for all \(k\in\mathbb N\) and any \(w\in\{0,1\}^k\) we have
\begin{multline*}
\lim_{n\rightarrow\infty} \frac1{\lvert F_n\rvert} \bigl\lvert\bigl\{m\in\mathbb N: \{m,m+1,\ldots,m+k-1\}\subset F_n\mbox{ and } \\
(x_m,x_{m+1},\ldots,x_{m+k-1})=w\bigr\}\bigr\rvert =2^{-k}.
\end{multline*}
Considering only Følner sequences does not pose a restriction, since there do not exist \((F_n)\)-normal sequences with respect to non-Følner sequences \((F_n)\).
As the next step, the authors consider countably infinite amenable cancellative semigroups \(G\); such groups admit (left) Følner sequences and are used in place of the index set \(\mathbb N\) of the sequence \(x\). More precisely, let \((F_n)\) be a Følner sequence in \(G\). Suppose that \(x=(x_g)_{g\in G}:G\rightarrow\{0,1\}\), that \(K\subset G\) is finite, and \(B\in\{0,1\}^K\). Set
\[
\mathsf N(B,x,F_n) =\bigl\lvert\bigl\{g\in G\cup\{e\}:(\text{ for all } h\in K)hg\in F_n\mbox{ and }x_{hg}=B(h)\bigr\}\bigr\rvert.
\]
A \((F_n)\)-normal sequence is defined by the property that for any nonempty finite \(K\subset G\) and every \(B\in\{0,1\}^K\),
\[
\lim_{n\rightarrow\infty}\frac 1{\lvert F_n\rvert} \mathsf N(B,x,F_n)=2^{-\lvert K\rvert}.
\]
Under the condition that \(\lvert F_n\rvert\) is strictly increasing (in particular), it is shown in Theorem~4.2 that \(\lambda\)-almost every \(x\in\{0,1\}^G\) is \((F_n)\)-normal, where \(\lambda\) is the uniform product measure on \(\{0,1\}^G\). In a topological sense however, the set of \((F_n)\)-normal sequences \(x\in\{0,1\}^G\) is small, that is, of first category (Proposition~4.7).
In Section~5, the authors generalize the Champernowne-construction of a normal number to this general setting.
In the remaining sections, special emphasis is given to the semigroups \((\mathbb N,+)\) and \((\mathbb N,\times)\). In Section~6, a more transparent construction of a Champernowne-like normal sequence is given for the case \((\mathbb N,\times)\). Section~7 is concerned with normal subsets of \(\mathbb N\), and combinatorial and Diophantine properties of such sets are proved. Finally, in Section~8, the existence of \((F_n)\)-normal Liouville numbers is established, and a set of such numbers is constructed.
Reviewer: Lukas Spiegelhofer (Wien)A multifractal analysis for cuspidal windings on hyperbolic surfaceshttps://www.zbmath.org/1483.111602022-05-16T20:40:13.078697Z"Jaerisch, Johannes"https://www.zbmath.org/authors/?q=ai:jaerisch.johannes"Kesseböhmer, Marc"https://www.zbmath.org/authors/?q=ai:kessebohmer.marc"Munday, Sara"https://www.zbmath.org/authors/?q=ai:munday.saraAuthors' abstract: In this paper, we investigate the multifractal decomposition of the limit set of a finitely generated, free Fuchsian group with respect to the mean cusp-winding number. We completely determine its multifractal spectrum by means of a certain free energy function and show that the Hausdorff dimension of sets consisting of limit points with the same scaling exponent coincides with the Legendre transform of this free energy function. As a by-product we generalize previously obtained results on the multifractal formalism for infinite iterated function systems to the setting of infinite graph directed Markov systems.
Reviewer: Alexey Ustinov (Khabarovsk)Mixed multifractal spectra of Birkhoff averages for non-uniformly expanding one-dimensional Markov maps with countably many brancheshttps://www.zbmath.org/1483.111622022-05-16T20:40:13.078697Z"Jaerisch, Johannes"https://www.zbmath.org/authors/?q=ai:jaerisch.johannes"Takahasi, Hiroki"https://www.zbmath.org/authors/?q=ai:takahasi.hirokiThe aim of the present paper is to describe conditional variational formulas for mixed multifractal spectra of Birkhoff averages of countably many observables in terms of the Hausdorff dimension of invariant probability measures for non-uniformly expanding one-dimensional Markov maps with a countable (finite or infinite) number of branches and finitely many neutral periodic points. To this direction by using the above mentioned results the authors managed to exhibit new fractal-geometric results for backward continued fraction expansions of real numbers regarding a question of Pollicot as well as formulas for multi-cusp winding spectra for the Bowen-Series maps associated with finitely generated free Fuchsian groups with the parabolic elements.
Reviewer: Chryssoula Ganatsiou (Larissa)Weighted Hurwitz numbers, \(\tau\)-functions, and matrix integralshttps://www.zbmath.org/1483.140592022-05-16T20:40:13.078697Z"Harnad, J."https://www.zbmath.org/authors/?q=ai:harnad.johnThis paper deals with weighted Hurwitz Numbers, \(\tau\)-functions and matrix integrals. The basis elements spanning the Sato Grassmannian element corresponding to the KP \(\tau\)-function that serves as generating function for rationally weighted Hurwitz numbers are shown to be Meijer \(G\)-functions. Using their Mellin-Barnes integral representation the \(\tau\)-function, evaluated at the trace invariants of an externally coupled matrix, is expressed as a matrix integral. Using the Mellin-Barnes integral transform of an infinite product of \(G\)-functions, a similar matrix integral representation is given for the KP \(\tau\)-function that serves as generating function for quantum weighted Hurwitz numbers. This paper is organized as follows: Section 1 is devoted to Hurwitz numbers: classical and weighted, and Section 2 to hypergeometric \(\tau\)-functions as generating functions for weighted Hurwitz numbers. In theses Sections the author gives a brief review of this theory, together with two illustrative examples: rational and quantum weighted Hurwitz numbers. In Section 3, it is shown how evaluation of such \(\tau\)-functions at the trace invariants of a finite matrix may be expressed either as a Wronskian determinant or as a matrix integral.
For the entire collection see [Zbl 1471.81009].
Reviewer: Ahmed Lesfari (El Jadida)Multi-component universal character hierarchy and its polynomial tau-functionshttps://www.zbmath.org/1483.140602022-05-16T20:40:13.078697Z"Li, Chuanzhong"https://www.zbmath.org/authors/?q=ai:li.chuanzhong|li.chuanzhong.1The aim of this paper is to consider more general polynomial tau functions than universal characters of the UC hierarchy in [\textit{T. Tsuda}, Commun. Math. Phys. 248, No. 3, 501--526 (2004; Zbl 1233.37042)] which can be treated as a zero-mode of a certain generating tau function and the method used here is also a little different from what Tsuda have done in the above paper about UC hierarchy. Also the author generalizes these results of UC hierarchy to a multicomponent UC hierarchy. He finds that the polynomial tau-function in terms of universal characters of the universal character(UC) hierarchy which can be treated as a zero mode of an appropriate combinatorial generating function. After that, he defines a multi-component UC hierarchy and obtains polynomial tau-functions of the multi-component UC hierarchy. The paper is organized as follows : the first section is an introduction to the subject and formulation of the main result. Section 2 is devoted to symmetric functions. Section 3 deals with universal character and universal character hierarchy. Section 4 is devoted to polynomial tau-functions of the UC hierarchy and Section 5 to polynomial \(\tau\)-functions of the \(s\)-component UC hierarchy.
Reviewer: Ahmed Lesfari (El Jadida)Imaginary cone and reflection subgroups of Coxeter groupshttps://www.zbmath.org/1483.200742022-05-16T20:40:13.078697Z"Dyer, Matthew J."https://www.zbmath.org/authors/?q=ai:dyer.matthew-jLet \(V\) be an \(\mathbb{R}\)-vector space equipped with a symmetric bilinear form \(\langle -,- \rangle\). Suppose \((\Phi, \Pi)\) is a based root system in \(V\) with associated Coxeter system \((W,S)\). Denote
\[
\mathscr{C} = \{v \in V \mid \langle v, \alpha \rangle \ge 0, \forall \alpha \in \Pi\}, \text{ and } \mathscr{K} = (\mathbb{R}_{\ge 0} \Pi) \cap (- \mathscr{C}).
\]
Define the imaginary cone \(\mathscr{Z}\) to be
\[
\mathscr{Z} = \bigcup_{w \in W} w \mathscr{K}.
\]
This extends the notion for Kac-Moody Lie algebras studied in [\textit{V. G. Kac}, Infinite dimensional Lie algebras. Cambridge etc.: Cambridge University Press (1990; Zbl 0716.17022)]. The paper under review provides a survey on the imaginary cone, emphasizing its relationship with reflection subgroups. There are four main results in this paper, which were unknown previously, listed as follows.
Theorem 6.3./Theorem 12.2. Let \(W^\prime\) be a reflection subgroup of \(W\), then \(\mathscr{Z}_{W^\prime} \subseteq \mathscr{Z}\), where \(\mathscr{Z}_{W^\prime}\) is the imaginary cone of \(W^\prime\).
Theorem 7.6. Suppose \(W\) is irreducible, infinite, and of finite rank. Then \(\overline{\mathscr{Z}}\) is the unique non-zero \(W\)-invariant closed pointed cone contained in \(\mathbb{R}_{\ge 0} \Pi\).
Theorem 10.3. (sketched) Suppose \(W\) is of finite rank. (a) The imaginary cone and the Tits cone is a dual pair. (b) The lattice (i.e. poset) formed by faces of \(\mathscr{Z}\) is isomorphic to the lattice formed by facial subgroups without finite components. (c) The face lattice of \(\mathscr{Z}\) is dual to that of the Tits cone. (d) If \(W^\prime\) is a facial subgroup without finite components, then its imaginary cone and its Tits cone can be described explicitly by each other.
Theorem 12.3. One has \(\mathscr{Z} = \mathbb{R}_{\ge 0} (\bigcup_{W^\prime \in \daleth} \mathscr{Z}_{W^\prime})\), where \(\daleth\) is the set of dihedral reflection subgroups of \(W\).
Besides, the hyperbolic and universal cases are discussed in Section 9. In Section 13, some motivations and applications are presented, including the dominance order, limit roots, etc. In particular, the author mentioned that a weakened version of the boundedness conjecture on Lusztig's \(a\)-function can be proved, but no more details are given. There is a list of notations at the end, which is helpful in reading.
Reviewer: Hongsheng Hu (Beijing)Corrigendum to: ``Effective equidistribution of translates of large submanifolds in semisimple homogeneous spaces''https://www.zbmath.org/1483.220082022-05-16T20:40:13.078697Z"Ubis, Adrián"https://www.zbmath.org/authors/?q=ai:ubis.adrianFrom the text: We wish to issue a corrigendum for our paper [ibid. 2017, No. 18, 5629--5666 (2017; Zbl 1405.22014)]. As the paper is currently written, there is an error in the proof of Proposition 3.5. Precisely, we assume that \(f^v_* (g\Gamma) = \tilde{f}_*(u_v g\Gamma)\overline{\tilde{f}_*}(u_v g\Gamma)\) has vanishing integral, while the truth is that this integral is just small due to mixing (but not necessarily zero). This error will affect the definition of mixing function (Definition 3.4), and in turn the proofs and statements of Propositions 6.2 and 9.4, which will change the constants in the statements of Theorems 1.3 and 3.6.Welcome to real analysis. Continuity and calculus, distance and dynamicshttps://www.zbmath.org/1483.260012022-05-16T20:40:13.078697Z"Kennedy, Benjamin B."https://www.zbmath.org/authors/?q=ai:kennedy.benjamin-bPublisher's description: Welcome to Real Analysis is designed for use in an introductory undergraduate course in real analysis. Much of the development is in the setting of the general metric space. The book makes substantial use not only of the real line and \(n\)-dimensional Euclidean space, but also sequence and function spaces. Proving and extending results from single-variable calculus provides motivation throughout. The more abstract ideas come to life in meaningful and accessible applications. For example, the contraction mapping principle is used to prove an existence and uniqueness theorem for solutions of ordinary differential equations and the existence of certain fractals; the continuity of the integration operator on the space of continuous functions on a compact interval paves the way for some results about power series.
The exposition is exceedingly clear and well-motivated. There are a wide variety of exercises and many pedagogical innovations. For example, each chapter includes Reading Questions so that students can check their understanding. In addition to the standard material in a first real analysis course, the book contains two concluding chapters on dynamical systems and fractals as an illustration of the power of the theory developed.Pointwise equicontinuity of Zadeh's extension of an interval maphttps://www.zbmath.org/1483.260252022-05-16T20:40:13.078697Z"Sun, Taixiang"https://www.zbmath.org/authors/?q=ai:sun.taixiang"Su, Guangwang"https://www.zbmath.org/authors/?q=ai:su.guangwang"Qin, Bin"https://www.zbmath.org/authors/?q=ai:qin.binSummary: Let \(I\) be a compact interval and \(f:I\longrightarrow I\) be continuous. Assume that \(\mathcal{F}(I)\) is the set of fuzzy numbers on \(I\), and that \(\hat{f}:\mathcal{F}(I)\longrightarrow\mathcal{F}(I)\) is the Zadeh's extension of \(f\), and that \(\overset{\leftarrow}{\lim}\{\mathcal{F}(I),\hat{f}\}\) is the inverse limit space of \((\mathcal{F}(I),\hat{f})\), and \(\sigma_{\hat{f}}:\overset{\leftarrow}{\lim}\{\mathcal{F}(I),\hat{f}\}\longrightarrow\overset{\leftarrow}{\lim}\{\mathcal{F}(I),\hat{f}\}\) is the left shift map. In this paper, we study the pointwise equicontinuity of \(\hat{f}\) and show that the following statements are equivalent: (1) \(\hat{f}\) is pointwise equicontinuous. (2) For some infinite subsequence \(S\) oositive integers, \(\hat{f}\) is \(S\) pointwise equicontinuous. (3) \(\{\hat{f}^{2n}\}f p_{n=1}^\infty\) is uniformly convergent on \(\mathcal{F}(I)\). (4) \(\sigma_{\hat{f}}\) is a periodic map with period 2.Substituting the typical compact sets into a power serieshttps://www.zbmath.org/1483.280052022-05-16T20:40:13.078697Z"Nagy, Donát"https://www.zbmath.org/authors/?q=ai:nagy.donatSummary: The Minkowski sum and Minkowski product can be considered as the addition and multiplication of subsets of \(\mathbb R\). If we consider a compact subset \(K \subseteq [0,1]\) and a power series \(f\) which is absolutely convergent on \([0, 1]\), then we may use these operations and the natural topology of the space of compact sets to substitute the compact set \(K\) into the power series \(f\). Changhao Chen studied this kind of substitution in the special case of polynomials and showed that if we substitute the typical compact set \(K \subseteq [0,1]\) into a polynomial, we get a set of Hausdorff dimension 0. We generalize this result and show that the situation is the same for power series where the coefficients converge to zero quickly. On the other hand we also show a large class of power series where the result of the substitution has Hausdorff dimension one.On the spatial Julia set generated by fractional Lotka-Volterra system with noisehttps://www.zbmath.org/1483.280112022-05-16T20:40:13.078697Z"Wang, Yupin"https://www.zbmath.org/authors/?q=ai:wang.yupin"Liu, Shutang"https://www.zbmath.org/authors/?q=ai:liu.shutang"Li, Hui"https://www.zbmath.org/authors/?q=ai:li.hui.3|li.hui.1|li.hui.4|li.hui.5|li.hui|li.hui.2"Wang, Da"https://www.zbmath.org/authors/?q=ai:wang.daSummary: This paper investigates the structures and properties of the spatial Julia set generated by a fractional complex Lotka-Volterra system with noise. The influence of several types of dynamic noise upon the system's Julia set is quantitatively analyzed through the Julia deviation index. Then, the symmetry of the Julia set is discussed and the symmetrical structure destruction caused by noise is studied. Numerical simulations are presented to further verify the correctness and effectiveness of the main theoretical results.On the formal principle for curves on projective surfaceshttps://www.zbmath.org/1483.320112022-05-16T20:40:13.078697Z"Pereira, Jorge Vitório"https://www.zbmath.org/authors/?q=ai:pereira.jorge-vitorio"Thom, Olivier"https://www.zbmath.org/authors/?q=ai:thom.olivierSummary: We prove that the formal completion of a complex projective surface along a rigid smooth curve with trivial normal bundle determines the birational equivalence class of the surface.Transmission dynamics of fractional order Typhoid fever model using Caputo-Fabrizio operatorhttps://www.zbmath.org/1483.340172022-05-16T20:40:13.078697Z"Shaikh, Amjad S."https://www.zbmath.org/authors/?q=ai:shaikh.amjad-salim"Sooppy Nisar, Kottakkaran"https://www.zbmath.org/authors/?q=ai:sooppy-nisar.kottakkaranSummary: In this manuscript, we develop existence, uniqueness and stability criteria for fractional order Typhoid fever model having Caputo-Fabrizio operator by using fixed point theory. This approach of the fractional derivative is relatively new for such kind of biological models. We have also obtained the first accessible approximate solutions for a proposed model by utilizing iterative Laplace transform method. This technique is a combination of one of the reliable method known as new iterative method and Laplace transform method. Finally, we have evaluated parameters that portray the conduct of illness and present the numerical simulations using plots.Dynamics of non-autonomous oscillator with a controlled phase and frequency of external forcinghttps://www.zbmath.org/1483.340522022-05-16T20:40:13.078697Z"Krylosova, D. A."https://www.zbmath.org/authors/?q=ai:krylosova.d-a"Seleznev, E. P."https://www.zbmath.org/authors/?q=ai:seleznev.eugene-p"Stankevich, N. V."https://www.zbmath.org/authors/?q=ai:stankevich.nataliya-vladimirovnaSummary: The dynamics of a non-autonomous oscillator in which the phase and frequency of the external force depend on the dynamical variable is studied. Such a control of the phase and frequency of the external force leads to the appearance of complex chaotic dynamics in the behavior of oscillator. A hierarchy of various periodic and chaotic oscillations is observed. The structure of the space of control parameters is studied. It is shown there are oscillatory modes similar to those of a non-autonomous oscillator with a potential in the form of a periodic function in the system dynamics, but there are also significant differences. Physical experiments of such systems are implemented.Investigation of dynamical properties in hysteresis-based a simple chaotic waveform generator with two stable equilibriumhttps://www.zbmath.org/1483.340582022-05-16T20:40:13.078697Z"Joshi, Manoj"https://www.zbmath.org/authors/?q=ai:joshi.manoj"Ranjan, Ashish"https://www.zbmath.org/authors/?q=ai:ranjan.ashishSummary: This research article describes a novel simple chaotic oscillator using bistable operation to generate chaotic waveform. In this design, chaos generation uses differential hysteresis phenomena of an Operational Amplifier (Op-Amp) with tank circuit. The behavior of the proposed chaotic system is investigated in terms of basic dynamical characteristics viz. equilibrium point stability, divergence, Lyapunov exponents, influence of initial condition, routes of chaos, basin of attraction and phase portraits by using theoretical analysis in MATLAB. We observed that proposed chaotic system belongs to the class of hidden attractor with two stable equilibrium points without quadratic or multiplying term that reduced the circuit complexity. Finally, an experimental investigation of the proposed design is performed that validates the theoretical and PSPICE results.Existence of two-point oscillatory solutions of a relay nonautonomous system with multiple eigenvalue of a real symmetric matrixhttps://www.zbmath.org/1483.340612022-05-16T20:40:13.078697Z"Yevstafyeva, V. V."https://www.zbmath.org/authors/?q=ai:yevstafyeva.victoria-v|yevstafyeva.vistoria-vSummary: We study an \(n\)-dimensional system of ordinary differential equations with hysteresis type relay nonlinearity and a periodic perturbation function on the right-hand side. It is supposed that the matrix of the system is real and symmetric and, moreover, it has an eigenvalue of multiplicity two. In the phase space of the system, we consider continuous bounded oscillatory solutions with two fixed points and the same time of return to each of these points. For these solutions, we prove the existence and nonexistence theorems. For a three-dimensional system, these results are illustrated by a numerical example.External localized harmonic influence on an incoherence cluster of chimera stateshttps://www.zbmath.org/1483.340662022-05-16T20:40:13.078697Z"Shepelev, I. A."https://www.zbmath.org/authors/?q=ai:shepelev.igor-aleksandrovich"Vadivasova, T. E."https://www.zbmath.org/authors/?q=ai:vadivasova.tatyana-evgenevna|vadivasova.tatiana-eSummary: We study impacts of external harmonic forces on chimera states in an ensemble of chaotic Rössler oscillators with nonlocal interaction. The main attention is paid to control the spatial structure by applying a targeted localized excitation on an incoherence cluster. This influence on a phase chimera enables us to eliminate the incoherence cluster and to realize the regime with a piecewise smooth spatial profile. The mechanism of elimination of the incoherence cluster of the phase chimera consists in-phase synchronization of all oscillators within the region of influence of the external force. This phenomenon is observed for a sufficiently wide range of the external force frequency, especially when its value is less than the natural frequency. Increasing the external force amplitude can lead to two scenarios depending on the dynamics of individual oscillators. In the case of regular dynamics, a strong force induces another type of the incoherence cluster within the region of the external force influence. The oscillator dynamics within this region becomes chaotic. Thus, the features of this cluster are similar to those for the incoherence cluster of an amplitude chimera. When the dynamics is chaotic, the force can cause the system to switch to the regime of a metastable spatial distribution with a qualitatively different character at different time intervals. It is impossible to eliminate the incoherence cluster of the amplitude chimera by means of the localized harmonic influence for any values of its parameters. The destruction of the amplitude chimera structure under the influence of the external force leads either to the intermittent regime or to inducing the stable incoherence cluster.Mathematical modeling of the impact of temperature variations and immigration on malaria prevalence in Nigeriahttps://www.zbmath.org/1483.340682022-05-16T20:40:13.078697Z"Ukwajunor, Eunice E."https://www.zbmath.org/authors/?q=ai:ukwajunor.eunice-e"Akarawak, Eno E. E."https://www.zbmath.org/authors/?q=ai:akarawak.eno-e-e"Abiala, Israel Olutunji"https://www.zbmath.org/authors/?q=ai:abiala.israel-olutunjiGlobal stability in a three-species Lotka-Volterra cooperation model with seasonal successionhttps://www.zbmath.org/1483.340702022-05-16T20:40:13.078697Z"Xie, Xizhuang"https://www.zbmath.org/authors/?q=ai:xie.xizhuang"Niu, Lin"https://www.zbmath.org/authors/?q=ai:niu.linSummary: In this paper, we focus on a three-species Lotka-Volterra cooperation model with seasonal succession. The Floquet multipliers of all nonnegative periodic solutions of such a time-periodic system are estimated via the stability analysis of equilibria. By Brouwer fixed point theorem and the connecting orbits theorem, it is proved that there admits a unique positive periodic solution under appropriate conditions. Furthermore, sharp global asymptotical stability criteria for extinction and coexistence are established. Compared to the classical three-species Lotka-Volterra cooperation model, the introduction of seasonal succession may lead to species' extinction. Finally, some numerical examples are given to illustrate the effectiveness of our theoretical results.A novel amplitude control method for constructing nested hidden multi-butterfly and multiscroll chaotic attractorshttps://www.zbmath.org/1483.340862022-05-16T20:40:13.078697Z"Wu, Qiujie"https://www.zbmath.org/authors/?q=ai:wu.qiujie"Hong, Qinghui"https://www.zbmath.org/authors/?q=ai:hong.qinghui"Liu, Xiaoyang"https://www.zbmath.org/authors/?q=ai:liu.xiaoyang"Wang, Xiaoping"https://www.zbmath.org/authors/?q=ai:wang.xiaoping"Zeng, Zhigang"https://www.zbmath.org/authors/?q=ai:zeng.zhigangSummary: A novel amplitude control method (ACM) is proposed to construct multiple self-excited or hidden attractors by scaling partial or total variables without changing their dynamic and topological properties. Various attractors including nested attractor, axisymmetric attractor, and centrosymmetric attractor can be obtained by multiplying signals with different amplitudes. An universal pulse control module is designed to realize the amplitude scale. Different number of scrolls can be adjusted by regulating the pulse signals without redesigning the nonlinear circuit. The classical Lorenz system and Jerk system are employed as examples to generate nested hidden multi-butterfly and multiscroll attractors. Some novel properties of ACM, such as nested morphology, amplitude modulation, and constant Lyapunov exponential spectrum, are analyzed theoretically and simulated numerically. The circuit design and PSpice simulation results are implemented to verify the availability and feasibility of the proposed approach.Periodic solutions for a nonautonomous mathematical model of hematopoietic stem cell dynamicshttps://www.zbmath.org/1483.341142022-05-16T20:40:13.078697Z"Adimy, Mostafa"https://www.zbmath.org/authors/?q=ai:adimy.mostafa"Amster, Pablo"https://www.zbmath.org/authors/?q=ai:amster.pablo"Epstein, Julián"https://www.zbmath.org/authors/?q=ai:epstein.julianAuthors' abstract: The main purpose of this paper is to study the existence of periodic solutions for a nonautonomous differential-difference system describing the dynamics of hematopoietic stem cell (HSC) population under some external periodic regulatory factors at the cellular cycle level. The starting model is a nonautonomous system of two age-structured partial differential equations describing the HSC population in quiescent \((G_0)\) and proliferating (\(G_1\), \(S\), \(G_2\) and \(M\)) phase. We are interested in the effects of periodically time varying coefficients due for example to circadian rhythms or to the periodic use of certain drugs, on the dynamics of HSC population. The method of characteristics reduces the age-structured model to a nonautonomous differential-difference system. We prove under appropriate conditions on the parameters of the system, using topological degree techniques and fixed point methods, the existence of periodic solutions of our model.
Reviewer: Jiří Šremr (Brno)Asymptotic autonomy of bi-spatial attractors for stochastic retarded Navier-Stokes equationshttps://www.zbmath.org/1483.350442022-05-16T20:40:13.078697Z"Zhang, Qiangheng"https://www.zbmath.org/authors/?q=ai:zhang.qiangheng"Li, Yangrong"https://www.zbmath.org/authors/?q=ai:li.yangrongSummary: We establish semi-convergence of a non-autonomous bi-spatial random attractor towards to an autonomous attractor under the topology of the regular space when time-parameter goes to infinity, where the criteria are given by forward compactness of the attractor in the terminal space as well as forward convergence of the random dynamical system in the initial space. We then apply to both non-autonomous and autonomous stochastic 2D Navier-Stokes equations with general delays (including variable and distribution delays). The forward-pullback asymptotic compactness in the space of continuous Sobolev-valued functions is proved by the method of spectrum decomposition.Singularities and heteroclinic connections in complex-valued evolutionary equations with a quadratic nonlinearityhttps://www.zbmath.org/1483.351192022-05-16T20:40:13.078697Z"Jaquette, Jonathan"https://www.zbmath.org/authors/?q=ai:jaquette.jonathan"Lessard, Jean-Philippe"https://www.zbmath.org/authors/?q=ai:lessard.jean-philippe"Takayasu, Akitoshi"https://www.zbmath.org/authors/?q=ai:takayasu.akitoshiSummary: In this paper, we consider the dynamics of solutions to complex-valued evolutionary partial differential equations (PDEs) and show existence of heteroclinic orbits from nontrivial equilibria to zero via computer-assisted proofs. We also show that the existence of unbounded solutions along unstable manifolds at the equilibrium follows from the existence of heteroclinic orbits. Our computer-assisted proof consists of three separate techniques of rigorous numerics: an enclosure of a local unstable manifold at the equilibria, a rigorous integration of PDEs, and a constructive validation of a trapping region around the zero equilibrium.A Riemann-Hilbert problem approach to infinite gap Hill's operators and the Korteweg-de Vries equationhttps://www.zbmath.org/1483.351452022-05-16T20:40:13.078697Z"McLaughlin, Kenneth T.-R."https://www.zbmath.org/authors/?q=ai:mclaughlin.kenneth-d-t-r"Nabelek, Patrik V."https://www.zbmath.org/authors/?q=ai:nabelek.patrik-vThis paper concerns the inverse problem for the one-dimensional Schrödinger operator \(L\colon H^2(\mathbb{R})^2\to L^2(\mathbb{R})\),
\[
L=-\frac{\partial^2}{\partial x^2} + u(x)
\]
with a real-valued and periodic potential \(u(x)\). In this case, the spectrum of \(L\) is a half-line in \(\mathbb{R}\) with a finite or countably infinite number of open intervals (called ``gaps'') removed. The emphasis in this work is on the infinite-gap potentials, i.e., the potentials \(u(x)\) for which the spectrum of \(L\) has infinitely many gaps. The authors first present a Riemann-Hilbert problem (RHP) for a \(2\times 2\) matrix-valued unknown, whose data consists of the spectral data associated with the operator \(L\) with the potential \(u(x)\). The first row of the solutions of this RHP is unique and that is sufficient to determine \(u(x)\) (uniquely) from any solution of this RHP.
The article then extends the result to the case when the spatially periodic potential \(u=u(x,t)\) let to evolve in time such that it satisfies the Korteweg-de Vries (KdV) equation in the form \(u_t - 6uu_x +u_{xxx} = 0\). This provides a Riemann-Hilbert problem representation for the solution of the Cauchy problem for the KdV equation with smooth infinite-gap initial data \(u_0(x) = u(x, 0)\). With the approach taken here, determination of \(u(x,t)\) for given \((x,t)\) does not rely on the Matveev-Its formula, hence avoids the necessity of the explicit calculation of the Riemann matrix in the setting of an infinite-genus hyperelliptic Riemann surface. The paper also takes a ``dressing'' approach in Section 7: It starts with the aforementioned RHP representation and determines conditions on the data for the RHP (i.e., on candidate spectral data) so that the resulting solution \(u(x,t)\) of the KdV equation is periodic in \(x\) or periodic in \(t\).
Reviewer: Deniz Bilman (Cincinnati)Long-time asymptotics of the focusing Kundu-Eckhaus equation with nonzero boundary conditionshttps://www.zbmath.org/1483.351462022-05-16T20:40:13.078697Z"Wang, Deng-Shan"https://www.zbmath.org/authors/?q=ai:wang.dengshan"Guo, Boling"https://www.zbmath.org/authors/?q=ai:guo.boling"Wang, Xiaoli"https://www.zbmath.org/authors/?q=ai:wang.xiaoli.2|wang.xiaoli.1The Kundu-Eckhaus equation
\[
iq_t+\frac{1}{2}q_{xx}+|q|^2q+2\beta |q|^4q-2i\beta(|q|^2)_x+q=0
\]
is considered, with the initial condition
\[
q(x,0)\sim Ae^{i(\mu x+\theta_\pm)} \text{ when } x\to\pm\infty.
\]
Here \(\beta, A(>0),\mu\), and \(\theta_\pm\) are real constants.
The long-time asymptotics of the solution is established in three different sectors depending on the magnitude \(\xi=x/t\): plane wave sector \(\xi<\xi_1=\mu-2\beta A^2-2\sqrt{2}A\), plane wave sector \(\xi>\xi_2=\mu-2\beta A^2+2\sqrt{2}A\), and modulated genus 1 elliptic wave sector \(\xi_1<\xi<\xi_2\).
Reviewer: Ilya Spitkovsky (Williamsburg)A determining form for the 2D Rayleigh-Bénard problemhttps://www.zbmath.org/1483.351602022-05-16T20:40:13.078697Z"Cao, Yu"https://www.zbmath.org/authors/?q=ai:cao.yu"Jolly, Michael S."https://www.zbmath.org/authors/?q=ai:jolly.michael-s"Titi, Edriss S."https://www.zbmath.org/authors/?q=ai:titi.edriss-salehSummary: We construct a determining form for the 2D Rayleigh-Bénard (RB) system in a strip with solid horizontal boundaries, in the cases of no-slip and stress-free boundary conditions. The determining form is an ODE in a Banach space of trajectories whose steady states comprise the long-time dynamics of the RB system. In fact, solutions on the global attractor of the RB system can be further identified through the zeros of a scalar equation to which the ODE reduces for each initial trajectory. The twist in this work is that the trajectories are for the velocity field only, which in turn determines the corresponding trajectories of the temperature.Long-time asymptotics for the focusing Fokas-Lenells equation in the solitonic region of space-timehttps://www.zbmath.org/1483.351752022-05-16T20:40:13.078697Z"Cheng, Qiaoyuan"https://www.zbmath.org/authors/?q=ai:cheng.qiaoyuan"Fan, Engui"https://www.zbmath.org/authors/?q=ai:fan.enguiSummary: We study the long-time asymptotic behavior of the focusing Fokas-Lenells (FL) equation
\[ u_{x t} + \alpha \beta^2 u - 2 i \alpha \beta u_x - \alpha u_{x x} - i \alpha \beta^2 | u |^2 u_x = 0\]
with generic initial data in a Sobolev space which supports bright soliton solutions. The FL equation is an integrable generalization of the well-known Schrodinger equation, and also linked to the derivative Schrodinger model, but it exhibits several different characteristics from theirs. (i) The Lax pair of the FL equation involves an additional spectral singularity at \(k = 0\). (ii) Four stationary phase points will appear during asymptotic analysis, which require a more detailed necessary description to obtain the long-time asymptotics of the focusing FL equation. Based on the Riemann-Hilbert problem for the initial value problem of the focusing FL equation, we show that inside any fixed time-spatial cone
\[ \mathcal{C} ( x_1 , x_2 , v_1 , v_2 ) = \{ ( x , t ) \in \mathbb{R}^2 | x = x_0 + v t , x_0 \in [ x_1 , x_2 ] , v \in [ v_1 , v_2 ] \},\]
the long-time asymptotic behavior of the solution \(u(x, t)\) for the focusing FL equation can be characterized with an \(N(\mathcal{I})\)-soliton on discrete spectrums and a leading order term \(\mathcal{O}( t^{- 1 / 2})\) on continuous spectrum up to a residual error order \(\mathcal{O}( t^{- 3 / 4})\). The main tool is a \(\overline{\partial} \)-generalization of the Deift-Zhou nonlinear steepest descent method.The lattice sine-Gordon equation as a superposition formula for an NLS-type systemhttps://www.zbmath.org/1483.351762022-05-16T20:40:13.078697Z"Demskoi, Dmitry K."https://www.zbmath.org/authors/?q=ai:demskoi.dmitry-kSummary: We treat the lattice sine-Gordon equation and two of its generalised symmetries as a compatible system. Elimination of shifts from the two symmetries of the lattice sine-Gordon equation yields an integrable NLS-type system. An auto-Bäcklund transformation and a superposition formula for the NLS-type system is obtained by elimination of shifts from the lattice sine-Gordon equation and its down-shifted version. We use the obtained formulae to calculate a superposition of two and three elementary solutions.Analytical modeling of electrical solitons in a nonlinear transmission line using Schamel-Korteweg deVries equationhttps://www.zbmath.org/1483.351812022-05-16T20:40:13.078697Z"Aziz, Farah"https://www.zbmath.org/authors/?q=ai:aziz.farah"Asif, Ali"https://www.zbmath.org/authors/?q=ai:asif.ali"Bint-e-Munir, Fatima"https://www.zbmath.org/authors/?q=ai:bint-e-munir.fatimaSummary: Nonlinear transmission lines (NLTLs) provide an effective means of experimental investigation of nonlinear dispersive media paving the way for accurate modeling of electrical solitons. In this work, an appropriate analytical model is proposed to analyze and predict electrical solitons in an NLTL. The model is based on a new form of nonlinearity which utilizes fractional powers to render an accurate representation of the Capacitance-Voltage (C-V) curve, even for relatively higher voltages. A complete analytical procedure is adopted to obtain the soliton solution in an NLTL. Nonlinear analysis results in a Schamel type Korteweg deVries (S-KdV) equation, instead of a conventional KdV equation. An extended version of the Tanh method is employed to find the solution of S-KdV which yields narrower solitons of Sech\(^4\) shape. The proposed model is validated experimentally by using a specially designed NLTL (100 sections) comprising of linear inductors and nonlinear capacitances. Square pulses with a repetition rate of 100 kHz are used to generate the solitons. The comparative analysis shows excellent profile matching between theoretically predicted and experimentally obtained solitons. Furthermore, in consistence with the prediction of the proposed model, the product of the Amplitude and fourth power of the Width \((A \times W^4)\) is found to be constant in experimental results. This model proposed in the present work can be of help in understanding and predicting the behavior of electrical solitons in NLTLs.Matrix solitons solutions of the modified Korteweg-de Vries equationhttps://www.zbmath.org/1483.351822022-05-16T20:40:13.078697Z"Carillo, Sandra"https://www.zbmath.org/authors/?q=ai:carillo.sandra"Lo Schiavo, Mauro"https://www.zbmath.org/authors/?q=ai:lo-schiavo.mauro"Schiebold, Cornelia"https://www.zbmath.org/authors/?q=ai:schiebold.corneliaSummary: Nonlinear non-abelian Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) equations and their links via Bäcklund transformations are considered. The focus is on the construction of soliton solutions admitted by matrix modified Korteweg-de Vries equation. Matrix equations can be viewed as a specialisation of operator equations in the finite dimensional case when operators admit a matrix representation. Bäcklund transformations allow to reveal structural properties [the first and third authors, J. Math. Phys. 50, No. 7, 073510, 14 p. (2009; Zbl 1256.37036)] enjoyed by non-commutative KdV-type equations, such as the existence of a recursion operator. Operator methods combined with Bäcklund transformations allow to construct explicit solution formulae [the first and third authors, ibid. 52, No. 5, 053507, 21 p. (2011; Zbl 1317.35220)]. The latter are adapted to obtain solutions admitted by the \(2 \times 2\) and \(3 \times 3\) matrix mKdV equation. Some of these matrix solutions are visualised to show the solitonic behaviour they exhibit. A further key tool used to obtain the presented results is an ad hoc construction of computer algebra routines to implement non-commutative computations.
For the entire collection see [Zbl 1464.70003].New traveling wave solutions and interesting bifurcation phenomena of generalized KdV-mKdV-like equationhttps://www.zbmath.org/1483.351842022-05-16T20:40:13.078697Z"Chen, Yiren"https://www.zbmath.org/authors/?q=ai:chen.yiren"Li, Shaoyong"https://www.zbmath.org/authors/?q=ai:li.shaoyongSummary: Using the bifurcation method of dynamical systems, we investigate the nonlinear waves and their limit properties for the generalized KdV-mKdV-like equation. We obtain the following results: (i) three types of new explicit expressions of nonlinear waves are obtained. (ii) Under different parameter conditions, we point out these expressions represent different waves, such as the solitary waves, the 1-blow-up waves, and the 2-blow-up waves. (iii) We revealed a kind of new interesting bifurcation phenomenon. The phenomenon is that the 1-blow-up waves can be bifurcated from 2-blow-up waves. Also, we gain other interesting bifurcation phenomena. We also show that our expressions include existing results.Integral complex modified Korteweg-de Vries (Icm-KdV) equationshttps://www.zbmath.org/1483.351872022-05-16T20:40:13.078697Z"Velasco-Juan, M."https://www.zbmath.org/authors/?q=ai:velasco-juan.m"Fujioka, J."https://www.zbmath.org/authors/?q=ai:fujioka.jorgeSummary: In this article we examine different ways of introducing nonlocal effects in the complex modified Korteweg-de Vries (cmKdV) equation. Four possible nonlocal generalizations of the cmKdV equation are presented. In these variants of the cmKdV equation the nonlinear term has been replaced by one, two, or three integral terms. Three of these variants can be obtained by the least action principle, and they possess different equivalent Lagrangians. The relationship between these Lagrangians is entirely new. Therefore, a novel type of \textit{Lagrangian equivalence} has been found. As Fubini's theorem is needed to prove the equivalence of these Lagrangians, we call them ``F-equivalent''. Noether's theorem is applied to these integral equations, and different conserved quantities are obtained. The behavior of solitary waves in these models is also studied. It was found that one of these models has soliton-like solutions. Numerical solutions show that these solitons are stable, and the amplitudes of perturbed solitons attain constant values after a short transient period. Finally, a variational method was applied to get a better understanding of the behavior of these pulses during the transient periods.Asymptotic behaviour of the solutions for a weakly damped anisotropic sixth-order Schrödinger type equation in \(\mathbb{R}^2\)https://www.zbmath.org/1483.351922022-05-16T20:40:13.078697Z"Alouini, Brahim"https://www.zbmath.org/authors/?q=ai:alouini.brahimSummary: We study the long-time behaviour of the solutions to a nonlinear damped anisotropic sixth-order Schrödinger type equation in \(\mathbb{R}^2\) that reads
\[
u_t +i\Delta u-i \left(\partial_y^4 u-\partial_y^6 u\right) +ig(|u|^2)u+\gamma u = f\, ,\quad (t,(x,y))\in \mathbb{R}\times \mathbb{R}^2.
\]
We prove that this behaviour is described by the existence of regular global attractor in an anisotropic Sobolev space with finite fractal dimension.Generalized Adler-Moser polynomials and multiple vortex rings for the Gross-Pitaevskii equationhttps://www.zbmath.org/1483.351942022-05-16T20:40:13.078697Z"Ao, Weiwei"https://www.zbmath.org/authors/?q=ai:ao.weiwei"Huang, Yehui"https://www.zbmath.org/authors/?q=ai:huang.yehui"Liu, Yong"https://www.zbmath.org/authors/?q=ai:liu.yong.1"Wei, Juncheng"https://www.zbmath.org/authors/?q=ai:wei.junchengThreshold scattering for the focusing NLS with a repulsive Dirac delta potentialhttps://www.zbmath.org/1483.351962022-05-16T20:40:13.078697Z"Ardila, Alex H."https://www.zbmath.org/authors/?q=ai:ardila.alex-hernandez"Inui, Takahisa"https://www.zbmath.org/authors/?q=ai:inui.takahisaSummary: We establish the scattering of solutions to the focusing mass supercritical nonlinear Schrödinger equation with a repulsive Dirac delta potential
\[
i \partial_t u + \partial_x^2 u + \gamma \delta(x) u + | u |^{p - 1} u = 0, \quad (t, x) \in \mathbb{R} \times \mathbb{R},
\]
at the mass-energy threshold, namely, when \(E_\gamma( u_0) [ M ( u_0 ) ]^\sigma = E_0(Q) [ M ( Q ) ]^\sigma\) where \(u_0 \in H^1(\mathbb{R})\) is the initial data, \(Q\) is the ground state of the free NLS on the real line \(\mathbb{R}\), \(E_\gamma\) is the energy, \(M\) is the mass and \(\sigma = (p + 3) /(p - 5)\). We also prove failure of the uniform space-time bounds at the mass-energy threshold.Scattering in the weighted \( L^2 \)-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearityhttps://www.zbmath.org/1483.351982022-05-16T20:40:13.078697Z"Bensouilah, Abdelwahab"https://www.zbmath.org/authors/?q=ai:bensouilah.abdelwahab"Dinh, Van Duong"https://www.zbmath.org/authors/?q=ai:dinh.van-duong"Majdoub, Mohamed"https://www.zbmath.org/authors/?q=ai:majdoub.mohamedSummary: We investigate the defocusing inhomogeneous nonlinear Schrödinger equation
\[ i \partial_tu + \Delta u = |x|^{-b} \left(\mathrm{e}^{\alpha|u|^2} - 1- \alpha |u|^2 \right) u, \quad u(0) = u_0, \quad x \in \mathbb{R}^2, \]
with \( 0<b<1 \) and \( \alpha = 2\pi(2-b) \). First we show the decay of global solutions by assuming that the initial data \( u_0 \) belongs to the weighted space \( \Sigma(\mathbb{R}^2) = \{u\in H^1(\mathbb{R}^2) : |x|u\in L^2(\mathbb{R}^2)\} \). Then we combine the local theory with the decay estimate to obtain scattering in \( \Sigma \) when the Hamiltonian is below the value \( \frac{2}{(1+b)(2-b)} \).The soliton solutions and long-time asymptotic analysis for an integrable variable coefficient nonlocal nonlinear Schrödinger equationhttps://www.zbmath.org/1483.352002022-05-16T20:40:13.078697Z"Chen, Guiying"https://www.zbmath.org/authors/?q=ai:chen.guiying"Xin, Xiangpeng"https://www.zbmath.org/authors/?q=ai:xin.xiangpeng"Zhang, Feng"https://www.zbmath.org/authors/?q=ai:zhang.fengSummary: An integrable variable coefficient nonlocal nonlinear Schrödinger equation (NNLS) is studied; by employing the Hirota's bilinear method, the bilinear form is obtained, and the \(N\)-soliton solutions are constructed. In addition, some singular solutions and period solutions of the addressed equation with specific coefficients are shown. Finally, under certain conditions, the asymptotic behavior of the two-soliton solution is analyzed to prove that the collision of the two-soliton is elastic.A note on global existence for the Zakharov system on \( \mathbb{T} \)https://www.zbmath.org/1483.352042022-05-16T20:40:13.078697Z"Compaan, E."https://www.zbmath.org/authors/?q=ai:compaan.erinSummary: We show that the one-dimensional periodic Zakharov system is globally well-posed in a class of low-regularity Fourier-Lebesgue spaces. The result is obtained by combining the I-method with Bourgain's high-low decomposition method. As a corollary, we obtain probabilistic global existence results in \( L^2 \)-based Sobolev spaces. We also obtain global well-posedness in \( H^{\frac12+} \times L^2 \), which is sharp (up to endpoints) in the class of \( L^2 \)-based Sobolev spaces.Lax pair, conservation laws, Darboux transformation, breathers and rogue waves for the coupled nonautonomous nonlinear Schrödinger system in an inhomogeneous plasmahttps://www.zbmath.org/1483.352072022-05-16T20:40:13.078697Z"Ding, Cui-Cui"https://www.zbmath.org/authors/?q=ai:ding.cui-cui"Gao, Yi-Tian"https://www.zbmath.org/authors/?q=ai:gao.yitian"Deng, Gao-Fu"https://www.zbmath.org/authors/?q=ai:deng.gao-fu"Wang, Dong"https://www.zbmath.org/authors/?q=ai:wang.dong.5|wang.dong.7|wang.dong.1|wang.dong.3|wang.dong.2|wang.dong.6|wang.dong.4|wang.dong.8|wang.dongSummary: Plasmas are believed to be possibly ``the most abundant form of ordinary matter in the Universe''. In this paper, a coupled nonautonomous nonlinear Schrödinger system is investigated, which describes the propagation of two envelope solitons in a weakly inhomogeneous plasma with the \(t\)-dependent linear and parabolic density profiles and nonconstant collisional damping. Lax pair with the nonisospectral parameter and infinitely-many conservation laws are derived. Based on the Lax pair, the \(N\)th-step Darboux transformation is constructed. Utilizing the \(N\)th-step Darboux transformation, we obtain the breather and rogue wave solutions, and find that the amplitude of the nonzero background is nonconstant and dependent on the inhomogeneous coefficients in the system under investigation. Characteristics of the breathers and rogue waves are discussed, and effects of the inhomogeneous coefficients on the breathers and rogue waves are analyzed. Breathers and rogue waves with the dark or bright soliton together are also constructed and their characteristics are discussed. We find that the dark and bright solitons can coexist and generate the breather-like waves.Quadratic lifespan and growth of Sobolev norms for derivative Schrödinger equations on generic torihttps://www.zbmath.org/1483.352092022-05-16T20:40:13.078697Z"Feola, Roberto"https://www.zbmath.org/authors/?q=ai:feola.roberto"Montalto, Riccardo"https://www.zbmath.org/authors/?q=ai:montalto.riccardoSummary: We consider a family of Schrödinger equations with unbounded Hamiltonian quadratic nonlinearities on a generic tori of dimension \(d \geq 1\). We study the behavior of high Sobolev norms \(H^s, s \gg 1\), of solutions with initial conditions in \(H^s\) whose \(H^\rho \)-Sobolev norm, \(1 \ll \rho \ll s\), is smaller than \(\varepsilon \ll 1\). We provide a control of the \(H^s\)-norm over a time interval of order \(O( \varepsilon^{- 2})\). Due to the lack of conserved quantities controlling high Sobolev norms, the key ingredient of the proof is the construction of a modified energy equivalent to the ``low norm'' \( H^\rho \) (when \(\rho\) is sufficiently high) over a nontrivial time interval \(O( \varepsilon^{- 2})\). This is achieved by means of normal form techniques for quasi-linear equations involving para-differential calculus. The main difficulty is to control the possible loss of derivatives due to the small divisors arising form three waves interactions. By performing ``tame'' energy estimates we obtain upper bounds for higher Sobolev norms \(H^s\).Quasi-invariance of low regularity Gaussian measures under the gauge map of the periodic derivative NLShttps://www.zbmath.org/1483.352102022-05-16T20:40:13.078697Z"Genovese, Giuseppe"https://www.zbmath.org/authors/?q=ai:genovese.giuseppe"Lucà, Renato"https://www.zbmath.org/authors/?q=ai:luca.renato"Tzvetkov, Nikolay"https://www.zbmath.org/authors/?q=ai:tzvetkov.nikolaySummary: The periodic DNLS gauge is an anticipative map with singular generator which revealed crucial in the study of the periodic derivative NLS. We prove quasi-invariance of the Gaussian measure on \(L^2(\mathbb{T})\) with covariance \([1+(-\Delta)^s]^{- 1}\) under these transformations for any \(s > \frac{1}{2}\). This extends previous achievements by \textit{A. R. Nahmod} et al. [Math. Res. Lett. 18, No. 5, 875--887 (2011; Zbl 1250.60018)] and the first author et al. [Math. Ann. 374, No. 3--4, 1075--1138 (2019; Zbl 1420.35354)], who proved the result for integer values of the regularity parameter \(s\).On three-wave interaction Schrödinger systems with quadratic nonlinearities: global well-posedness and standing waveshttps://www.zbmath.org/1483.352222022-05-16T20:40:13.078697Z"Pastor, Ademir"https://www.zbmath.org/authors/?q=ai:pastor.ademirSummary: Reported here are results concerning the global well-posedness in the energy space and existence and stability of standing-wave solutions for 1-dimensional three-component systems of nonlinear Schrödinger equations with quadratic nonlinearities. For two particular systems we are interested in, the global well-posedness is established in view of the a priori bounds for the local solutions. The standing waves are explicitly obtained and their spectral stability is studied in the context of Hamiltonian systems. For more general Hamiltonian systems, the existence of standing waves is accomplished with a variational approach based on the Mountain Pass Theorem. Uniqueness results are also provided in some very particular cases.Dynamics of threshold solutions for energy critical NLS with inverse square potentialhttps://www.zbmath.org/1483.352292022-05-16T20:40:13.078697Z"Yang, Kai"https://www.zbmath.org/authors/?q=ai:yang.kai"Zeng, Chongchun"https://www.zbmath.org/authors/?q=ai:zeng.chongchun"Zhang, Xiaoyi"https://www.zbmath.org/authors/?q=ai:zhang.xiaoyiGinzburg-Landau patterns in circular and spherical geometries: vortices, spirals, and attractorshttps://www.zbmath.org/1483.352322022-05-16T20:40:13.078697Z"Dai, Jia-Yuan"https://www.zbmath.org/authors/?q=ai:dai.jia-yuan"Lappicy, Phillipo"https://www.zbmath.org/authors/?q=ai:lappicy.phillipoThe authors consider the (time-dependent) Ginzburg-Landau equation (containing a positive bifurcation parameter) on compact surfaces of revolution such as the unit disk or the unit 2-sphere and investigate how topological structure of the surface affects the dynamics of vortex solutions. They first show that all the bifurcation curves of time-independent vortex solutions (which are called vortex equilibria and satisfy the elliptic version of the Ginzburg-Landau equation) are global. Then the existence of the spiral wave solutions for the complex version of the considered Ginzburg-Landau equation is proved by perturbing vortex solutions. The authors finally construct global attractor of vortex equilibria. They prove the results by adapting the shooting method (usually used in this context) to the new situation. In this way, the needed hyperbolicity of vortex equilibria can be established.
Reviewer: Catalin Popa (Iaşi)The essential spectrum of periodically stationary solutions of the complex Ginzburg-Landau equationhttps://www.zbmath.org/1483.352342022-05-16T20:40:13.078697Z"Zweck, John"https://www.zbmath.org/authors/?q=ai:zweck.john-w"Latushkin, Yuri"https://www.zbmath.org/authors/?q=ai:latushkin.yuri"Marzuola, Jeremy L."https://www.zbmath.org/authors/?q=ai:marzuola.jeremy-l"Jones, Christopher K. R. T."https://www.zbmath.org/authors/?q=ai:jones.christopher-k-r-tSummary: We establish the existence and regularity properties of a monodromy operator for the linearization of the cubic-quintic complex Ginzburg-Landau equation about a periodically stationary (breather) solution. We derive a formula for the essential spectrum of the monodromy operator in terms of that of the associated asymptotic linear differential operator. This result is obtained using the theory of analytic semigroups under the assumption that the Ginzburg-Landau equation includes a spectral filtering (diffusion) term. We discuss applications to the stability of periodically stationary pulses in ultrafast fiber lasers.Graphop mean-field limits for Kuramoto-type modelshttps://www.zbmath.org/1483.352702022-05-16T20:40:13.078697Z"Gkogkas, Marios Antonios"https://www.zbmath.org/authors/?q=ai:gkogkas.marios-antonios"Kuehn, Christian"https://www.zbmath.org/authors/?q=ai:kuhn.christianOn a periodic age-structured mosquito population model with spatial structurehttps://www.zbmath.org/1483.352882022-05-16T20:40:13.078697Z"Lv, Yunfei"https://www.zbmath.org/authors/?q=ai:lv.yunfei"Pei, Yongzhen"https://www.zbmath.org/authors/?q=ai:pei.yongzhen"Yuan, Rong"https://www.zbmath.org/authors/?q=ai:yuan.rongSummary: This paper deals with a general age-structured model with diffusion. The existence and uniqueness of solutions of the equivalent integral equation are obtained in light of the contraction mapping theorem. By taking the mosquito population growth as a motivating example, we derive a periodic stage-structured model with diffusion, intra-specific competition and periodic delay. Next, we show that the solution is globally bounded for the setup we chose. Then, the basic reproduction number \(R_0\) for this model is introduced to establish the threshold dynamics on mosquito extinction and persistence in terms of \(R_0\). In the case where intra-specific competition among immature individuals is ignored, the adult equation is decoupled from the full equations, and the global stability of the positive periodic solution is then obtained by introducing a suitable phase space on which the periodic semiflow is eventually strongly monotone and strictly subhomogeneous.A reaction-diffusion model for salmonella transmission within an industrial hens house with distributed resistance to salmonella carrier statehttps://www.zbmath.org/1483.352962022-05-16T20:40:13.078697Z"Zongo, Pascal"https://www.zbmath.org/authors/?q=ai:zongo.pascal"Beaumont, Catherine"https://www.zbmath.org/authors/?q=ai:beaumont.catherineSummary: To understand how the animals' repartition within the hen house combined with their genetic heterogeneity in host resistance to Salmonella infection affects the dynamics of Salmonellosis epidemics, we propose a spatio-temporal model for Salmonella transmission within a flock of genetic heterogeneous animals that differ according to their level of genetic resistance to Salmonella carrier state \(\theta \in [0, \Theta)\), \(\Theta > 0\). We then define the basic reproduction ratio \(\mathcal{R}_0\) that is a function of \(\mathbf{r}_0(\theta)\) that may be interpreted as the weight of transmission \textit{bacteria-to-hens of type} \(\theta\)-\textit{to-bacteria}, where hens of type \(\theta\) represent the animals having the same level of resistance \(\theta\). We show that \(\mathcal{R}_0\) serves as a threshold parameter that predicts whether Salmonella will spread or not. Furthermore, in the case where all the parameters are spatially independent, we obtain an explicit formula for \(\mathcal{R}_0\), and we show that the disease will stabilize at a positive steady state that is globally attractive when \(\mathcal{R}_0>1\). According to the results of the analysis of the impact of heterogeneities, we argue that the severity of disease transmission is largely depending on the choice of the initial distribution of genetical fowls within a spatially heterogeneous environment.
For the entire collection see [Zbl 1476.34004].Optimisation of the total population size with respect to the initial condition for semilinear parabolic equations: two-scale expansions and symmetrisationshttps://www.zbmath.org/1483.353042022-05-16T20:40:13.078697Z"Mazari, Idriss"https://www.zbmath.org/authors/?q=ai:mazari.idriss"Nadin, Grégoire"https://www.zbmath.org/authors/?q=ai:nadin.gregoire"Toledo Marrero, Ana Isis"https://www.zbmath.org/authors/?q=ai:toledo-marrero.ana-isisAttractor dimension estimates for dynamical systems: theory and computation. Dedicated to Gennady Leonovhttps://www.zbmath.org/1483.370012022-05-16T20:40:13.078697Z"Kuznetsov, Nikolay"https://www.zbmath.org/authors/?q=ai:kuznetsov.nikolay-v|kuznetsov.nikolay-germanovich"Reitmann, Volker"https://www.zbmath.org/authors/?q=ai:reitmann.volkerThis book covers some special topics in the theory of dynamical systems, with emphasis on attractor dimension estimates and investigations on global attractors and invariant sets by means of Lyapunov functions and adapted metrics.
It is interesting and very well written. Mostly, chapters are self-contained and rich of detailed explanations. Many powerful computational tools and algorithms provide a solid numerical background for the study of attractor dimensions. The authors present MATLAB programs for the computation of the dimensions and of Lyapunov exponents, and interestingly, they also visualize these quantities graphically. The effectiveness of introducing the tool of Lyapunov functions into the study of dimensional features is shown for several concrete dynamical systems, such as the Hénon map, Lorenz and Rössler systems, as well as their generalizations that one finds in various physical applications.
This book is structured in three parts and each part consists of some chapters covering some special topics.
In Part I, the authors provide the basic concepts of attractor theory, exterior products and dimension theory which are related to Lyapunov functions, dissipativity, homoclinic orbits, the Yakubovich-Kalman frequency theorem, the frequency-domain estimation of singular values, topological dimension, Hausdorff dimension, fractal dimension, topological entropy and dimension-like features.
In Part II, the authors mention dimension properties of dynamical systems in Euclidean spaces. They provide dimension estimates for almost periodic flows. This part covers estimates of the topological dimension, of the Hausdorff dimension and of the fractal dimension for invariant sets of some concrete physical systems. Further, estimates of the Lyapunov dimension are given. Analytical formulas for the exact Lyapunov dimension of well-known dynamical systems are studied. Among these are: the Hénon map, the Lorenz system, the Glukhovsky-Dolzhansky system, the Yang-Tigan system and the Shimizu-Morioka system. Computation of attractors and Lyapunov dimension are also provided.
In Part III, the authors explore dimension properties for dynamical systems on manifolds. This part provides basic concepts for dimension estimation on manifolds. The authors describe the Hausdorff dimension estimates for invariant sets of vector fields, the Lyapunov dimension as upper bound of the fractal dimension, the Hausdorff dimension estimates by use of a tubular Carathéodory structure and applications to stability theory. Dimension and entropy estimates for global attractors of cocycles on manifolds are discussed as well. Dimension estimates are discussed also for non-injective smooth maps, piecewise \(C^{1}\)-maps and maps with special singularity sets.
In each part, the authors justify the results by providing theoretical and numerical examples and applications. To make the material self-contained, the authors present at the end of the book some basic concepts about differential manifolds, Riemannian manifolds, degree theory, homology theory, geometric measure theory and hyperbolicity in dynamical systems.
Overall, this book contains advanced material on attractor dimension estimates for dynamical systems. This is definitely suitable for researchers in applied mathematics and computational theory of dynamical systems.
Reviewer: Mohammad Sajid (Buraydah)Scaling laws in dynamical systemshttps://www.zbmath.org/1483.370022022-05-16T20:40:13.078697Z"Leonel, Edson Denis"https://www.zbmath.org/authors/?q=ai:leonel.edson-denisThe author presents and combines various approaches and results on nonlinear dynamical systems with special emphasis on the scaling formalism.
The titles of the chapters are quite informative and they reflect not only the structure of the book but also its content:
1. Introduction, 2. One-Dimensional Mappings, 3. Some Dynamical Properties for the Logistic Map, 4. The Logistic-Like Map, 5. Introduction to Two Dimensional Mappings, 6. A Fermi Accelerator Model; 7. Dissipation in the Fermi-Ulam Model; 8. Dynamical Properties for a Bouncer Model; 9. Localization of Invariant Spanning Curves; 10. Chaotic Diffusion in Non-Dissipative Mappings; 11. Scaling on a Dissipative Standard Mapping; 12. Introduction to Billiard Dynamics; 13. Time Dependent Billiards; 14. Suppression of Fermi Acceleration in the Oval Billiard; 15. A Thermodynamic Model for Time Dependent Billiards.
The book provides useful introductory materials, including exercises, for undergraduate and graduate students who wish to have an overview of common scaling properties an their related methodologies in mathematics, physics, mechanical and control engineering.
Reviewer: Vladimir Sobolev (Samara)Dr. Tien-Yien Li's three seminal papershttps://www.zbmath.org/1483.370032022-05-16T20:40:13.078697Z"Ding, Jiu"https://www.zbmath.org/authors/?q=ai:ding.jiuSummary: We present the most important mathematical contributions of Dr. Tien-Yien Li by discussing his three celebrated papers.Mixing sets for rigid transformationshttps://www.zbmath.org/1483.370042022-05-16T20:40:13.078697Z"Ryzhikov, V. V."https://www.zbmath.org/authors/?q=ai:ryzhikov.valery-vSummary: It is shown that, for any infinite set \(M\subset\mathbb N\) of density zero, there exists a rigid measure-preserving transformation of a probability space which is mixing along \(M\). As examples, Gaussian actions and Poisson suspensions over infinite rank-one constructions are considered. Analogues of the obtained result for group actions and a method not using Gaussian and Poisson suspensions are also discussed.Relative entropy and the Pinsker product formula for sofic groupshttps://www.zbmath.org/1483.370052022-05-16T20:40:13.078697Z"Hayes, Ben"https://www.zbmath.org/authors/?q=ai:hayes.benFrom the abstract: ``We continue our study of the outer Pinsker factor for probability measure-preserving actions of sofic groups. Using the notion of local and doubly empirical convergence developed by Austin we prove that in many cases the outer Pinsker factor of a product action is the product of the outer Pinsker factors. Our results are parallel to those of Seward for Rokhlin entropy. We use these Pinsker product formulas to show that if \(X\) is a compact group, and \(G\) is a sofic group with \(G\curvearrowright X\) by automorphisms, then the outer Pinsker factor of \(G\curvearrowright (X,m_X)\) is given as a quotient by a \(G\)-invariant, closed, normal subgroup of \(X\). We use our results to show that if \(G\) is sofic and \(f\in M_n (\mathbb{Z}(G))\) is invertible as a convolution operator \(\ell^2 (G)^{\oplus n}\to \ell^2 (G)^{\oplus n}\), then the action of \(G\) on the Pontryagin dual of \(\mathbb{Z} (G)^{\oplus n}/\mathbb{Z} (G)^{\oplus n}f\) has completely positive measure-theoretic entropy with respect to the Haar measure.''
The definition of sofic group is given and other auxiliary notions are recalled. The relative entropy for actions of sofic groups is defined and its main properties are discussed. The relative outer Pinsker algebra is defined. Some preliminaries on local and doubly empirical convergence, the definition of strong soficity, and applications to actions on compact groups by automorphisms, are given. Actions which are strongly sofic with respect to any sofic approximation of a countable, discrete, and sofic group, are considered with examples. Certain permanence properties of strong soficity are proven. The author also gives a product formula for outer Pinsker factors of strongly sofic actions and uses it in the study of algebraic actions. It is shown that ``the given definition of (upper) relative entropy for actions of sofic groups agrees with the usual definition when the group is amenable''.
Reviewer: Symon Serbenyuk (Kyïv)Furstenberg systems of bounded multiplicative functions and applicationshttps://www.zbmath.org/1483.370062022-05-16T20:40:13.078697Z"Frantzikinakis, Nikos"https://www.zbmath.org/authors/?q=ai:frantzikinakis.nikos"Host, Bernard"https://www.zbmath.org/authors/?q=ai:host.bernardLet \(\mathbb{U}\) be the complex unit disc and \(R\) be a homeomorphism of a compact metric space \(Y.\) One of the main results of the paper states the following. Let \(f:\mathbb{N}\to\mathbb{U}\) be a strongly aperiodic multiplicative function, and \((Y,R)\) has zero topological entropy and at most countably many ergodic invariant measures. Then for each \({y\in Y}\) and every \({g\in C(Y)}\) \[ \lim_{N\to\infty}\frac{1}{\log N}\sum_{n=1}^N\frac{g(R^ny)f(n)}{n}=0. \]
The proof is based on a structural result for measure-preserving systems naturally associated with any collection of multiplicative functions (joint Furstenberg systems). In other words, it is proved that a joint Furstenberg system is a factor of a system with no irrational spectrum and with ergodic components isomorphic to direct products of infinite-step nilsystems and Bernoulli systems. The authors use some identity for multiplicative functions due to \textit{T. Tao} and \textit{J. Teräväinen} [Duke Math. J. 168, No. 11, 1977--2027 (2019; Zbl 1436.11115)].
The general strategy of the article is similar to that used by the authors in [Ann. Math. (2) 187, No. 3, 869--931 (2018; Zbl 1400.11129)], where Möbius and Liouville functions were considered.
Reviewer: Ivan Podvigin (Novosibirsk)Spread out random walks on homogeneous spaceshttps://www.zbmath.org/1483.370072022-05-16T20:40:13.078697Z"Prohaska, Roland"https://www.zbmath.org/authors/?q=ai:prohaska.rolandThe setting considered here is that of a homogeneous space \(X=G/\Gamma\) for a \(\sigma\)-compact locally compact metrizable group \(G\) and a discrete subgroup \(\Gamma<G\), with a Borel probability measure \(\mu\) on \(G\) used to define a random walk on \(X\). That is, each step of the random walk chooses an element \(g\in G\) according to \(\mu\) and then moves \(x\in X\) to \(gx\in X\). Here the Markov chain theory is used to carry out a careful analysis under the assumption that the increment function is spread out. In the lattice (finite volume) case a complete picture of the asymptotics of the \(n\)-step distribution is found, and they are shown to equidistribute to Haar measure. Situations in which this equidistribution is exponentially fast or locally uniform relative to the initial point are studied. In the case of infinite volume the recurrence is shown and it is proved a ratio limit theorem for symmetric spread out random walks on homogeneous spaces under a growth condition.
Reviewer: Thomas B. Ward (Newcastle)On eventually always hitting pointshttps://www.zbmath.org/1483.370082022-05-16T20:40:13.078697Z"Ganotaki, Charis"https://www.zbmath.org/authors/?q=ai:ganotaki.charis"Persson, Tomas"https://www.zbmath.org/authors/?q=ai:persson.tomasSummary: We consider dynamical systems \((X,T,\mu)\) which have exponential decay of correlations for either Hölder continuous functions or functions of bounded variation. Given a sequence of balls \((B_n)_{n=1}^\infty\), we give sufficient conditions for the set of eventually always hitting points to be of full measure. This is the set of points \(x\) such that for all large enough \(m\), there is a \(k < m\) with \(T^k (x) \in B_m\). We also give an asymptotic estimate as \(m\rightarrow\infty\) on the number of \(k < m\) with \(T^k (x) \in B_m\). As an application, we prove for almost every point \(x\) an asymptotic estimate on the number of \(k \le m\) such that \(a_k\ge m^t\), where \(t\in (0,1)\) and \(a_k\) are the continued fraction coefficients of \(x\).Invariant states on noncommutative torihttps://www.zbmath.org/1483.370092022-05-16T20:40:13.078697Z"Bambozzi, Federico"https://www.zbmath.org/authors/?q=ai:bambozzi.federico"Murro, Simone"https://www.zbmath.org/authors/?q=ai:murro.simone"Pinamonti, Nicola"https://www.zbmath.org/authors/?q=ai:pinamonti.nicolaAuthors' abstract: For any number \(h\) such that \(\hbar :=h/2\pi\) is irrational and any skew-symmetric, non-degenerate bilinear form \(\sigma : \mathbb{Z}^{2g}\times \mathbb{Z}^{2g} \to \mathbb{Z}\), let be \(\mathcal{A}^h_{g,\sigma}\) be the twisted group \(\ast\)-algebra \(\mathbb{C}[\mathbb{Z}^{2g}]\) and consider the ergodic group of \(\ast\)-automorphisms of \(\mathcal{A}^h_{g,\sigma }\) induced by the action of the symplectic group \(\mathrm{Sp} (\mathbb{Z}^{2g}, \sigma)\). We show that the only \(\text{Sp} (\mathbb{Z}^{2g}, \sigma)\)-invariant state on \(\mathcal{A}^h_{g,\sigma}\) is the trace state \(\tau\).
Reviewer: Michael Skeide (Campobasso)Spectral and ergodic properties of completely positive maps and decoherencehttps://www.zbmath.org/1483.370102022-05-16T20:40:13.078697Z"Fidaleo, Francesco"https://www.zbmath.org/authors/?q=ai:fidaleo.francesco"Ottomano, Federico"https://www.zbmath.org/authors/?q=ai:ottomano.federico"Rossi, Stefano"https://www.zbmath.org/authors/?q=ai:rossi.stefanoSummary: In an attempt to propose more general conditions for decoherence to occur, we study spectral and ergodic properties of unital, completely positive maps on not necessarily unital \(C^\ast\)-algebras, with a particular focus on gapped maps for which the transient portion of the arising dynamical system can be separated from the persistent one. After some general results, we first devote our attention to the abelian case by investigating the unital \(\ast\)-endomorphisms of, in general non-unital, \(C^\ast\)-algebras, and their spectral structure. The finite-dimensional case is also investigated in detail, and examples are provided of unital completely positive maps for which the persistent part of the associated dynamical system is equipped with the new product making it into a \(C^\ast\)-algebra, and the map under consideration restricts to a unital \(\ast\)-automorphism for this new \(C^\ast\)-structure, thus generating a conservative dynamics on that persistent part.Koopman operators and the \(3x+1\)-dynamical systemhttps://www.zbmath.org/1483.370112022-05-16T20:40:13.078697Z"Leventides, John"https://www.zbmath.org/authors/?q=ai:leventides.john"Poulios, Costas"https://www.zbmath.org/authors/?q=ai:poulios.costasAn étale equivalence relation on a tiling space arising from a two-sided subshift and associated \(C^{\ast}\)-algebrashttps://www.zbmath.org/1483.370122022-05-16T20:40:13.078697Z"Matsumoto, Kengo"https://www.zbmath.org/authors/?q=ai:matsumoto.kengoSummary: A \(\lambda\)-graph bisystem \(\mathcal{L}\) consists of a pair \((\mathcal{L}^- ,\mathcal{L}^+)\) of two labelled Bratteli diagrams, that presents a two-sided subshift \(\Lambda_{\mathcal{L}}\). We will construct a compact totally disconnected metric space consisting of tilings of a two-dimensional half plane from a \(\lambda\)-graph bisystem. The tiling space has a certain AF-equivalence relation written \(R_{\mathcal{L}}\) with a natural shift homeomorphism \(\sigma_{\mathcal{L}}\) coming from the shift homeomorphism \(\sigma_{\Lambda_{\mathcal{L}}}\) on the subshift \(\Lambda_{\mathcal{L}}\). The equivalence relation \(R_{\mathcal{L}}\) yields an AF-algebra \(\mathcal{F}_{\mathcal{L}}\) with an automorphism \(\rho_{\mathcal{L}}\) induced by \(\sigma_{\mathcal{L}}\). We will study invariance of the étale equivalence relation \(R_{\mathcal{L}}\), the groupoid \(\mathcal{G}_{\mathcal{L}} =R_{\mathcal{L}} \rtimes_{\sigma_{\mathcal{L}}} \mathbb{Z}\) and the groupoid \(C^{\ast}\)-algebras \(C^{\ast} (R_{\mathcal{L}}), C^{\ast} (\mathcal{G}_{\mathcal{L}})\) under topological conjugacy of the presenting two-sided subshifts.Further discussion on Kato's chaos in set-valued discrete systemshttps://www.zbmath.org/1483.370132022-05-16T20:40:13.078697Z"Li, Risong"https://www.zbmath.org/authors/?q=ai:li.risong"Lu, Tianxiu"https://www.zbmath.org/authors/?q=ai:lu.tianxiu"Chen, Guanrong"https://www.zbmath.org/authors/?q=ai:chen.guanrong"Yang, Xiaofang"https://www.zbmath.org/authors/?q=ai:yang.xiaofangSummary: For a compact metric space \(Y\) and a continuous map \(g:Y\rightarrow Y\), the collective accessibility and collectively Kato chaotic of the dynamical system \((Y, g)\) were defined. The relations between topologically weakly mixing and collective accessibility, or strong accessibility, or strongly Kato chaos were studied. Some common properties of \(g\) and \(\overline{g}\) were given. Where \(\overline{g}: \kappa(Y)\rightarrow \kappa(Y)\) is defined as \(\overline{g}(B)=g(B)\) for any \(B\in\kappa(Y)\), and \(\kappa(Y)\) is the collection of all nonempty compact subsets of \(Y\). Moreover, it is proved that \(g\) is collectively accessible (or strongly accessible) if and only if \(\overline{g}\) in \(w^e\)-topology is collectively accessible (or strongly accessible).Complete regularity of Ellis semigroups of \(\mathbb{Z}\)-actionshttps://www.zbmath.org/1483.370142022-05-16T20:40:13.078697Z"Barge, Marcy"https://www.zbmath.org/authors/?q=ai:barge.marcy-m"Kellendonk, Johannes"https://www.zbmath.org/authors/?q=ai:kellendonk.johannesSummary: It is shown that the Ellis semigroup of a \(\mathbb{Z}\)-action on a compact totally disconnected space is completely regular if and only if forward proximality coincides with forward asymptoticity and backward proximality coincides with backward asymptoticity. Furthermore, the Ellis semigroup of a \(\mathbb{Z}\)- or \(\mathbb{R}\)-action for which forward proximality and backward proximality are transitive relations is shown to have at most two left minimal ideals. Finally, the notion of near simplicity of the Ellis semigroup is introduced and related to the above.Max-min theorems for weak containment, square summable homoclinic points, and completely positive entropyhttps://www.zbmath.org/1483.370152022-05-16T20:40:13.078697Z"Hayes, Ben"https://www.zbmath.org/authors/?q=ai:hayes.benSummary: We prove a max-min theorem for weak containment in the context of algebraic actions. Specifically, we show that given an algebraic action \(G\curvearrowright X\), there is a maximal, closed \(G\)-invariant subgroup \(Y\) of \(X\) so that \(G\curvearrowright(Y,m_Y)\) is weakly contained in a Bernoulli shift. This subgroup is also the minimal closed subgroup so that any action weakly contained in a Bernoulli shift is \(G\curvearrowright X/Y\)-ergodic ``in the presence of \(G\curvearrowright X\).'' We give several applications, including a major simplification of the proof that measure entropy equals topological entropy for principal algebraic actions whose associated convolution operator is injective. We also deduce from our techniques that algebraic actions whose square summable homoclinic group is dense have completely positive entropy when the acting group is sofic.Nilpotent Cantor actionshttps://www.zbmath.org/1483.370162022-05-16T20:40:13.078697Z"Hurder, Steven"https://www.zbmath.org/authors/?q=ai:hurder.steven-e"Lukina, Olga"https://www.zbmath.org/authors/?q=ai:lukina.olgaA nilpotent Cantor action is a minimal equicontinuous action of a finitely generated group \(\Gamma\) on a Cantor space \(X\), where \(\Gamma\) contains a finitely generated nilpotent subgroup \(\Gamma_0\) of finite index. Nilpotent Cantor actions arise in the classification of renormalizable groups; that is, finitely generated groups which admit a proper self-embedding with image of finite index. These groups arise in the study of laminations with the shape of a compact manifold, and in the classification of generalized Hirsch foliations.
The authors prove that any effective or faithful action of a finitely generated group on a Cantor space, which is continuously orbit equivalent to a nilpotent Cantor action, must itself be a nilpotent Cantor action. As an application, they obtain new invariants of nilpotent Cantor actions under continuous orbit equivalence. The virtual nilpotency class \(vc(\Gamma)\) of a finitely generated virtually nilpotent group \(\Gamma\) is defined as the length of a central series for a torsion-free nilpotent subgroup of finite index. The second main result is that two finitely generated groups admitting continuously orbit equivalent effective Cantor actions have the same virtual nilpotency class.
Reviewer: Mahender Singh (Sahibzada Ajit Singh Nagar)Asymptotic expansivityhttps://www.zbmath.org/1483.370172022-05-16T20:40:13.078697Z"Lee, K."https://www.zbmath.org/authors/?q=ai:lee.keonhee"Morales, C. A."https://www.zbmath.org/authors/?q=ai:morales.carlos-arnoldo"Villavicencio, H."https://www.zbmath.org/authors/?q=ai:villavicencio.h-andresSummary: We study continuous maps of metric spaces for which two nearby orbits are asymptotic (termed \textit{asymptotically expansive} for short). We also analyze the \textit{bi-asymptotically expansive homeomorphisms} namely asymptotically expansive homeomorphisms with asymptotically expansive inverse. Indeed, we obtain necessary and sufficient conditions for asymptotic expansivity, characterize the asymptotically expansive homeomorphisms of the circle, prove a spectral decomposition theorem and estimate the entropy through the growth rate of the periodic orbits.Continuum-wise expansive measureshttps://www.zbmath.org/1483.370182022-05-16T20:40:13.078697Z"Shin, Bomi"https://www.zbmath.org/authors/?q=ai:shin.bomiThe fundamental notion of expansiveness for homeomorphisms on metric spaces, introduced as ``unstable'' in [\textit{W. R. Utz}, Proc. Am. Math. Soc. 1, 769--774 (1950; Zbl 0040.09903)] has been generalized, weakened, and extended in multiple ways. Among these notions, \textit{H. Kato} [Can. J. Math. 45, No. 3, 576--598 (1993; Zbl 0797.54047)] introduced continuum-wise cw-expansiveness, \textit{B. Carvalho} and \textit{W. Cordeiro} [J. Differ. Equations 261, No. 6, 3734--3755 (2016; Zbl 1360.37026)]
introduced \(N\)-expansiveness, \textit{J. Li} and \textit{R. Zhang} [J. Dyn. Differ. Equations 29, No. 3, 877--894 (2017; Zbl 1379.37021)] introduced countable expansiveness. In a different direction the expansive property was extended to Borel measures, roughly connected to the idea of the probability of a pair of orbits remaining close to within a prescribed radius being zero. Here the notion of continuum-wise expansivity is extended to measures, giving the notion of cw-expansive measure. It is shown that the set of expansive measures is strictly contained in the set of cw-expansive measures, and the cw-expansive measures for equicontinuous homeomorphisms are characterized in terms of nonlocally connected points. Other properties for interval and circle settings are studied and analogues of \(N\)-expansiveness are also introduced.
Reviewer: Thomas B. Ward (Newcastle)Ubiquity of entropies of intermediate factorshttps://www.zbmath.org/1483.370192022-05-16T20:40:13.078697Z"Mcgoff, Kevin"https://www.zbmath.org/authors/?q=ai:mcgoff.kevin"Pavlov, Ronnie"https://www.zbmath.org/authors/?q=ai:pavlov.ronnieSummary: We consider topological dynamical systems \((X, T)\), where \(X\) is a compact metrizable space and \(T\) denotes an action of a countable amenable group \(G\) on \(X\) by homeomorphisms. For two such systems \((X, T)\) and \((Y, S)\) and a factor map \(\pi : X \to Y\), an intermediate factor is a topological dynamical system \((Z, R)\) for which \(\pi\) can be written as a composition of factor maps \(\psi : X \to Z\) and \(\varphi : Z \to Y\). In this paper, we show that for any countable amenable group \(G\), for any \(G\)-subshifts \((X, T)\) and \((Y, S)\), and for any factor map \(\pi : X \to Y\), the set of entropies of intermediate subshift factors is dense in the interval \([ h(Y, S), h(X, T)]\). As a corollary, we also prove that if \((X, T)\) and \((Y, S)\) are zero-dimensional \(G\)-systems, then the set of entropies of intermediate zero-dimensional factors is equal to the interval \([ h(Y,S), h(X,T)]\). Our proofs rely on a generalized Marker Lemma that may be of independent interest.Asymptotic decoupling and weak Gibbs measures for finite alphabet shift spaceshttps://www.zbmath.org/1483.370202022-05-16T20:40:13.078697Z"Pfister, C.-E."https://www.zbmath.org/authors/?q=ai:pfister.charles-edouard"Sullivan, W. G."https://www.zbmath.org/authors/?q=ai:sullivan.wayne-gThe paper deals with equilibrium states for continuous functions on a large class of finite-alphabet shift spaces. The authors study the decoupling condition on shift spaces and the space of functions of bounded total oscillations on shift spaces. Their properties and examples are presented. Let \(A\) be a finite set and \(L=\mathbb{Z}^d\). As the main result, the authors prove that if a shift space \(X\subset A^{L}\) satisfies the decoupling condition and \(\phi\) is a function with bounded total oscillations on \(X\), then an equilibrium measure \(\nu\) for \(\phi\) is a weak Gibbs measure for \(\phi-P(\phi)\) where \(P(\phi)\) is the topological pressure of \(\phi\). Then they obtain a full large-deviation principle for the empirical measures on \((X, \nu)\). They prove that if \(X\) is a shift space satisfying the decoupling condition then the ergodic measures on \(X\) are entropy dense. An example of a function of bounded total oscillations not satisfying the Bowen property is provided.
Reviewer: Yuki Yayama (Chiilán)Mean equicontinuity and mean sensitivity on cellular automatahttps://www.zbmath.org/1483.370212022-05-16T20:40:13.078697Z"De Los Santos Baños, Luguis"https://www.zbmath.org/authors/?q=ai:de-los-santos-banos.luguis"García-Ramos, Felipe"https://www.zbmath.org/authors/?q=ai:garcia-ramos.felipeSummary: We show that a cellular automaton (or shift-endomorphism) on a transitive subshift is either almost equicontinuous or sensitive. On the other hand, we construct a cellular automaton on a full shift (hence a transitive subshift) that is neither almost mean equicontinuous nor mean sensitive.Chaos and ergodicity are decidable for linear cellular automata over \((\mathbb{Z}/m\mathbb{Z})^n\)https://www.zbmath.org/1483.370222022-05-16T20:40:13.078697Z"Dennunzio, Alberto"https://www.zbmath.org/authors/?q=ai:dennunzio.alberto"Formenti, Enrico"https://www.zbmath.org/authors/?q=ai:formenti.enrico"Grinberg, Darij"https://www.zbmath.org/authors/?q=ai:grinberg.darij"Margara, Luciano"https://www.zbmath.org/authors/?q=ai:margara.lucianoSummary: We prove that important properties describing complex behaviours as ergodicity, chaos, topological transitivity, and topological mixing, are decidable for one-dimensional linear cellular automata (LCA) over \((\mathbb{Z}/m\mathbb{Z})^n\) (Theorem 6 and Corollary 7), a large and important class of cellular automata (CA) which are able to exhibit the complex behaviours of general CA and are used in applications. In particular, we provide a decidable characterization of ergodicity, which is known to be equivalent to all the above mentioned properties, in terms of the characteristic polynomial of the matrix associated with LCA. We stress that the setting of LCA over \((\mathbb{Z}/m\mathbb{Z})^n\) with \(n > 1\) is more expressive, gives rise to much more complex dynamics, and is more difficult to deal with than the already investigated case \(n = 1\). The proof techniques from [\textit{M. Itô} et al., J. Comput. Syst. Sci. 27, 125--140 (1983; Zbl 0566.68047); \textit{G. Manzini} and \textit{L. Margara}, Theor. Comput. Sci. 221, No. 1--2, 157--177 (1999; Zbl 0930.68090)] used when \(n = 1\) for obtaining decidable characterizations of dynamical and ergodic properties can no longer be exploited when \(n > 1\) for achieving the same goal. Indeed, in order to get the decision algorithm (Algorithm 1) we need to prove a non trivial result of abstract algebra (Theorem 5) which is also of interest in its own.
We also illustrate the impact of our results in real-world applications concerning the important and growing domain of cryptosystems which are often based on one-dimensional LCA over \((\mathbb{Z}/m\mathbb{Z})^n\) with \(n > 1\). As a matter of facts, since cryptosystems have to satisfy the so-called confusion and diffusion properties (ensured by ergodicity and chaos, respectively, of the involved LCA) Algorithm \(^*1\) turns out to be an important tool for building chaotic/ergodic one-dimensional linear CA over \((\mathbb{Z}/m\mathbb{Z} )^n\) and, hence, for improving the existing methods based on them.Phase space classification of an Ising cellular automaton: the Q2R modelhttps://www.zbmath.org/1483.370232022-05-16T20:40:13.078697Z"Montalva-Medel, Marco"https://www.zbmath.org/authors/?q=ai:montalva-medel.marco"Rica, Sergio"https://www.zbmath.org/authors/?q=ai:rica.sergio"Urbina, Felipe"https://www.zbmath.org/authors/?q=ai:urbina.felipeSummary: An exact classification of the different dynamical behaviors that exhibits the phase space of a reversible and conservative cellular automaton, the so-called Q2R model, is shown in this paper. Q2R is a cellular automaton which is a dynamical variation of the Ising model in statistical physics and whose space of configurations grows exponentially with the system size. As a consequence of the intrinsic reversibility of the model, the phase space is composed only by configurations that belong to a fixed point or a cycle. In this work, we classify them in four types accordingly to well differentiated topological characteristics. Three of them -- which we call of type S-I, S-II, and S-III -- share a symmetry property, while the fourth, which we call of type AS does not. Specifically, we prove that any configuration of Q2R belongs to one of the four previous types of cycles. Moreover, at a combinatorial level, we can determine the number of cycles for some small periods which are almost always present in the Q2R. Finally, we provide a general overview of the resulting decomposition of the arbitrary size Q2R phase space and, in addition, we realize an exhaustive study of a small Ising system \((4\times 4)\) which is thoroughly analyzed under this new framework, and where simple mathematical tools are introduced in order to have a more direct understanding of the Q2R dynamics and to rediscover known properties like the energy conservation.About new properties of recurrent motions and minimal sets of dynamical systemshttps://www.zbmath.org/1483.370242022-05-16T20:40:13.078697Z"Afanas'ev, Aleksandr Petrovich"https://www.zbmath.org/authors/?q=ai:afanasev.aleksandr-petrovich"Dzyuba, Sergeĭ Mikhaĭlovich"https://www.zbmath.org/authors/?q=ai:dzyuba.sergei-mikhailovichSummary: The article presents a new property of recurrent motions of dynamical systems. Within a compact metric space, this property establishes the relation between motions of general type and recurrent motions. Besides, this property establishes rather simple behaviour of recurrent motions, thus naturally corroborating the classical definition given in the monograph [\textit{V. V. Nemytskii} and \textit{V. V. Stepanov}, Qualitative theory of differential equations. New Jersey: Princeton University Press (1960; Zbl 0089.29502)].
Actually, the above-stated new property of recurrent motions was announced, for the first time, in the earlier article by the same authors [Differ. Equ. 41, No. 11, 1544--1549 (2005; Zbl 1124.37010); translation from Differ. Uravn. 41, No. 11, 1469--1474 (2005)]. The very same article provides a short proof for the corresponding theorem. The proof in question turned out to be too schematic. Moreover, it (the proof) includes a range of obvious gaps.
Some time ago it was found that, on the basis of this new property, it is possible to show that within a compact metric space \(\alpha \)- and \(\omega \)-limit sets of each and every motion are minimal. Therefore, within a compact metric space each and every motion, which is positively (negatively) stable in the sense of Poisson, is recurrent.
Those results are of obvious significance. They clearly show the reason why, at present, there are no criteria for existence of non-recurrent motions stable in the sense of Poisson. Moreover, those results show the reason why the existing attempts of creating non-recurrent motions, stable in the sense of Poisson, on compact closed manifolds turned out to be futile. At least, there are no examples of such motions.
The key point of the new property of minimal sets is the stated new property of recurrent motions. That is why here, in our present article, we provide a full and detailed proof for that latter property.
For the first time, the results of the present study were reported on the 28th of January, 2020 at a seminar of Dobrushin Mathematics Laboratory at the Institute for Information Transmission Problems named after A. A. Kharkevich of the Russian Academy of Sciences.On totally periodic \(\omega \)-limit sets for monotone maps on regular curveshttps://www.zbmath.org/1483.370252022-05-16T20:40:13.078697Z"Mchaalia, Amira"https://www.zbmath.org/authors/?q=ai:mchaalia.amiraSummary: An \(\omega \)-limit set of a continuous self-mapping of a compact metric space \(X\) is said to be totally periodic if all of its points are periodic. In [Chaos Solitons Fractals 75, 91--95 (2015; Zbl 1352.37048)] \textit{G. Askri} and \textit{I. Naghmouchi} proved that if \(f\) is a one-to-one continuous self mapping of a regular curve, then every totally periodic \(\omega \)-limit set of \(f\) is finite. This also holds whenever \(f\) is a monotone map of a local dendrite by \textit{H. Abdelli} in [Chaos Solitons Fractals 71, 66--72 (2015; Zbl 1352.37123)]. In this paper we generalize these results to monotone maps on regular curves. On the other hand, we give some remarks related to expansivity and totally periodic \(\omega \)-limit sets for every continuous map on compact metric space.Localization of the chain recurrent set using shape theory and symbolical dynamicshttps://www.zbmath.org/1483.370262022-05-16T20:40:13.078697Z"Shoptrajanov, M."https://www.zbmath.org/authors/?q=ai:shoptrajanov.martinSummary: The main aim of this paper is localization of the chain recurrent set in shape theoretical framework. Namely, using the intrinsic approach to shape from [\textit{N. Shekutkovski}, Topol. Proc. 39, 27--39 (2012; Zbl 1215.54008)] we present a result which claims that under certain conditions the chain recurrent set preserves local shape properties. We proved this result in [\textit{N. Shekutkovski} and \textit{M. Shoptrajanov}, Topology Appl. 202, 117--126 (2016; Zbl 1341.54021)] using the notion of a proper covering. Here we give a new proof using the Lebesque number for a covering and verify this result by investigating the symbolical image of a couple of systems of differential equations following [\textit{G. Osipenko}, Differ. Uravn. Protsessy Upr. 1998, No. 4, 59--74 (1998; Zbl 07039068)].Entropy on noncompact setshttps://www.zbmath.org/1483.370272022-05-16T20:40:13.078697Z"Cánovas, Jose S."https://www.zbmath.org/authors/?q=ai:canovas.jose-sSummary: In this paper we review and explore the notion of topological entropy for continuous maps defined on non compact topological spaces which need not be metrizable. We survey the different notions, analyze their relationship and study their properties. Some questions remain open along the paper.Genericity of infinite entropy for maps with low regularityhttps://www.zbmath.org/1483.370282022-05-16T20:40:13.078697Z"De Faria, Edson"https://www.zbmath.org/authors/?q=ai:de-faria.edson"Hazard, Peter"https://www.zbmath.org/authors/?q=ai:hazard.peter"Tresser, Charles"https://www.zbmath.org/authors/?q=ai:tresser.charlesSummary: For bi-Lipschitz homeomorphisms of a compact manifold it is known that topological entropy is always finite. For compact manifolds of dimension two or greater, we show that in the closure of the space of bi-Lipschitz homeomorphisms, with respect to either the Hölder or the Sobolev topologies, topological entropy is generically infinite. We also prove versions of the\(C^1\)-Closing Lemma in either of these spaces. Finally, we give examples of homeomorphisms with infinite topological entropy which are Hölder and/or Sobolev of every exponent.Measures of intermediate entropies for star vector fieldshttps://www.zbmath.org/1483.370292022-05-16T20:40:13.078697Z"Li, Ming"https://www.zbmath.org/authors/?q=ai:li.ming.2|li.ming.6|li.ming|li.ming.7|li.ming.1|li.ming.8|li.ming.4|li.ming.5|li.ming.9|li.ming.3"Shi, Yi"https://www.zbmath.org/authors/?q=ai:shi.yi"Wang, Shirou"https://www.zbmath.org/authors/?q=ai:wang.shirou"Wang, Xiaodong"https://www.zbmath.org/authors/?q=ai:wang.xiaodong.4The authors investigate star vector fields on \(d\)-dimensional closed Riemannian manifolds including multisingular hyperbolic vector fields and Lorenz attractors. In particular, they prove that every star vector field has the intermediate entropy property and show the lower semicontinuity of the topological entropy of star vector fields. To obtain their main results, the authors study the approximation of the topological entropy of a star vector field by means of the topological entropy of hyperbolic horseshoes and prove the intermediate entropy property for suspension flows over shifts of finite type generated by irreducible matrices.
Reviewer: Yuki Yayama (Chiilán)The dynamics of nonautonomous dynamical systems with the large deviations theoremhttps://www.zbmath.org/1483.370302022-05-16T20:40:13.078697Z"Tang, Yanjie"https://www.zbmath.org/authors/?q=ai:tang.yanjie"Yin, Jiandong"https://www.zbmath.org/authors/?q=ai:yin.jiandongLet \(X\) be a compact metric space. For each \(n\in \mathbb{N}\) let \(f_n : X \rightarrow X\) be a continuous map and \(\mathcal{F}=\{f_i \}_{i\geq 0}\) denote the sequence \((f_0,f_1, f_2,\dots, f_n,\dots)\). The pair \((X, \mathcal{F})\) is called a nonautonomous discrete dynamical system. Let \(\mathcal{B}(X)\) be the \(\sigma\)-algebra of Borel subsets of \(X\) and \(\mu\) be a probability measure on the measurable space \((X,\mathcal{B}(X))\). The authors introduce the concepts of large deviations and expansive measure for these systems. Then they prove that if the pair \((\mathcal{F}, \mu)\) satisfies the large deviations theorem and \((X,\mathcal{B}(X))\) is topologically strongly ergodic, then \((X,\mathcal{B}(X))\) is ergodically sensitive.
Secondly, the authors prove that if the pair \((\mathcal{F}, \mu)\) satisfies the large deviations theorem in the sequence \(\{s_j\}_{j=1}^{\infty}\), then \((X, \mathcal{F})\) is topologically ergodic, where \(\{s_j\}_{j=1}^{\infty}\subset \mathbb{Z}^{+}\) is an increasing sequence. Further, it is proved that if the pair \((\mathcal{F}, \mu)\) satisfies the large deviations theorem and each measurable set with positive measure with respect to \(\mu\) has a nonempty interior and \(\mathcal{F}\) is topologically strongly ergodic, then \(\mu\) is expansive.
Reviewer: Hasan Akin (Trieste)Reiterative distributional chaos in non-autonomous discrete systemshttps://www.zbmath.org/1483.370312022-05-16T20:40:13.078697Z"Yin, Zongbin"https://www.zbmath.org/authors/?q=ai:yin.zongbin"Xiang, Qiaomin"https://www.zbmath.org/authors/?q=ai:xiang.qiaomin"Wu, Xinxing"https://www.zbmath.org/authors/?q=ai:wu.xinxingThe authors study several types of distributional chaos (DC) and reiterative distributional chaos (RDC) for discrete dynamical systems. They prove that RDC of type \(2\frac{1}{2}\) and type 2 are equivalent for linear operators on Banach spaces. The authors present a basic relation between RDC1 operators and RDC\(2\frac{1}{2}\) operators. Let \(f : X\rightarrow X\) be a continuous linear operator on a Banach space \(X\). It is proved that \(f\) is RDC1 if and only if it is RDC\(2\frac{1}{2}\).
Then, the authors investigate the iterative invariance of various types of RDC for nonautonomous discrete systems. Let \(f_{1,\infty} = \{f_i \}_{i\geq 1}\) be a sequence of self-maps of a metric space \(X\). It is proved that an equicontinuous nonautonomous system \((X,f_{1,\infty})\) exhibits RDC of type \(i\) with \(i\in \{1, 1+ , 2, 2\frac{1}{2},2\frac{1}{2}-\}\), if and only if its \(k\)-th iteration \(f^{[k]}_{1,\infty}\) exhibits RDC of type \(i\) for any \(k\geq 2\)
Reviewer: Hasan Akin (Trieste)Steady Euler flows and Beltrami fields in high dimensionshttps://www.zbmath.org/1483.370322022-05-16T20:40:13.078697Z"Cardona, Robert"https://www.zbmath.org/authors/?q=ai:cardona.robertSummary: Using open books, we prove the existence of a non-vanishing steady solution to the Euler equations for some metric in every homotopy class of non-vanishing vector fields of any odd-dimensional manifold. As a corollary, any such field can be realized in an invariant submanifold of a contact Reeb field on a sphere of high dimension. The solutions constructed are geodesible and hence of Beltrami type, and can be modified to obtain chaotic fluids. We characterize Beltrami fields in odd dimensions and show that there always exist volume-preserving Beltrami fields which are neither geodesible nor Euler flows for any metric. This contrasts with the three-dimensional case, where every volume-preserving Beltrami field is a steady Euler flow for some metric. Finally, we construct a non-vanishing Beltrami field (which is not necessarily volume-preserving) without periodic orbits in every manifold of odd dimension greater than three.Complete integrability of diffeomorphisms and their local normal formshttps://www.zbmath.org/1483.370332022-05-16T20:40:13.078697Z"Jiang, Kai"https://www.zbmath.org/authors/?q=ai:jiang.kai"Stolovitch, Laurent"https://www.zbmath.org/authors/?q=ai:stolovitch.laurentLet \(\Phi\) be a local diffeomorphism on \(\mathbb{K}^n\) (where \(\mathbb{K}\) is \(\mathbb{R}\) or \(\mathbb{C}\)) having the origin as its isolated fixed point. This diffeomorphism is called integrable if there exists \(p\geq1\) pairwise commuting (germs of) diffeomorphisms \(\Phi_1=\Phi\), \(\Phi_2\), \(\dots\), \(\Phi_p\) with \(d\Phi_i(0)=A_i\) and \(q=n-p\) common first integrals \(F_1,\dots,F_q\) of those diffeomorphisms such that:
(i) The diffeomorphisms \(\Phi_i\) are independent, i.e., the matrices \(\{\ln A_i\}\), \(i=1,\dots,p\), are linearly independent over \(\mathbb{K}^n\) (for \(\mathbb{K}=\mathbb{C}\) independence is required for the families of all possible logarithms);
(ii) The first integrals \(F_j\) are functionally independent almost everywhere.
In this case we say that \((\Phi_1,\dots,\Phi_p,F_1,\dots,F_q)\) is a discrete completely integrable system of type \((p,q)\).
The authors consider the normal form problem of a commutative family of germs of analytic or smooth diffeomorphisms at a fixed point and give sufficient conditions which ensure that such an integrable family can be transformed into a normal form by an analytic or smooth transformation if the initial diffeomorphisms are respectively analytic or smooth.
Reviewer: Boris S. Kruglikov (Tromsø)Dynamical zeta functions of Reidemeister typehttps://www.zbmath.org/1483.370342022-05-16T20:40:13.078697Z"Fel'shtyn, Alexander"https://www.zbmath.org/authors/?q=ai:felshtyn.alexander"Ziętek, Malwina"https://www.zbmath.org/authors/?q=ai:zietek.malwinaSummary: In this paper we study dynamical representation theory zeta functions counting numbers of fixed irreducible representations for iterations of group endomorphism. The rationality and functional equation for these zeta functions are proven for several classes of groups. We prove Pólya-Carlson dichotomy between rationality and a natural boundary for analytic behavior of the Reidemeister zeta functions for a large class of automorphisms of infinitely generated abelian groups. We also establish the connection between the Reidemeister zeta function and dynamical representation theory zeta functions under restriction of endomorphism to a subgroup.Fixed points of the Ruelle-Thurston operator and the Cauchy transformhttps://www.zbmath.org/1483.370352022-05-16T20:40:13.078697Z"Levin, Genadi"https://www.zbmath.org/authors/?q=ai:levin.genadi-mSummary: We give necessary and sufficient conditions for a function in a naturally appearing function space to be a fixed point of the Ruelle-Thurston operator associated to a rational function (see Lemma 2.1). The proof uses essentially a 2020 paper of the author et al. [Nonlinearity 33, No. 8, 3970--4012 (2020; Zbl 1453.37044)]. As an immediate consequence, in Theorem 1 and Lemma 2.2 we revisit Theorem 1 and Lemma 5.2 of the author [in: Frontiers in complex dynamics. In celebration of John Milnor's 80th birthday. Based on a conference, Banff, Canada, February 2011. Princeton, NJ: Princeton University Press. 163--196 (2014; Zbl 1348.37075)].Phase transitions on the Markov and Lagrange dynamical spectrahttps://www.zbmath.org/1483.370362022-05-16T20:40:13.078697Z"Lima, Davi"https://www.zbmath.org/authors/?q=ai:lima.davi"Moreira, Carlos Gustavo"https://www.zbmath.org/authors/?q=ai:moreira.carlos-gustavo-t-de-aGiven a surface \(M\), a \(C^2\) diffeomorphism \(\varphi\) on \(M\), and a continuous observable \(f:M\to\mathbb{R}\), the Lagrange value of a point \(x\in M\) is the limit of the suprema of the observable value along the orbit of \(x\), that is, \(\limsup_{n\to\infty}f(\varphi^n(x))\), the Markov value of \(x\) is the suprema of the observable value along the orbit of \(x\), that is, \(\sup_{n\in\mathbb{Z}}f(\varphi^n(x))\). The Lagrange spectrum is the union of the Lagrange values, while the Markov spectrum is the union of the Markov values. Given a diffeomorphism \(\varphi\) with a horseshoe, with Hausdorff dimension bigger than \(1\), there exist a residual subset from an open neighborhood of \(\varphi\) and a residual subset of the smooth observable functions such that the existence of transition points for the Lagrange spectrum and the Markov spectrum can be proved. The Hausdorff dimension of the subsets of the spectrum jumps from values less than one to one.
Reviewer: Xu Zhang (Weihai)Conformality for a robust class of non-conformal attractorshttps://www.zbmath.org/1483.370372022-05-16T20:40:13.078697Z"Pozzetti, Maria Beatrice"https://www.zbmath.org/authors/?q=ai:pozzetti.maria-beatrice"Sambarino, Andrés"https://www.zbmath.org/authors/?q=ai:sambarino.andres"Wienhard, Anna"https://www.zbmath.org/authors/?q=ai:wienhard.anna-katharinaThe authors study the Hausdorff dimension of limit sets for Anosov representations. In doing so, the framework of hyperconvex representations is extended and a convergence property for them -- an analogue of a differentiability property -- is established. As an application of this convergence notion, it is proved that the Hausdorff dimension of the limit set of a hyperconvex representation is equal to a suitably chosen critical exponent. More specifically, the paper covers Anosov representations, thus providing an upper bound for the Hausdorff dimension of limit sets, a discussion about local conformality and \((q,p,r)\)-hyperconvexity, and investigating differentiability properties and examples of locally conformal representations. Finally, some fundamental groups of surfaces are discussed.
Reviewer: Christian Pötzsche (Klagenfurt)A new class of Lyapunov functions for stability analysis of singular dynamical systems. Elements of \(p\)-regularity theoryhttps://www.zbmath.org/1483.370382022-05-16T20:40:13.078697Z"Evtushenko, Yu. G."https://www.zbmath.org/authors/?q=ai:evtushenko.yuri-g"Tret'yakov, A. A."https://www.zbmath.org/authors/?q=ai:tretyakov.alexey-aSummary: A new approach is proposed for studying the stability of dynamical systems in the case when traditional Lyapunov functions are ineffective or not applicable for research at all. The main tool used to analyze degenerate systems is the so-called \(p\)-factor Lyapunov function, which makes it possible to reduce the original problem to a new one based on constructions of \(p\)-regularity theory. An example of a meaningful application of the considered method is given.Global trace formula for ultra-differentiable Anosov flowshttps://www.zbmath.org/1483.370392022-05-16T20:40:13.078697Z"Jézéquel, Malo"https://www.zbmath.org/authors/?q=ai:jezequel.maloAssume smooth vector field \(V\) on a smooth manifold generates an Anosov flow \((\phi_t)_{t\in\mathbb{R}}\). One of the profound connections between these two objects arises via various trace formulas relating distributions defined by suitably weighted sums over spectral data from a differential operator associated to \(V\) and suitably weighted sums over periodic orbits of \((\phi_t)\). There are considerable technical difficulties to this however, in part because the associated differential operator is not elliptic and so may have wild spectrum on the space of \(L^2\) functions. One approach due to \textit{O. Butterley} and \textit{C. Liverani} [J. Mod. Dyn. 1, No. 2, 301--322 (2007; Zbl 1144.37011); J. Mod. Dyn. 7, No. 2, 255--267 (2013; Zbl 1328.37017)] is to find a scale of anisotropic Banach spaces of distributions on the manifold and use these to define a suitable Ruelle spectrum for \(V\). Many technical difficulties remain, and here the results found by the author in [J. Spectr. Theory 10, No. 1, 185--249 (2020; Zbl 1442.37038)] are used to show that the trace formula conjectured by \textit{S. Dyatlov} and \textit{M. Zworski} [Ann. Sci. Éc. Norm. Supér. (4) 49, No. 3, 543--577 (2016; Zbl 1369.37028)] holds for Anosov flows under a smoothness condition more stringent than \(C^{\infty}\) but less stringent than being in the Gevrey class.
Reviewer: Thomas B. Ward (Newcastle)Weak\(\ast\) and entropy approximation of nonhyperbolic measures: a geometrical approachhttps://www.zbmath.org/1483.370402022-05-16T20:40:13.078697Z"Díaz, Lorenzo J."https://www.zbmath.org/authors/?q=ai:diaz.lorenzo-justiniano"Gelfert, Katrin"https://www.zbmath.org/authors/?q=ai:gelfert.katrin"Santiago, Bruno"https://www.zbmath.org/authors/?q=ai:santiago.brunoLet \(M\) be a closed Riemannian manifold. The authors consider the set \(\mathrm{RTPH}^1(M)\), namely the set of those \(C^1\) diffeomorphisms \(f\) of \(M\) so that there is a \(C^1\) neighborhood \(\mathcal V_f\) of \(f\) such that each diffeomorphism in \(\mathcal V_f\) is nonhyperbolic, has a partially hyperbolic splitting with one-dimensional central bundle and whose strong stable and strong unstable foliations are both minimal. Under these assumptions they show that there exists a \(C^1\) open and dense subset of \(\mathrm{RTPH}^1(M)\) such that every nonhyperbolic ergodic measure (i.e., with zero central exponent) can be approximated in the weak\(\ast\) topology and in entropy by measures supported in basic sets with positive (negative) central Lyapunov exponent.
This result extends a classical result by \textit{A. Katok} [Publ. Math., Inst. Hautes Étud. Sci. 51, 137--173 (1980; Zbl 0445.58015)]
which states that ergodic hyperbolic measures can be approximated in the weak\(\ast\) topology by measures supported in basic sets.
The authors also show that for an open and dense subset in \(\mathrm{RTPH}^1(M)\) any nonhyperbolic ergodic measure belongs to the intersection of the convex hulls of the measures with positive central exponent and with negative central exponent.
Reviewer: Miguel Paternain (Montevideo)Topological obstructions for robustly transitive endomorphisms on surfaceshttps://www.zbmath.org/1483.370412022-05-16T20:40:13.078697Z"Lizana, C."https://www.zbmath.org/authors/?q=ai:lizana.cristina"Ranter, W."https://www.zbmath.org/authors/?q=ai:ranter.wagnerThe authors show that every robustly transitive surface endomorphism displaying critical points is a partially hyperbolic endomorphism and that the only surfaces that might admit robustly transitive endomorphisms are either the torus \(\mathbb{T}^2\) or the Klein bottle \(\mathbb{K}^2\). They also prove that the action of a transitive endomorphism admitting a dominated splitting in the first homology group of the surface has at least one eigenvalue with modulus larger than one.
Reviewer: Miguel Paternain (Montevideo)Koszul information geometry and Souriau Lie group thermodynamicshttps://www.zbmath.org/1483.370422022-05-16T20:40:13.078697Z"Barbaresco, Frédéric"https://www.zbmath.org/authors/?q=ai:barbaresco.fredericSummary: The François Massieu 1869 idea to derive some mechanical and thermal properties of physical systems from ``Characteristic Functions'', was developed by Gibbs and Duhem in thermodynamics with the concept of potentials, and introduced by Poincaré in probability. This paper deals with generalization of this Characteristic Function concept by Jean-Louis Koszul in Mathematics and by Jean-Marie Souriau in Statistical Physics. The Koszul-Vinberg Characteristic Function (KVCF) on convex cones will be presented as cornerstone of ``Information Geometry'' theory, defining Koszul Entropy as Legendre transform of minus the logarithm of KVCF, and Fisher Information Metrics as hessian of these dual functions, invariant by their automorphisms. In parallel, Souriau has extended the Characteristic Function in Statistical Physics looking for other kinds of invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat and momentum as pure geometrical objects. In covariant Souriau model, Gibbs equilibriums states are indexed by a geometric parameter, the Geometric (Planck) Temperature, with values in the Lie algebra of the dynamical Galileo/Poincaré groups, interpreted as a space-time vector, giving to the metric tensor a null Lie derivative. Fisher Information metric appears as the opposite of the derivative of Mean ``Moment map'' by geometric temperature, equivalent to a Geometric Capacity or Specific Heat. These elements has been developed by author.
For the entire collection see [Zbl 1470.00021].On generalized compensation functions for factor maps between shift spaces on countable alphabetshttps://www.zbmath.org/1483.370432022-05-16T20:40:13.078697Z"Lacalle, Camilo"https://www.zbmath.org/authors/?q=ai:lacalle.camilo"Yayama, Yuki"https://www.zbmath.org/authors/?q=ai:yayama.yukiFor one-sided shifts on finite alphabets \((X,\sigma_X)\) and \((Y,\sigma_Y)\), with spaces of invariant Borel probability measures \(M(X,\sigma_X)\) and \(M(Y,\sigma_Y)\) and a continuous factor map \(\pi\colon (X,\sigma_X)\to(Y,\sigma_Y)\), a continuous real-valued function \(F\) on \(X\) is said to be a compensation function if
\[
\sup_{\mu\in M(X,\sigma_X)} \left\{h_{\mu}(\sigma_X)+\int(F+f\circ\pi){\mathrm{d}}\mu\right\} = \sup_{\nu\in M(Y,\sigma_Y)} \left\{h_{\nu}(\sigma_Y)+\int f{\mathrm{d}}\nu\right\}
\]
for all continuous functions \(f\) on \(Y\). This is said to be a saturated compensation function if \(F=G\circ\pi\) for some continuous function \(G\) on \(Y\). Compensation functions may be used to explore the relationship between the equilibrium states for \(F+f\circ\pi\) and those of \(f\). Here the setting is changed to shifts with countable alphabets for a particular class of one-block factor maps. The thermodynamic formalism for sequences on countable shifts is used to generalize existing results on factor maps between compact systems to this sort of non-compact setting.
Reviewer: Thomas B. Ward (Newcastle)Extremal parameters and their duals for boundary maps associated to Fuchsian groupshttps://www.zbmath.org/1483.370442022-05-16T20:40:13.078697Z"Abrams, Adam"https://www.zbmath.org/authors/?q=ai:abrams.adamLet \(M\) be a closed, oriented, compact surface of genus \(g \geq 2\) and constant negative curvature. This surface is a quotient of the unit disc \(\mathbb{D}\) by a finitely generated Fuchsian group of the first kind acting freely on \(\mathbb{D}.\) The boundary of the disc will be denoted by \(\mathbb{S}\). The surface \(M\) admits a fundamental polygon \(\mathscr{F}\) of \(8g-4\) sides with an appropriated identification. The sides satisfy the extension condition of \textit{R. Bowen} and \textit{C. Series} [Publ. Math., Inst. Hautes Étud. Sci. 50, 153--170 (1979; Zbl 0439.30033)], namely the geodesic extensions of these segments never intersect the interior of the tiling sets \(\gamma\mathscr{F}\), \(\gamma \in \Gamma\). Let \(P_i\) and \(Q_{i+1}\) with the subindices taken \(\text{mod}\ 8g-4\), the endpoints of the geodesic which contains the side \(i\) of \(\mathscr{F}\). Let us denote by \(T_i\) the generators of \(\Gamma\) associated to \(\mathscr{F}\). The Bowen-Series boundary map is generalized by means of a set of \(8g-4\) parameters, \(\overline{A}=\{A_1, A_2, \dots, A_{8g-4}\}\) with \(A_i \in [P_i,Q_i]\) defining a boundary map \(f_{\overline{A}}: \mathbb{S} \rightarrow \mathbb{S}\) by \(f_{\overline{A}}(x)=T_i(x)\) if \(x\in[A_i,A_{i+1})\). The map \(F_{\overline{A}}:\mathbb{S}\times \mathbb{S} \setminus \Delta \rightarrow \mathbb{S}\times \mathbb{S} \setminus \Delta\), where \(\Delta=\{(x,x): x \in \mathbb{S}\}\) defined as \(F_{\overline{A}}(u,w) = (T_i(u), T_i(w))\) if \((u,w) \in [A_i, A_i+1)\) is bijective when restricted to a suitable domain \(\Omega_{\overline{A}}\). This map \(F_{\overline{A}}:\Omega_{\overline{A}}\rightarrow \Omega_{\overline{A}}\) is called the natural extension of the boundary map. The set \(\overline{A}\) is said to be extremal if \(A_i \in \{P_i,Q_i\}\) for all \(1 \leq i \leq 8g- 4.\)
In this paper, the author describes arithmetic cross-sections for geodesic flows on surfaces \(M\) as above. In Section 2, he shows that if the boundary map parameters are extremal then the natural extension map has a domain with finite rectangular structure, and the associated arithmetic cross-section is parameterized by this set. This domain is characterized in Theorem 10. By means of this construction, the geodesic flow can be represented as a special flow over a symbolic system of coding sequences. In Section 3, dual parameters are considered. It is proved in Theorem 25 that every extremal parameter choice has a dual parameter choice for which the natural extension of the boundary map also has a finite rectangular structure domain. Several explicit examples and nice figures are provided throughout the paper with \(g=2\).
Reviewer: Ernesto Martínez (Madrid)New multi-scroll attractors obtained via Julia set mappinghttps://www.zbmath.org/1483.370452022-05-16T20:40:13.078697Z"Atangana, Abdon"https://www.zbmath.org/authors/?q=ai:atangana.abdon"Bouallegue, Ghaith"https://www.zbmath.org/authors/?q=ai:bouallegue.ghaith"Bouallegue, Kais"https://www.zbmath.org/authors/?q=ai:bouallegue.kaisSummary: The Cobra attractor have attracted very recently and the model has been investigated using classical differential operators with integer and non-integer order. The model, in the case of fractional differential operator, is able to replicate indeed the Cobra for some values of fractional order. On the other hand, Julia set has been used for many purposes, in this paper; we develop a procedure that combines some chaotic attractors with the Julia set mapping to obtain multi-roll attractors. Using our algorithm, we obtained for the first time a lung of human being.Energy analysis of Sprott-A system and generation of a new Hamiltonian conservative chaotic system with coexisting hidden attractorshttps://www.zbmath.org/1483.370462022-05-16T20:40:13.078697Z"Jia, Hongyan"https://www.zbmath.org/authors/?q=ai:jia.hongyan"Shi, Wenxin"https://www.zbmath.org/authors/?q=ai:shi.wenxin"Wang, Lei"https://www.zbmath.org/authors/?q=ai:wang.lei.7|wang.lei.9|wang.lei.15|wang.lei.17|wang.lei.18|wang.lei.6|wang.lei.11|wang.lei.5|wang.lei.19|wang.lei.16|wang.lei.8|wang.lei.14|wang.lei|wang.lei.4"Qi, Guoyuan"https://www.zbmath.org/authors/?q=ai:qi.guoyuanSummary: The paper firstly investigates energy cycle of the Sprott-A system by transforming the Sprott-A system into the Kolmogorov-type system. We found the dynamics of the Sprott-A system are influenced by the change along the energy exchange between the conservative energy and the external supplied energy. And the action of the external supplied torque is the main reason that the Sprott-A system generates chaos. Secondly, based on energy analysis of the Sprott-A system, a new four-dimension (4-D) chaotic system is obtained. The new 4-D chaotic system is a conservative system with a constant Hamiltonian energy. Besides, it is also a no-equilibrium system, this means that the new 4-D chaotic system can exhibit hidden characteristics. Further, the coexisting hidden attractors are found when selecting different initial points. Finally, the new 4-D chaotic system is implemented by FPGA, and the coexisting attractors observed are consistent with those found in numerical analysis, which in experiment verifies the existence of coexisting hidden attractors of the new 4-D chaotic system from physical point of view.A new megastable nonlinear oscillator with infinite attractorshttps://www.zbmath.org/1483.370472022-05-16T20:40:13.078697Z"Leutcho, Gervais Dolvis"https://www.zbmath.org/authors/?q=ai:leutcho.gervais-dolvis"Jafari, Sajad"https://www.zbmath.org/authors/?q=ai:jafari.sajad"Hamarash, Ibrahim Ismael"https://www.zbmath.org/authors/?q=ai:hamarash.ibrahim-ismael"Kengne, Jacques"https://www.zbmath.org/authors/?q=ai:kengne.jacques"Tabekoueng Njitacke, Zeric"https://www.zbmath.org/authors/?q=ai:njitacke.zeric-tabekoueng"Hussain, Iqtadar"https://www.zbmath.org/authors/?q=ai:hussain.iqtadarSummary: Dynamical systems with megastable properties are very rare in the literature. In this paper, we introduce a new two-dimensional megastable dynamical system with a line of equilibria, having an infinite number of stable states. By modifying this new system with temporally-periodic forcing term, a new two-dimensional non-autonomous nonlinear oscillator capable to generate an infinite number of coexisting limit cycle attractors, torus attractors and, strange attractors is constructed. The analog implementation of the new megastable oscillator is investigated to further support numerical analyses and henceforth validate the mathematical model.Hyperchaos in 3-D piecewise smooth mapshttps://www.zbmath.org/1483.370482022-05-16T20:40:13.078697Z"Patra, Mahashweta"https://www.zbmath.org/authors/?q=ai:patra.mahashweta"Banerjee, Soumitro"https://www.zbmath.org/authors/?q=ai:banerjee.soumitroSummary: In this paper, we show various ways of the occurrence of a hyperchaotic orbit in 3D piecewise linear normal form maps. We show that hyperchaotic orbit can be born from a periodic orbit or a quasiperiodic orbit in various ways like-(a) a direct transition to a hyperchaotic orbit from a periodic orbit or a from a quasiperiodic orbit through border collision bifurcation; (b) a transition from a periodic orbit to a hyperchaotic orbit via quasiperiodic and chaotic orbit; (c) a transition from a mode-locked periodic orbit to a hyperchaotic orbit via higher dimensional torus. We also show bifurcations where a hyperchaotic orbit bifurcates to a different hyperchaotic orbit or a three-piece hyperchaotic orbit. We further show period increment with the coexistence of hyperchaotic attractors. Moreover, we numerically calculate the existence region of a hyperchaotic orbit in the parameter space region.On the finiteness of attractors for one-dimensional maps with discontinuitieshttps://www.zbmath.org/1483.370492022-05-16T20:40:13.078697Z"Brandão, P."https://www.zbmath.org/authors/?q=ai:brandao.p-a|brandao.paulo"Palis, J."https://www.zbmath.org/authors/?q=ai:palis.jacob-jun"Pinheiro, V."https://www.zbmath.org/authors/?q=ai:pinheiro.vilton|pinheiro.vladiaThe main result of this paper is that every piecewise \(C^3\) interval map with negative Schwarzian derivative has at most only a finite number of attractors. Earlier related results were given by several authors, e.g., see [\textit{A. M. Blokh} and \textit{M. Yu. Lyubich}, Ergodic Theory Dyn. Syst. 9, No. 4, 751--758 (1989; Zbl 0665.58024); Ann. Sci. Éc. Norm. Supér. (4) 24, No. 5, 545--573 (1991; Zbl 0790.58024); \textit{M. Lyubich}, ``Ergodic theory for smooth one dimensional dynamical systems'', Preprint, \url{arXiv:math/9201286}].
The authors also derive further results about attractors of contracting Lorenz maps. These results extend those in [\textit{M. St. Pierre}; Diss. Math. 382, 134 p. (1999; Zbl 0956.37006); \textit{S. van Strien} and \textit{E. Vargas}, J. Am. Math. Soc. 17, No. 4, 749--782 (2004; Zbl 1073.37043)].
Reviewer: Steve Pederson (Atlanta)There are no \(\sigma\)-finite absolutely continuous invariant measures for multicritical circle mapshttps://www.zbmath.org/1483.370502022-05-16T20:40:13.078697Z"de Faria, Edson"https://www.zbmath.org/authors/?q=ai:de-faria.edson"Guarino, Pablo"https://www.zbmath.org/authors/?q=ai:guarino.pabloThe authors state that it is well known that every multicritical circle map without periodic orbits admits a unique invariant Borel probability measure which is purely singular with respect to Lebesgue measure. Their main purpose is to understand whether or not such a map can leave invariant an infinite \(\sigma\)-finite invariant measure which is absolutely continuous with respect to Lebesgue measure. They show that the answer is negative using a result of \textit{Y. Katznelson} [J. Anal. Math. 31, 1--18 (1977; Zbl 0346.28012)].
Reviewer: Steve Pederson (Atlanta)Genericity of chaos for colored graphshttps://www.zbmath.org/1483.370512022-05-16T20:40:13.078697Z"Lijó, Ramón Barral"https://www.zbmath.org/authors/?q=ai:barral-lijo.ramon"Nozawa, Hiraku"https://www.zbmath.org/authors/?q=ai:nozawa.hirakuSummary: To each colored graph one can associate its closure in the universal space of isomorphism classes of pointed colored graphs, and this subspace can be regarded as a generalized subshift. Based on this correspondence, we introduce two definitions for chaotic (colored) graphs, one of them analogous to Devaney's. We show the equivalence of our two novel definitions of chaos, proving their topological genericity in various subsets of the universal space.On the relation between action and linkinghttps://www.zbmath.org/1483.370522022-05-16T20:40:13.078697Z"Senior, David Bechara"https://www.zbmath.org/authors/?q=ai:senior.david-bechara"Hryniewicz, Umberto L."https://www.zbmath.org/authors/?q=ai:hryniewicz.umberto-l"Salomão, Pedro A. S."https://www.zbmath.org/authors/?q=ai:salomao.pedro-a-sThe authors focus on numerical invariants of contact forms in dimension three.
Some questions related to the application of these invariants in geometry of contact forms (periodic orbits, Reeb flows, contact volume, systolic norm, Reinhardt domains, Seifert surfaces, etc.) are formulated. The main result is so-called Action-Linking Lemma that contains an important identity for every adapted Seifert surface in a contact manifold.
Reviewer: Mihail Banaru (Smolensk)On the cycles of components of disconnected Julia setshttps://www.zbmath.org/1483.370532022-05-16T20:40:13.078697Z"Cui, Guizhen"https://www.zbmath.org/authors/?q=ai:cui.guizhen"Peng, Wenjuan"https://www.zbmath.org/authors/?q=ai:peng.wenjuanThe main goal of the paper is to carry out the study of cycles of components of disconnected Julia sets. For any integers \(d\geq3\) and \(n\geq1\), the authors construct a hyperbolic rational map of degree \(d\) with \(n\) cycles of the connected components of its Julia set except single points and Jordan curves.
Reviewer: Haifeng Chu (Xi'an)Escaping Fatou components of transcendental self-maps of the punctured planehttps://www.zbmath.org/1483.370542022-05-16T20:40:13.078697Z"Martí-Pete, David"https://www.zbmath.org/authors/?q=ai:marti-pete.davidHolomorphic self-maps of the Riemann sphere are the most well-studied families of systems in complex dynamics (rational dynamics), followed by self-maps of the once punctured plane (transcendental dynamics). This paper concerns the study of holomorphic self-maps of the twice punctured Riemann sphere, a subject still in its early days.
In transcendental dynamics, an important role is played by the escaping set, which consists of those points which iterate to the essential singularity of the function. In the present setting, however, there are two essential singularities. The main result is that any possible way of escaping is possible for a wandering Fatou component. The author's main technique is approximation theory.
Reviewer: Kirill Lazebnik (New Haven)Exponential polynomials with Fatou and non-escaping sets of finite Lebesgue measurehttps://www.zbmath.org/1483.370552022-05-16T20:40:13.078697Z"Wolff, Mareike"https://www.zbmath.org/authors/?q=ai:wolff.mareikeSummary: We give conditions ensuring that the Fatou set and the complement of the fast escaping set of an exponential polynomial \(f\) both have finite Lebesgue measure. Essentially, these conditions are designed such that \(|f(z)|\geq \exp (|z|^\alpha)\) for some \(\alpha>0\) and all \(z\) outside a set of finite Lebesgue measure.Entropy of real rational surface automorphismshttps://www.zbmath.org/1483.370562022-05-16T20:40:13.078697Z"Diller, Jeffrey"https://www.zbmath.org/authors/?q=ai:diller.jeffrey"Kim, Kyounghee"https://www.zbmath.org/authors/?q=ai:kim.kyoungheeLet \(g \colon Z\rightarrow Z\) be a continuous transformation of a compact metric space \(Z\). Its topological entropy \(\mathrm{h}_{\mathrm{top}}(g)\) is a number which measures the rate at which the dynamics induced by \(g\) creates distinct orbits, when observed with an arbitrarily small, but positive, scale (see [\textit{S. Cantat}, in: Frontiers in complex dynamics. In celebration of John Milnor's 80th birthday. Based on a conference, Banff, Canada, February 2011. Princeton, NJ: Princeton University Press. 463--514 (2014; Zbl 1345.37001)]).
In this paper the authors construct certain surfaces \(X\) as the blow-up of \(\mathbb{P}^2_{\mathbb{C}}\) at a number of suitably chosen real points such that a birational map \(\hat{f}\) of \(\mathbb{P}^2\) of the form \(\hat{f}=T\circ \tau\), where \(T\) is a real automorphism of \(\mathbb{P}^2\) and \(\tau\) the standard quadratic transformation, lifts to an automorphism \(f\) of \(X\). The automorphism \(f\) defines an automorphism \(f_{\mathbb{R}}\) of \(X(\mathbb{R})\), the real submanifold of \(X\). Then the authors compare the topological entropies \(\mathrm{h}_{\mathrm{top}}(f_{\mathbb{R}})\) and \(\mathrm{h}_{\mathrm{top}}(f)\). In particular, they present explicit examples where \(\mathrm{h}_{\mathrm{top}}(f_{\mathbb{R}})= \mathrm{h}_{\mathrm{top}}(f)\) and others where \(\mathrm{h}_{\mathrm{top}}(f_{\mathbb{R}})< \mathrm{h}_{\mathrm{top}}(f)\).
Reviewer: Nikolaos Tziolas (Lefkosía)Asymptotic upper bound for tangential speed of parabolic semigroups of holomorphic self-maps in the unit dischttps://www.zbmath.org/1483.370572022-05-16T20:40:13.078697Z"Cordella, Davide"https://www.zbmath.org/authors/?q=ai:cordella.davideThe author studies continuous semigroups of holomorphic maps in the unit disc \((\Phi_t)_{t\ge 0}\). For a non-elliptic semigroup, \textit{F. Bracci} [Ann. Univ. Mariae Curie-Skłodowska, Sect. A 73, No. 2, 21--43 (2019; Zbl 1436.30007)] introduced and studied three kinds of speeds: the total speed, the orthogonal speed, and the tangential speed. The tangential speed \(v^T(t)\) is related to the slope of convergence of orbits to the Denjoy-Wolff point of the semigroup. In the present paper, the author proves a conjecture in [loc. cit.] claiming that \(\limsup_{t\to\infty}\left(v^T(t)-\frac 12\log t\right)<\infty\) holds for parabolic semigroups.
Reviewer: Barbara Drinovec Drnovsek (Ljubljana)A converse landing theorem in parameter spaceshttps://www.zbmath.org/1483.370582022-05-16T20:40:13.078697Z"Deniz, Aslı"https://www.zbmath.org/authors/?q=ai:deniz.asliSummary: In this article, we prove that for several one-dimensional holomorphic families of holomorphic maps, in the parameter plane, there exists a local piece of a curve that lands at a given parabolic parameter, in the spirit of well-known results about the quadratic and the exponential families. We also show that, under some assumptions, this general result partially answers the existence and landing questions of ray structures in the parameter planes for holomorphic families of transcendental entire maps.Two-parameter unfolding of a parabolic point of a vector field in \(\mathbb{C}\) fixing the originhttps://www.zbmath.org/1483.370592022-05-16T20:40:13.078697Z"Rousseau, Christiane"https://www.zbmath.org/authors/?q=ai:rousseau.christianeThe author studies the bifurcation of a family of polynomial vector fields \(\dot{z}=z(z^k+\epsilon_1 z+\epsilon_0)\) in the complex plane with two parameters. The powerful tool of the ``periodgon'' is used to describe the bifurcation diagrams. The periodgon is a new invariant introduced by the author and \textit{A.Chéritat}
[``Generic 1-parameter pertubations of a vector field with a singular point of codimension \(k\)'', Preprint, \url{arXiv:1701.03276}]
to characterize a polynomial vector field up to a rotation of order \(k\) if it is monic and centered. Later, it has been generalized in [\textit{M. Klimeš} and the author, Conform. Geom. Dyn. 22, 141--184 (2018; Zbl 1403.37057)] to all generic polynomial vector fields. With the help of this tool, the bifurcations of parabolic points and homoclinic loops through infinity are investigated. For a generic 2-parameter unfolding of a parabolic point of codimension \(k\) preserving the origin, the author raises an interesting question of whether there exists a unique normal form in which the parameters are uniquely defined (i.e., canonically defined) if the origin is fixed in the unfolding.
Reviewer: Kwok-wai Chung (Hong Kong)Shilnikov-type dynamics in three-dimensional piecewise smooth mapshttps://www.zbmath.org/1483.370602022-05-16T20:40:13.078697Z"Roy, Indrava"https://www.zbmath.org/authors/?q=ai:roy.indrava"Patra, Mahashweta"https://www.zbmath.org/authors/?q=ai:patra.mahashweta"Banerjee, Soumitro"https://www.zbmath.org/authors/?q=ai:banerjee.soumitroSummary: We show the existence of Shilnikov-type dynamics and bifurcation behaviour in general discrete three-dimensional piecewise smooth maps and give analytical results for the occurence of such dynamical behaviour. Our main example in fact shows a `two-sided' Shilnikov dynamics, i.e. simultaneous looping and homoclinic intersection of the one-dimensional eigenmanifolds of fixed points on both sides of the border. We also present two complementary methods to analyse the return time of an orbit to the border: one based on recursion and another based on complex interpolation.Knudsen diffusivity in random billiards: spectrum, geometry, and computationhttps://www.zbmath.org/1483.370612022-05-16T20:40:13.078697Z"Chumley, Timothy"https://www.zbmath.org/authors/?q=ai:chumley.timothy"Feres, Renato"https://www.zbmath.org/authors/?q=ai:feres.renato"Garcia German, Luis Alberto"https://www.zbmath.org/authors/?q=ai:garcia-german.luis-albertoSingularities of invariant densities for random switching between two linear ODEs in 2Dhttps://www.zbmath.org/1483.370622022-05-16T20:40:13.078697Z"Bakhtin, Yuri"https://www.zbmath.org/authors/?q=ai:bakhtin.yuri-yu"Hurth, Tobias"https://www.zbmath.org/authors/?q=ai:hurth.tobias"Lawley, Sean D."https://www.zbmath.org/authors/?q=ai:lawley.sean-d"Mattingly, Jonathan C."https://www.zbmath.org/authors/?q=ai:mattingly.jonathan-cGradient flow of the stochastic relaxation on a generic exponential familyhttps://www.zbmath.org/1483.370632022-05-16T20:40:13.078697Z"Malagò, Luigi"https://www.zbmath.org/authors/?q=ai:malago.luigi"Pistone, Giovanni"https://www.zbmath.org/authors/?q=ai:pistone.giovanniSummary: We study the natural gradient flow of the expected value \(E_p [f]\) of an objective function \(f\) for \(p\) in an exponential family. We parameterize the exponential family with the expectation parameters and we show that the dynamical system associated to the natural gradient flow can be extended outside the marginal polytope.
For the entire collection see [Zbl 1470.00021].A random dynamical systems perspective on isochronicity for stochastic oscillationshttps://www.zbmath.org/1483.370642022-05-16T20:40:13.078697Z"Engel, Maximilian"https://www.zbmath.org/authors/?q=ai:engel.maximilian"Kuehn, Christian"https://www.zbmath.org/authors/?q=ai:kuhn.christianThe authors study the problem of defining isochrons for stochastic oscillations. They propose a new approach for finding stochastic isochrons as sections of equal expected return times versus the idea of considering eigenfunctions of the backward Kolmogorov operator. The authors introduce a new rigorous definition of stochastic isochrons as random stable manifolds for random periodic solutions with noise-dependent period. This allows them to establish a random version of isochron maps whose level sets coincide with the random stable manifolds. Finally, the relations between the random dynamical systems interpretation and the equal expected return time approach via averaged quantities are discussed.
Reviewer: Chao Wang (Kunming)Stability of stochastic hybrid systems with Markovian switched controllers associated with a transfer functionhttps://www.zbmath.org/1483.370652022-05-16T20:40:13.078697Z"Imzegouan, Chafai"https://www.zbmath.org/authors/?q=ai:imzegouan.chafai"Bouzahir, Hassane"https://www.zbmath.org/authors/?q=ai:bouzahir.hassane"Benaid, Brahim"https://www.zbmath.org/authors/?q=ai:benaid.brahimSummary: This paper is concerned with stochastic stability analysis for a kind of stochastic systems with Markovian switched controllers associated with a transfer function. Stability in probability and asymptotic stability in probability are discussed based on the Lyapunov method and the stochastic analysis. Sufficient conditions on a state-space representation of the system which guarantee stability are given. Two numerical examples are provided to illustrate the usefulness of our theoretical results.Explicit symmetries of the Kepler Hamiltonianhttps://www.zbmath.org/1483.370662022-05-16T20:40:13.078697Z"Knörrer, Horst"https://www.zbmath.org/authors/?q=ai:knorrer.horstSummary: Using quaternions, we give explicit formulas for the global symmetries of the three-dimensional Kepler problem. The regularizations of the Kepler problem that are based on the Hopf map and on stereographic projections, respectively, are interpreted in terms of these symmetries.
For the entire collection see [Zbl 1456.14003].A note on the commutator of Hamiltonian vector fieldshttps://www.zbmath.org/1483.370672022-05-16T20:40:13.078697Z"Żołądek, Henryk"https://www.zbmath.org/authors/?q=ai:zoladek.henrykSummary: We present two proofs of the Jacobi identity for the Poisson bracket on a symplectic manifold.
For the entire collection see [Zbl 1456.14003].Hamiltonian pitchfork bifurcation in transition across index-1 saddleshttps://www.zbmath.org/1483.370682022-05-16T20:40:13.078697Z"Lyu, Wenyang"https://www.zbmath.org/authors/?q=ai:lyu.wenyang"Naik, Shibabrat"https://www.zbmath.org/authors/?q=ai:naik.shibabrat"Wiggins, Stephen"https://www.zbmath.org/authors/?q=ai:wiggins.stephenSummary: We study the effect of changes in the parameters of a two dimensional potential energy surface on the phase space structures relevant for chemical reaction dynamics. The changes in the potential energy are representative of chemical reactions such as isomerization between two structural conformations or dissociation of a molecule with an intermediate. We present a two degrees of freedom (DOF) quartic Hamiltonian that shows pitchfork bifurcation when the parameters are varied and we derive the bifurcation criteria relating the parameters. Next, we describe the phase space structures -- unstable periodic orbits and its associated invariant manifolds, and phase space dividing surfaces -- for the systems that can show trajectories undergo reaction defined as crossing of a potential energy barrier. Finally, we quantify the reaction dynamics for these systems by obtaining the directional flux and gap time distribution to illustrate the dependence on total energy and coupling strength between the two degrees of freedom.On the Lyapunov instability in Newtonian dynamicshttps://www.zbmath.org/1483.370692022-05-16T20:40:13.078697Z"Burgos, J. M."https://www.zbmath.org/authors/?q=ai:burgos.juan-manuel"Maderna, E."https://www.zbmath.org/authors/?q=ai:maderna.ezequiel"Paternain, M."https://www.zbmath.org/authors/?q=ai:paternain.miguelThe authors consider the classical problem of Lyapunov instability of equilibrium states for a finite-dimensional Newtonian system with a non-strict local minimum of the potential. For the cases when the local minimum of the potential energy is reached on the hypersurface of the configuration space, new results on the instability of equilibrium states are obtained.
Reviewer: Vladimir Sobolev (Samara)Geometrical classification of self-similar motion of two-dimensional three point vortex system by deviation curvature on Jacobi fieldhttps://www.zbmath.org/1483.370702022-05-16T20:40:13.078697Z"Hirakui, Yuma"https://www.zbmath.org/authors/?q=ai:hirakui.yuma"Yajima, Takahiro"https://www.zbmath.org/authors/?q=ai:yajima.takahiroSummary: In this study, we geometrically analyze the relation between a point vortex system and deviation curvatures on the Jacobi field. First, eigenvalues of deviation curvatures are calculated from relative distances of point vortices in a three point vortex system. Afterward, based on the assumption of self-similarity, time evolutions of eigenvalues of deviation curvatures are shown. The self-similar motions of three point vortices are classified into two types, expansion and collapse, when the relative distances vary monotonously. Then, we find that the eigenvalues of self-similarity are proportional to the inverse fourth power of relative distances. The eigenvalues of the deviation curvatures monotonically convergent to zero for expansion, whereas they monotonically diverge for collapse, which indicates that the strengths of interactions between point vortices related to the time evolution of spatial geometric structure in terms of the deviation curvatures. In particular, for collapse, the collision point becomes a geometric singularity because the eigenvalues of the deviation curvature diverge. These results show that the self-similar motions of point vortices are classified by eigenvalues of the deviation curvature. Further, nonself-similar expansion is numerically analyzed. In this case, the eigenvalues of the deviation curvature are nonmonotonous but converge to zero, suggesting that the motion of the nonself-similar three point vortex system is also classified by eigenvalues of the deviation curvature.Some remarks on high degree polynomial integrals of the magnetic geodesic flow on the two-dimensional torushttps://www.zbmath.org/1483.370712022-05-16T20:40:13.078697Z"Agapov, S. V."https://www.zbmath.org/authors/?q=ai:agapov.sergei-vadimovich"Valyuzhenich, A. A."https://www.zbmath.org/authors/?q=ai:valyuzhenich.aleksandr-andreevich"Shubin, V. V."https://www.zbmath.org/authors/?q=ai:shubin.v-vThis paper aims at fixing a gap in the proof of the main result from [the first two authors, Discrete Contin. Dyn. Syst. 39, No. 11, 6565--6583 (2019; Zbl 1432.37082)].
Consider conformal coordinates \((x,y)\) on the \(2\)-torus with metric \(ds^2=\Lambda(x,y)(dx^2+dy^2)\), and let \[H=\frac{p_x^2+p_y^2}{2 \Lambda(x,y)}.\] The magnetic gedesic flow is the Hamiltonian flow of \(H\) with respect to the magnetic Poisson bracket \[\{-,-\}_{\text{mg}} = \frac{\partial}{\partial x}\wedge \frac{\partial}{\partial p_x}+ \frac{\partial}{\partial y} \wedge \frac{\partial}{\partial p_y} + \Omega(x,y) \frac{\partial}{\partial p_x} \wedge \frac{\partial}{\partial p_y}\,,\] where \(\Omega(x,y)\, dx\wedge dy\) is a closed \(2\)-form defining a fixed nonzero magnetic field.
The main result of [loc. cit.] states that if the magnetic geodesic flow has a first integral \(F\) which is polynomial of degree \(N\) in the momenta \((p_x,p_y)\) and which has periodic analytic coefficients at \(N+2\) energy levels \(\{H=E_j\}\), \(E_1,\ldots, E_{N+2}\in \mathbb{R}\) pairwise distinct, then \(\Omega\) and \(\Lambda\) are functions of a unique variable. Furthermore \(H\) admits a first integral \(F_1\) which is linear in momenta. In Section 2 the authors outline the main steps of the proof given in [loc. cit.] and identify an argument which can fail. It is explained in Section 3 how to fix this issue without altering the statement of the important result that was presented.
Reviewer: Maxime Fairon (Glasgow)Superintegrable systems on moduli spaces of flat connectionshttps://www.zbmath.org/1483.370722022-05-16T20:40:13.078697Z"Arthamonov, S."https://www.zbmath.org/authors/?q=ai:arthamonov.semeon"Reshetikhin, N."https://www.zbmath.org/authors/?q=ai:reshetikhin.nikolai-yuLet \(G\) be a connected simple linear algebraic group over \(\mathbb{C}\). The aim of this paper is to construct superintegrable Hamiltonian systems on moduli spaces of flat \(G\)-connections over any oriented surface with nonempty boundary. Hamiltonians of such systems are traces of holonomies along non-intersecting non-self-intersecting curves. The construction naturally works in the same way for various real forms of \(G\), for example for compact simple Lie groups or for split real forms. The moduli space is a Poisson variety with Atiyah-Bott Poisson structure. Among particular cases of such systems are spin generalizations of Ruijsenaars-Schneider models.
This paper is organized as follows. In Section 1 the authors recall the definition of superintegrable (degenerately integrable) systems. There they also define the notion of affine superintegrable systems in the algebro-geometrical setting. Section 2 is an overview of basic notions about moduli spaces of flat connections on a surface. In this section they recall the definition of graph functions and the description of Poisson brackets between two such functions. In Section 3 they describe the main result, the construction of a family of Hamiltonian systems defined by a choice of non-intersecting, non-self-intersecting cycles and prove their superintegrability. At the end of this section they introduce the notion of a partial order on such systems. In Section 4 they explain how solutions to equations of motion of these superintegrable systems can be solved using the projection method. Section 5 has some genus one examples. In the conclusion the authors define a system on the space of chord diagrams which is conjectured to be superintegrable. They discuss the case of non-generic conjugation orbits and some further directions.
Reviewer: Ahmed Lesfari (El Jadida)Solvable dynamical systems in the plane with polynomial interactionshttps://www.zbmath.org/1483.370732022-05-16T20:40:13.078697Z"Calogero, Francesco"https://www.zbmath.org/authors/?q=ai:calogero.francesco-a"Payandeh, Farrin"https://www.zbmath.org/authors/?q=ai:payandeh.farrinSummary: In this paper we report a few examples of algebraically solvable dynamical systems characterized by \(2\) coupled Ordinary Differential Equations which read as follows:
\[
\dot{x}_n=P^{\left( n\right) }\left( x_1,x_2\right),\quad n=1,2,
\]
with \(P^{(n)}(x_1,x_2)\) specific polynomials of relatively low degree in the \(2\) dependent variables \(x_1\equiv x_1(t)\) and \(x_2\equiv x_2(t)\). These findings are obtained via a new twist of a recent technique to identify dynamical systems solvable by algebraic operations, themselves explicitly identified as corresponding to the time evolutions of the zeros of polynomials the coefficients of which evolve according to algebraically solvable (systems of) evolution equations.
For the entire collection see [Zbl 1456.14003].Bi-Hamiltonian structure of spin Sutherland models: the holomorphic casehttps://www.zbmath.org/1483.370742022-05-16T20:40:13.078697Z"Fehér, L."https://www.zbmath.org/authors/?q=ai:feher.laszlo-gyIn the previous work [Lett. Math. Phys. 110, No. 5, 1057--1079 (2020; Zbl 1445.37042)] the author developed a bi-Hamiltonian interpretation for a family of Sutherland spin models having hyperbolic and trigonometric form. The current paper expands this investigation to what the author calls the holomorphic spin Sutherland hierarchy. This consists of the holomorphic evolution equations of the form \(\dot{Q} = (L^k)_0 Q\), \(\dot{L} = [\mathcal{R}(Q)(L^k), L]\) for all \(k \in \mathbb{N}\). Here \(Q\) is an invertible \(n \times n\) complex diagonal matrix, \(L\) is an arbitrary \(n \times n\) complex matrix, and the subscript \(0\) means diagonal part. It is assumed that the eigenvalues of \(Q\) are distinct, so that the operator \(\mathcal{R}(Q) = 1/2 (\mathrm{Ad}_Q + \mathrm{id})( \mathrm{Ad}_Q - \mathrm{id})^{-1}\) (where \(\mathrm{Ad}_Q(X) = QXQ^{-1}\)) is a well-defined linear operator on the off-diagonal subspace of gl(\(n, \mathbb{C}).\)
The holomorphic spin Sutherland hierarchy is known to be a reduction of the natural integrable system on the cotangent bundle \(\mathcal{M} = T^* \mathrm{GL}(n, \mathbb{C})\). The author's main result is that the unreduced integrable system on \(\mathcal{M}\) leads to a bi-Hamiltonian structure for the spin Sutherland hierarchy after Poisson reduction. One of the two compatible Poisson structures is associated with the canonical cotangent bundle symplectic form on \(\mathcal{M}\), and the second one is constructed from the Semenov-Tian-Shansky Poisson bracket of the Heisenberg double of \(T^* \mathrm{GL}(n, \mathbb{C})\) with its standard Poisson-Lie group structure.
The author also shows that the bi-Hamiltonian structures of the hyperbolic and trigonometric real forms described in his previous work can be recovered from the holomorphic form.
Reviewer: William J. Satzer Jr. (St. Paul)The rigid body dynamics in an ideal fluid: Clebsch top and Kummer surfaceshttps://www.zbmath.org/1483.370752022-05-16T20:40:13.078697Z"Françoise, Jean-Pierre"https://www.zbmath.org/authors/?q=ai:francoise.jean-pierre"Tarama, Daisuke"https://www.zbmath.org/authors/?q=ai:tarama.daisukeSummary: This is an expository presentation of a completely integrable Hamiltonian system of Clebsch top under a special condition introduced by Weber. After a brief account of the geometric setting of the system, the structure of the Poisson commuting first integrals is discussed following the methods by \textit{F. Magri} and \textit{T. Skrypnyk} [``The Clebsch System'', Preprint, \url{arXiv:1512.04872}]. Introducing supplementary coordinates, a geometric connection to Kummer surfaces, a typical class of K3 surfaces, is mentioned and also the system is linearized on the Jacobian of a hyperelliptic curve of genus two determined by the system. Further some special solutions contained in some vector subspace are discussed. Finally, an explicit computation of the action variables is introduced.
For the entire collection see [Zbl 1456.14004].Orbital invariants of flat billiards bounded by arcs of confocal quadrics and containing focuseshttps://www.zbmath.org/1483.370762022-05-16T20:40:13.078697Z"Vedyushkina, V. V."https://www.zbmath.org/authors/?q=ai:vedyushkina.viktoriya-viktorovnaSummary: Rotation functions for flat billiards bounded by arcs of confocal quadrics and containing focuses are calculated. The orbital Bolsinov-Fomenko invariants of these dynamical systems are also calculated.Symplectic geometry of the Koopman operatorhttps://www.zbmath.org/1483.370772022-05-16T20:40:13.078697Z"Kozlov, V. V."https://www.zbmath.org/authors/?q=ai:kozlov.valerij-vasilievich|kozlov.vladimir-vasilievich|kozlov.victor-v|kozlov.vasilii-vasilevichSummary: We consider the Koopman operator generated by an invertible transformation of a space with a finite countably additive measure. If the square of this transformation is ergodic, then the orthogonal Koopman operator is a symplectic transformation on the real Hilbert space of square summable functions with zero mean. An infinite set of quadratic invariants of the Koopman operator is specified, which are pairwise in involution with respect to the corresponding symplectic structure. For transformations with a discrete spectrum and a Lebesgue spectrum, these quadratic invariants are functionally independent and form a complete involutive set, which suggests that the Koopman transform is completely integrable.Noncommutative cross-ratio and Schwarz derivativehttps://www.zbmath.org/1483.370782022-05-16T20:40:13.078697Z"Retakh, Vladimir"https://www.zbmath.org/authors/?q=ai:retakh.vladimir-s"Rubtsov, Vladimir"https://www.zbmath.org/authors/?q=ai:rubtsov.vladimir-n"Sharygin, Georgy"https://www.zbmath.org/authors/?q=ai:sharygin.georgy-iSummary: We present here a theory of noncommutative cross-ratio, Schwarz derivative and their connections and relations to the operator cross-ratio. We apply the theory to ``noncommutative elementary geometry'' and relate it to noncommutative integrable systems. We also provide a noncommutative version of the celebrated ``pentagramma mirificum''.
For the entire collection see [Zbl 1456.14004].Generating multicluster conservative chaotic flows from a generalized Sprott-A systemhttps://www.zbmath.org/1483.370792022-05-16T20:40:13.078697Z"Cang, Shijian"https://www.zbmath.org/authors/?q=ai:cang.shijian"Li, Yue"https://www.zbmath.org/authors/?q=ai:li.yue"Kang, Zhijun"https://www.zbmath.org/authors/?q=ai:kang.zhijun"Wang, Zenghui"https://www.zbmath.org/authors/?q=ai:wang.zenghuiSummary: In this paper, we propose a general structure of the generalized Sprott-A system based on the matrix form of the Sprott-A system. To investigate the multicluster chaotic flows derived from the general structure, several example systems are reported by modifying the Hamiltonian of the generalized Sprott-A system without changing its nonconstant state matrix. Through numerical simulations, it is interesting to find that the topology of the chaotic flows generated by the example systems has clusters of different numbers and shapes in phase space for the given different parameters and initial conditions, which are completely controlled by the Hamiltonian (i.e., completely closed isosurfaces). Moreover, the captured chaos is volume-conservative, which is verified by the sums of the corresponding Lyapunov exponents. Besides, we analyze the complexity of the example systems by the approximate entropy, sample entropy and fuzzy entropy, and find that increasing the number of conservative chaotic clusters may not enhance the complexities of the proposed systems.Moser's theorem for hyperbolic-type degenerate lower tori in Hamiltonian systemhttps://www.zbmath.org/1483.370802022-05-16T20:40:13.078697Z"Jing, Tianqi"https://www.zbmath.org/authors/?q=ai:jing.tianqi"Si, Wen"https://www.zbmath.org/authors/?q=ai:si.wenThe main result of the paper is about the existence of lower-dimensional tori for a class of degenerate Hamiltonian systems, that are also supposed to be only \(C^\ell\) smooth. This is a generalization of known results for which the degenerate case was considered within the analytic class, or the loss of analyticity was considered in the non-degenerate case.
The proof is ruled out in the framework of KAM theory. The loss of regularity is overcome via the Jackson-Moser-Zehnder analytic approximation, while the degeneracy is treated through normal form theory for quasi-homogeneous polynomial systems and the stability of equilibrium points of odd degree polynomials.
Some applications of the main theorem are given.
Reviewer: Stefano Marò (Pisa)Pencils of quadrics, billiard double-reflection and confocal incircular netshttps://www.zbmath.org/1483.370812022-05-16T20:40:13.078697Z"Dragović, Vladimir"https://www.zbmath.org/authors/?q=ai:dragovic.vladimir"Radnović, Milena"https://www.zbmath.org/authors/?q=ai:radnovic.milena"Ranomenjanahary, Roger Fidèle"https://www.zbmath.org/authors/?q=ai:ranomenjanahary.roger-fideleSummary: We present recent results about double reflection and incircular nets. The building blocks are pencils of quadrics, related billiards and quad graphs.
For the entire collection see [Zbl 1456.14003].Fifth-order generalized Heisenberg supermagnetic modelshttps://www.zbmath.org/1483.370822022-05-16T20:40:13.078697Z"Jiang, Nana"https://www.zbmath.org/authors/?q=ai:jiang.nana"Zhang, Meina"https://www.zbmath.org/authors/?q=ai:zhang.meina"Guo, Jiafeng"https://www.zbmath.org/authors/?q=ai:guo.jiafeng"Yan, Zhaowen"https://www.zbmath.org/authors/?q=ai:yan.zhaowenSummary: This paper is concerned with the construction of the fifth-order generalized Heisenberg supermagnetic models. We investigate the integrable structure and properties of the supersymmetric systems. We also establish their gauge equivalent equations with the gauge transformation for two quadratic constraints, i.e., the super fifth-order nonlinear Schrödinger equation and the fermionic fifth-order nonlinear Schrödinger equation, respectively.Nodal curves and a class of solutions of the Lax equation for shock clustering and Burgers turbulencehttps://www.zbmath.org/1483.370832022-05-16T20:40:13.078697Z"Li, Luen-Chau"https://www.zbmath.org/authors/?q=ai:li.luen-chauSummary: In this paper, we present an expository account of the work done in the last few years in understanding a matrix Lax equation which arises in the study of scalar hyperbolic conservation laws with spectrally negative pure-jump Markov initial data. We begin with its extension to general \(N\times N\) matrices, which is Liouville integrable on generic coadjoint orbits of a matrix Lie group. In the probabilistically interesting case in which the Lax operator is the generator of a pure-jump Markov process, the spectral curve is generically a fully reducible nodal curve. In this case, the equation is not Liouville integrable, but we can show that the flow is still conjugate to a straight line motion, and the equation is exactly solvable. En route, we establish a dictionary between an open, dense set of lower triangular generator matrices and algebro-geometric data which plays an important role in our analysis.
For the entire collection see [Zbl 1456.14003].Boussinesq hierarchy and bi-Hamiltonian geometryhttps://www.zbmath.org/1483.370842022-05-16T20:40:13.078697Z"Ortenzi, Giovanni"https://www.zbmath.org/authors/?q=ai:ortenzi.giovanni"Pedroni, Marco"https://www.zbmath.org/authors/?q=ai:pedroni.marcoThe authors study the Boussinesq hierarchy in the geometric context of bi-Hamiltonian manifolds. Specifically, they consider the matrix Boussinesq hierarchy on symplectic leaves and show that such a hierarchy can be projected on the quotient space, thus giving rise to the scalar Boussinesq hierarchy.
Reviewer: Jiao Wei (Zhengzhou)Negative times of the Davey-Stewartson integrable hierarchyhttps://www.zbmath.org/1483.370852022-05-16T20:40:13.078697Z"Pogrebkov, Andrei K."https://www.zbmath.org/authors/?q=ai:pogrebkov.andrei-kSummary: We use example of the Davey-Stewartson hierarchy to show that in addition to the standard equations given by Lax operator and evolutions of times with positive numbers, one can consider time evolutions with negative numbers and the same Lax operator. We derive corresponding Lax pairs and integrable equations.Trace ideal properties of a class of integral operatorshttps://www.zbmath.org/1483.370862022-05-16T20:40:13.078697Z"Gesztesy, Fritz"https://www.zbmath.org/authors/?q=ai:gesztesy.fritz"Nichols, Roger"https://www.zbmath.org/authors/?q=ai:nichols.rogerSummary: We consider a particular class of integral operators \(T_{\gamma,\delta}\) in \(L^2(\mathbb{R}^n),n\in\mathbb{N},n\geqslant 2\), with integral kernels \(T_{\gamma, \delta}(\cdot ,\cdot)\) bounded (Lebesgue) a.e. by
\[
|T_{\gamma,\delta}(x,y)|\leqslant C\langle x\rangle^{-\delta}|x-y|^{2\gamma -n}\langle y\rangle^{-\delta}, \quad x,y\in\mathbb{R}^n, x\neq y,
\]
for fixed \(C\in(0,\infty),0 < 2\gamma < n,\delta > \gamma\), and prove that
\[
T_{\gamma,\delta}\in\mathcal{B}_p\big(L^2(\mathbb{R}^n)\big)\text{ for } p > n/(2\gamma), p\geqslant 2.
\]
(Here \(\langle x\rangle:=(1+|x|^2)^{1/2},x\in\mathbb{R}^n\), and \(\mathcal{B}_p\) abbreviates the \(\ell^p\)-based trace ideal.) These integral operators (and their matrix-valued analogs) naturally arise in the study of multi-dimensional Schrödinger and Dirac-type operators and we describe an application to the case of massless Dirac-type operators.
For the entire collection see [Zbl 1456.14003].Algebro-geometric solutions of the coupled Chaffee-Infante reaction diffusion hierarchyhttps://www.zbmath.org/1483.370872022-05-16T20:40:13.078697Z"Yue, Chao"https://www.zbmath.org/authors/?q=ai:yue.chao"Xia, Tiecheng"https://www.zbmath.org/authors/?q=ai:xia.tie-chengFrom the abstract: ``The coupled Chaffee-Infante reaction diffusion (CCIRD) hierarchy associated with a \(3\times3\) matrix spectral problem is derived by
using two sets of the Lenard recursion gradients. Based on the characteristic polynomial of the Lax matrix for the CCIRD
hierarchy, we introduce a trigonal curve \(\mathcal{K}_{m-2}\) of arithmetic genus \(m-2\), from which the corresponding Baker-Akhiezer
function and meromorphic functions on \(\mathcal{K}_{m-2}\) are constructed. Then, the CCIRD equations are decomposed into Dubrovin-type ordinary differential equations. Furthermore, the theory of the trigonal curve and the properties of the three kinds of Abel
differentials are applied to obtain the explicit theta function representations of the Baker-Akhiezer function and the
meromorphic functions. In particular, algebro-geometric solutions for the entire CCIRD hierarchy are obtained.''
This paper is organized as follows. Section 1 is an introduction to the subject and summarizes the main results. In Section 2, the authors obtain the CCIRD hierarchy related to a \(3\times3\) matrix spectral problem based on the Lenard recursion equations. In Section 3, a trigonal curve \(\mathcal{K}_{m-2}\) of arithmetic genus \(m-2\) with three infinite points is introduced by the use of the characteristic polynomial of the Lax matrix for the stationary CCIRD equations, from which the stationary Baker-Akhiezer function and associated meromorphic functions are given on \(\mathcal{K}_{m-2}\). Then, the stationary CCIRD equations are decomposed into the system of Dubrovin-type ordinary differential equations. In Section 4, they present the explicit theta function representations of the stationary Baker-Akhiezer function, of the meromorphic functions, and, in particular, of the potentials for the entire stationary CCIRD hierarchy. In Section 5, the authors extend all the Baker-Akhiezer functions, the meromorphic functions, the Dubrovin-type equations, and the theta function representations dealt with in Sections 3 and 4 to the time-dependent case. Section 6 is devoted to some conclusions.
Reviewer: Ahmed Lesfari (El Jadida)Hamiltonian aspects of three-layer stratified fluidshttps://www.zbmath.org/1483.370882022-05-16T20:40:13.078697Z"Camassa, R."https://www.zbmath.org/authors/?q=ai:camassa.roberto"Falqui, G."https://www.zbmath.org/authors/?q=ai:falqui.gregorio"Ortenzi, G."https://www.zbmath.org/authors/?q=ai:ortenzi.giovanni"Pedroni, M."https://www.zbmath.org/authors/?q=ai:pedroni.marco"Ho, T. T. Vu"https://www.zbmath.org/authors/?q=ai:ho.t-t-vuThe authors study some structural properties of three-layer stratified incompressible Euler flows in the long-wave regime, especially for the case of a rigid upper lid constraint. The authors deduce that the long-wave dispersionless limit satisfies a system of quasi-linear equations, and further prove explicitly that the reduced system does not admit Riemann invariants. The phenomenon of effective pressure differentials implying the ``paradox'' of non-conservation of the horizontal momentum was first noticed by \textit{R. Camassa} et al. [J. Fluid Mech. 695, 330--340 (2012; Zbl 1250.76049)] in the two-layer case, and is proved to persist for an \(n\)-layered situation, with $n\geq 3$, and even be enhanced in some sense, for zero initial velocities, by scaling linearly with density differences, as opposed to quadratically as in the two-layer case. A natural Hamiltonian structure on the configuration spaces of effective three-layered fluid motions was derived by means of a geometric reduction process from the full two-dimensional Hamiltonian structure introduced by \textit{T. B. Benjamin} [J. Fluid Mech. 165, 445--474 (1986; Zbl 0595.76023)]. By using the canonical formalism, the Boussinesq limit and its solution with some special symmetries are discussed.
Reviewer: Cheng He (Beijing)Second-order conditional Lie-Bäcklund symmetry and differential constraint of radially symmetric diffusion systemhttps://www.zbmath.org/1483.370892022-05-16T20:40:13.078697Z"Wang, Jianping"https://www.zbmath.org/authors/?q=ai:wang.jianping.1"Ba, Huijing"https://www.zbmath.org/authors/?q=ai:ba.huijing"Liu, Yaru"https://www.zbmath.org/authors/?q=ai:liu.yaru"He, Longqi"https://www.zbmath.org/authors/?q=ai:he.longqi"Ji, Lina"https://www.zbmath.org/authors/?q=ai:ji.linaSummary: The classifications and reductions of radially symmetric diffusion system are studied due to the conditional Lie-Bäcklund symmetry method. We obtain the invariant condition, which is the so-called determining system and under which the radially symmetric diffusion system admits second-order conditional Lie-Bäcklund symmetries. The governing systems and the admitted second-order conditional Lie-Bäcklund symmetries are identified by solving the nonlinear determining system. Exact solutions of the resulting systems are constructed due to the compatibility of the original system and the admitted differential constraint corresponding to the invariant surface condition. For most of the cases, they are reduced to solving four-dimensional dynamical systems.Periodic peakon and smooth periodic solutions for KP-MEW(3,2) equationhttps://www.zbmath.org/1483.370902022-05-16T20:40:13.078697Z"Cai, Junning"https://www.zbmath.org/authors/?q=ai:cai.junning"Wei, Minzhi"https://www.zbmath.org/authors/?q=ai:wei.minzhi"He, Liping"https://www.zbmath.org/authors/?q=ai:he.lipingSummary: In this paper, we consider the KP-MEW(3,2) equation by the bifurcation theory of dynamical systems when integral constant is considered. The corresponding traveling wave system is a singular planar dynamical system with one singular straight line. The phase portrait for \(c<0\), \(0<c<1\), and \(c>1\) is drawn. Exact parametric representations of periodic peakon solutions and smooth periodic solution are presented.Quasi-periodic incompressible Euler flows in 3Dhttps://www.zbmath.org/1483.370912022-05-16T20:40:13.078697Z"Baldi, Pietro"https://www.zbmath.org/authors/?q=ai:baldi.pietro"Montalto, Riccardo"https://www.zbmath.org/authors/?q=ai:montalto.riccardoSummary: We prove the existence of time-quasi-periodic solutions of the incompressible Euler equation on the three-dimensional torus \(\mathbb{T}^3\), with a small time-quasi-periodic external force. The solutions are perturbations of constant (Diophantine) vector fields, and they are constructed by means of normal forms and KAM techniques for reversible quasilinear PDEs.Almost-periodic response solutions for a forced quasi-linear Airy equationhttps://www.zbmath.org/1483.370922022-05-16T20:40:13.078697Z"Corsi, Livia"https://www.zbmath.org/authors/?q=ai:corsi.livia"Montalto, Riccardo"https://www.zbmath.org/authors/?q=ai:montalto.riccardo"Procesi, Michela"https://www.zbmath.org/authors/?q=ai:procesi.michelaThis paper is mainly concerned with the existence of almost-periodic solutions for quasi-linear perturbations of the Airy equation. By combining the approach developed by
\textit{W. Craig} and \textit{C. E. Wayne} [Commun. Pure Appl. Math. 46, No. 11, 1409--1498 (1993; Zbl 0794.35104)] with a KAM reducibility scheme and pseudo-differential calculus on \(\mathbb{T}^{\infty}\), the authors establish some new results on the existence of this type of solutions for a quasi-linear PDE.
Reviewer: Chao Wang (Kunming)Lax pair for a novel two-dimensional latticehttps://www.zbmath.org/1483.370932022-05-16T20:40:13.078697Z"Kuznetsova, Maria N."https://www.zbmath.org/authors/?q=ai:kuznetsova.maria-nSummary: In paper by \textit{I. Habibullin} [Phys. Scr. 87, No. 6, Article ID 065005, 5 p. (2013; Zbl 1275.17044)]
and our joint paper [\textit{I. T. Habibullin} and the author, Theor. Math. Phys. 203, No. 1, 569--581 (2020; Zbl 1452.37072); translation from Teor. Mat. Fiz. 203, No. 1, 161--173 (2020)] the algorithm for classification of integrable equations with three independent variables was proposed. This method is based on the requirement of the existence of an infinite set of Darboux integrable reductions and on the notion of the characteristic Lie-Rinehart algebras. The method was applied for the classification of integrable cases of different subclasses of equations \(u_{n,xy} = f(u_{n+1},u_n,u_{n-1}, u_{n,x},u_{n,y})\) of special forms. Under this approach the novel integrable chain was obtained. In present paper we construct Lax pair for the novel chain. To construct the Lax pair, we use the scheme suggested in papers by \textit{E. V. Ferapontov} [Theor. Math. Phys. 110, No. 1, 68--77 (1997; Zbl 0919.35132); translation from Teor. Mat. Fiz. 110, No. 1, 86--97 (1997); et al., J. Phys. A, Math. Theor. 46, No. 24, Article ID 245207, 13 p. (2013; Zbl 1329.35263)]. We also study the periodic reduction of the chain.Optimal control of Clarke subdifferential type fractional differential inclusion with non-instantaneous impulses driven by Poisson jumps and its topological propertieshttps://www.zbmath.org/1483.370942022-05-16T20:40:13.078697Z"Durga, N."https://www.zbmath.org/authors/?q=ai:durga.nagarajan"Muthukumar, P."https://www.zbmath.org/authors/?q=ai:muthukumar.palanisamySummary: This article is devoted to studying the topological structure of a solution set for Clarke subdifferential type fractional non-instantaneous impulsive differential inclusion driven by Poisson jumps. Initially, for proving the solvability result, we use a nonlinear alternative of Leray-Schauder fixed point theorem, Gronwall inequality, stochastic analysis, a measure of noncompactness, and the multivalued analysis. Furthermore, the mild solution set for the proposed problem is demonstrated with nonemptyness, compactness, and, moreover, \(R_\delta\)-set. By employing Balder's theorem, the existence of optimal control is derived. At last, an application is provided to validate the developed theoretical results.Random attractors for non-autonomous fractional stochastic Ginzburg-Landau equations on unbounded domainshttps://www.zbmath.org/1483.370952022-05-16T20:40:13.078697Z"Shu, Ji"https://www.zbmath.org/authors/?q=ai:shu.ji"Zhang, Jian"https://www.zbmath.org/authors/?q=ai:zhang.jianSummary: This paper deals with the dynamical behavior of solutions for non-autonomous stochastic fractional Ginzburg-Landau equations driven by additive noise with \(\alpha\in(0,1)\). We prove the existence and uniqueness of tempered pullback random attractors for the equations in \(L^2(\mathbf{R}^3)\). In addition, we also obtain the upper semicontinuity of random attractors when the intensity of noise approaches zero. The main difficulty here is the noncompactness of Sobolev embeddings on unbounded domains. To solve this, we establish the pullback asymptotic compactness of solutions in \(L^2(\mathbf{R}^3)\) by the tail-estimates of solutions.Wong-Zakai approximations and attractors for fractional stochastic reaction-diffusion equations on unbounded domainshttps://www.zbmath.org/1483.370962022-05-16T20:40:13.078697Z"Sun, Yaqing"https://www.zbmath.org/authors/?q=ai:sun.yaqing"Gao, Hongjun"https://www.zbmath.org/authors/?q=ai:gao.hongjunSummary: In this paper, we investigate the Wong-Zakai approximations induced by a stationary process and the long term behavior of the fractional stochastic reaction-diffusion equation driven by a white noise. Precisely, one of the main ingredients in this paper is to establish the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximations of fractional stochastic reaction-diffusion equations. Thereafter the upper semi-continuity of attractors for the Wong-Zakai approximation of the equation as \(\delta\rightarrow0\) is proved.Parameter-free quantification of stochastic and chaotic signalshttps://www.zbmath.org/1483.370972022-05-16T20:40:13.078697Z"Lopes, S. R."https://www.zbmath.org/authors/?q=ai:lopes.sergio-roberto"Prado, T. L."https://www.zbmath.org/authors/?q=ai:prado.t-l"Corso, G."https://www.zbmath.org/authors/?q=ai:corso.gilberto"dos S. Lima, G. Z."https://www.zbmath.org/authors/?q=ai:dos-s-lima.g-z"Kurths, J."https://www.zbmath.org/authors/?q=ai:kurths.jurgenSummary: Recurrence entropy \((\mathcal{S})\) is a novel time series quantifier based on recurrence microstates. Here we show for the first time that \(\max(\mathcal{S})\) can be used as a \textit{parameter-free} quantifier of short and long-term memories (time-correlations) of stochastic and chaotic signals. \(\max(\mathcal{S})\) can also evaluate properties of the distribution function of the data set that are not related to time correlation. To illustrate this fact, we show that even shuffled versions of distinct time correlated stochastic signals lead to different, but coherently varying, values of \(\max(\mathcal{S})\). Such a property of \(\max(\mathcal{S})\) is associated to its ability to quantify in how many ways distinct short recurrence sequences can be found in time series. Applied to a deterministic dissipative system, the method brings evidence about the attractor properties and the degree of chaoticity. We conclude that the development of a new parameter-free quantifier of stochastic and chaotic time series can open new perspectives to stochastic data and deterministic time series analyses and may find applications in many areas of science.Visualizing the phase space of the HeI\(_2\) van der Waals complex using Lagrangian descriptorshttps://www.zbmath.org/1483.370982022-05-16T20:40:13.078697Z"Agaoglou, Makrina"https://www.zbmath.org/authors/?q=ai:agaoglou.makrina"García-Garrido, Víctor J."https://www.zbmath.org/authors/?q=ai:garcia-garrido.victor-j"Katsanikas, Matthaios"https://www.zbmath.org/authors/?q=ai:katsanikas.matthaios"Wiggins, Stephen"https://www.zbmath.org/authors/?q=ai:wiggins.stephenSummary: In this paper we demonstrate the capability of the method of Lagrangian descriptors to unveil the phase space structures that characterize transport in high-dimensional symplectic maps. In order to illustrate its use, we apply it to a four-dimensional symplectic map model that is used in chemistry to explore the nonlinear dynamics of van der Waals complexes. The advantage of this technique is that it allows us to easily and effectively extract the invariant manifolds that determine the dynamics of the system under study by means of examining the intersections of the underlying phase space structures with low-dimensional slices. With this approach, one can perform a full computational \textit{phase space tomography} from which three-dimensional representations of the higher-dimensional phase space can be systematically reconstructed. This analysis may be of much help for the visualization and understanding of the nonlinear dynamical mechanisms that take place in high-dimensional systems. In this context, we demonstrate how this tool can be used to detect whether the stable and unstable manifolds of the system intersect forming turnstile lobes that enclose a certain phase space volume, and the nature of their intersection.Adaptive Hamiltonian variational integrators and applications to symplectic accelerated optimizationhttps://www.zbmath.org/1483.370992022-05-16T20:40:13.078697Z"Duruisseaux, Valentin"https://www.zbmath.org/authors/?q=ai:duruisseaux.valentin"Schmitt, Jeremy"https://www.zbmath.org/authors/?q=ai:schmitt.jeremy-m"Leok, Melvin"https://www.zbmath.org/authors/?q=ai:leok.melvinSymplectic integrators with a variable stepsize usually do not preserve the energy. Nevertheless, the use of a variable stepsize could be useful in many instances. Motivated by these facts, the authors study the use of variable stepsize for specific classes of variational integrators. This is done by introducing a fictitious time \(\tau\), and defining an augmented Hamiltonian defined in terms of a ``monitor function'' which rules the time-step variation. The considered integrators turn out to be symplectic with constant time-step w.r.t. \(\tau\) and variable time-step w.r.t. the physical time \(t\). Applications to the Kepler problem and to accelerated optimization are presented.
Reviewer: Luigi Brugnano (Firenze)Painting the phase space of dissipative systems with Lagrangian descriptorshttps://www.zbmath.org/1483.371002022-05-16T20:40:13.078697Z"García-Garrido, Víctor J."https://www.zbmath.org/authors/?q=ai:garcia-garrido.victor-j"García-Luengo, Julia"https://www.zbmath.org/authors/?q=ai:garcia-luengo.juliaSummary: In this paper we apply the method of Lagrangian descriptors to explore the geometrical structures in phase space that govern the dynamics of dissipative systems. We demonstrate through many classical examples taken from the nonlinear dynamics literature that this mathematical technique can provide valuable information and insights to develop a more general and detailed understanding of the global behavior and underlying geometry of these systems. In order to achieve this goal, we analyze systems that display dynamical features such as hyperbolic points with different expansion and contraction rates, limit cycles, slow manifolds and strange attractors. Furthermore, we study how this technique can be used to detect transition ellipsoids that arise in Hamiltonian systems subject to dissipative forces, and which play a crucial role in characterizing trajectories that evolve across an index-1 saddle point of the underlying potential energy surface.Characterising stochastic fixed points and limit cycles for dynamical systems with additive noisehttps://www.zbmath.org/1483.371012022-05-16T20:40:13.078697Z"Biswas, Saranya"https://www.zbmath.org/authors/?q=ai:biswas.saranya"Rounak, Aasifa"https://www.zbmath.org/authors/?q=ai:rounak.aasifa"Perlikowski, Przemysław"https://www.zbmath.org/authors/?q=ai:perlikowski.przemyslaw"Gupta, Sayan"https://www.zbmath.org/authors/?q=ai:gupta.sayanSummary: This study focuses on characterising numerically the attractor volume in the state space of dynamical systems excited by additive white noise. A definition for stochastic attractors is introduced in terms of probability measure and numerical methodologies are presented to characterise them. The study is limited to investigating the effects of additive noise on fixed point and limit cycle attractors of the corresponding noise free system. The effect of noise intensity on the definition and validity of the proposed methodology is also discussed.Converging outer approximations to global attractors using semidefinite programminghttps://www.zbmath.org/1483.371022022-05-16T20:40:13.078697Z"Schlosser, Corbinian"https://www.zbmath.org/authors/?q=ai:schlosser.corbinian"Korda, Milan"https://www.zbmath.org/authors/?q=ai:korda.milanSummary: This paper develops a method for obtaining guaranteed outer approximations for global attractors of continuous and discrete time nonlinear dynamical systems. The method is based on a hierarchy of semidefinite programming problems of increasing size with guaranteed convergence to the global attractor. The approach taken follows an established line of reasoning, where we first characterize the global attractor via an infinite dimensional linear programming problem (LP) in the space of Borel measures. The dual to this LP is in the space of continuous functions and its feasible solutions provide guaranteed outer approximations to the global attractor. For systems with polynomial dynamics, a hierarchy of finite-dimensional sum-of-squares tightenings of the dual LP provides a sequence of outer approximations to the global attractor with guaranteed convergence in the sense of volume discrepancy tending to zero. The method is very simple to use and based purely on convex optimization. Numerical examples with the code available online demonstrate the method.Ergodic decay laws in Newtonian and relativistic chaotic scatteringhttps://www.zbmath.org/1483.371032022-05-16T20:40:13.078697Z"Fernández, Diego S."https://www.zbmath.org/authors/?q=ai:fernandez.diego-s"López, Álvaro G."https://www.zbmath.org/authors/?q=ai:lopez.alvaro-g"Seoane, Jesús M."https://www.zbmath.org/authors/?q=ai:seoane.jesus-m"Sanjuán, Miguel A. F."https://www.zbmath.org/authors/?q=ai:sanjuan.miguel-a-fSummary: In open Hamiltonian systems, the escape from a bounded region of phase space according to an exponential decay law is frequently associated with the existence of hyperbolic dynamics in such a region. Furthermore, exponential decay laws based on the ergodic hypothesis are used to describe escapes in these systems. However, we uncover that the presence of the set that governs the hyperbolic dynamics, commonly known as the chaotic saddle, invalidates the assumption of ergodicity. For the paradigmatic Hénon-Heiles system, we use both theoretical and numerical arguments to show that the escaping dynamics is non-ergodic independently of the existence of KAM tori, since the chaotic saddle, in whose vicinity trajectories are more likely to spend a finite amount of time evolving before escaping forever, is not utterly spread over the energy shell. Taking this into consideration, we provide a clarifying discussion about ergodicity in open Hamiltonian systems and explore the limitations of ergodic decay laws when describing escapes in this kind of systems. Finally, we generalize our claims by deriving a new decay law in the relativistic regime for an inertial and a non-inertial reference frames under the assumption of ergodicity, and suggest another approach to the description of escape laws in open Hamiltonian systems.Koopman resolvent: a Laplace-domain analysis of nonlinear autonomous dynamical systemshttps://www.zbmath.org/1483.371042022-05-16T20:40:13.078697Z"Susuki, Yoshihiko"https://www.zbmath.org/authors/?q=ai:susuki.yoshihiko"Mauroy, Alexandre"https://www.zbmath.org/authors/?q=ai:mauroy.alexandre"Mezić, Igor"https://www.zbmath.org/authors/?q=ai:mezic.igorThe finite-time expected deviation exponent for continuous dynamical systemshttps://www.zbmath.org/1483.371052022-05-16T20:40:13.078697Z"You, Guoqiao"https://www.zbmath.org/authors/?q=ai:you.guoqiaoSummary: In this paper, we introduce a concept called the finite-time \textit{expected} deviation exponent (FTEDE), which measures the expected separation rate of a particle with another initially infinitesimally close but randomly sampled particle over a finite time period. The proposed FTEDE can be viewed as a \textit{stochastic} version of the traditional finite-time Lyapunov exponent (FTLE) and is also a useful tool to measure the chaotic behaviors of continuous dynamical systems.Noise-induced kink propagation in shallow granular layershttps://www.zbmath.org/1483.371062022-05-16T20:40:13.078697Z"Jara-Schulz, Gladys"https://www.zbmath.org/authors/?q=ai:jara-schulz.gladys"Ferré, Michel A."https://www.zbmath.org/authors/?q=ai:ferre.michel-a"Falcón, Claudio"https://www.zbmath.org/authors/?q=ai:falcon.claudio"Clerc, Marcel G."https://www.zbmath.org/authors/?q=ai:clerc.marcel-gSummary: Out of equilibrium systems are characterized by exhibiting the coexistence of domains with complex spatiotemporal dynamics. Here, we investigate experimentally the noise-induced domain wall propagation on a one-dimensional shallow granular layer subjected to an air flow oscillating in time. We present results of the appearance of an effective drift as a function of the inclination of the experimental cell, which can be understood using a simple Langevin model to describe the dynamical evolution of these solutions via its pinning-depinning transition. The statistical characterization of displacements of the granular kink position is performed. The dynamics of the stochastic model shows a fairly good agreement with the experimental observations.Equilibria and their stability in networks with steep sigmoidal nonlinearitieshttps://www.zbmath.org/1483.371072022-05-16T20:40:13.078697Z"Duncan, William"https://www.zbmath.org/authors/?q=ai:duncan.william"Gedeon, Tomas"https://www.zbmath.org/authors/?q=ai:gedeon.tomas"Kokubu, Hiroshi"https://www.zbmath.org/authors/?q=ai:kokubu.hiroshi"Mischaikow, Konstantin"https://www.zbmath.org/authors/?q=ai:mischaikow.konstantin"Oka, Hiroe"https://www.zbmath.org/authors/?q=ai:oka.hiroeMathematical analysis of the global dynamics model for HIV infection of CD4\(^+\) T cells with treatment using Adomian decomposition approachhttps://www.zbmath.org/1483.371082022-05-16T20:40:13.078697Z"Gbadamosi, B."https://www.zbmath.org/authors/?q=ai:gbadamosi.b"Akinpelu, F. O."https://www.zbmath.org/authors/?q=ai:akinpelu.folake-oSummary: A compartmental epidemic model proposed by \textit{L. Wang} and \textit{M. Y. Li} [Math. Biosci. 200, No. 1, 44--57 (2006; Zbl 1086.92035)]
was investigated. A nonlinear incidence rate and treatment were taken into consideration. The bifurcation method introduced in [\textit{C. Castillo-Chavez} and \textit{B. Song}, Math. Biosci. Eng. 1, No. 2, 361--404 (2004; Zbl 1060.92041)] was being used to perform a bifurcation study, which is predicated on the use of the center manifold theory. The forward bifurcation was discovered. The Adomian Decomposition Method (ADM) was also used to approximate the solution of the problem's nonlinear system of differential equations. The computations were carried out using Maple, and the graphical results are given.Stability and bifurcation of an SIS epidemic model with saturated incidence rate and treatment functionhttps://www.zbmath.org/1483.371092022-05-16T20:40:13.078697Z"Naji, Raid kamel"https://www.zbmath.org/authors/?q=ai:naji.raid-kamel"Thirthar, Ashraf Adnan"https://www.zbmath.org/authors/?q=ai:thirthar.ashraf-adnanSummary: In this paper an SIS epidemic model with saturated incidence rate and treatment function is proposed and studied. The existence of all feasible equilibrium points is discussed. The local stability conditions of the disease free equilibrium point and endemic equilibrium point are established with the help of basic reproduction number. However the global stability conditions of these equilibrium points are established using Lyapunov method. The local bifurcation near the disease free equilibrium point is investigated. Hopf bifurcation condition, which may occurs around the endemic equilibrium point is obtained. The conditions of backward bifurcation and forward bifurcation near the disease free equilibrium point are also determined. Finally, numerical simulations are given to investigate the global dynamics of the system and confirm the obtained analytical results.Stability analysis of fractional order mathematical model of tumor-immune system interactionhttps://www.zbmath.org/1483.371102022-05-16T20:40:13.078697Z"Öztürk, Ilhan"https://www.zbmath.org/authors/?q=ai:ozturk.ilhan"Özköse, Fatma"https://www.zbmath.org/authors/?q=ai:ozkose.fatmaSummary: In this paper, a fractional-order model of tumor-immune system interaction has been considered. In modeling dynamics, the total population of the model is divided into three subpopulations: macrophages, activated macrophages and tumor cells. The effects of fractional derivative on the stability and dynamical behaviors of the solutions are investigated by using the definition of the Caputo fractional operator that provides convenience for initial conditions of the differential equations. The existence and uniqueness of the solutions for the fractional derivative is examined and numerical simulations are presented to verify the analytical results. In addition, our model is used to describe the kinetics of growth and regression of the B-lymphoma \(BCL_1\) in the spleen of mice. Numerical simulations are given for different choices of fractional order \(\alpha\) and the obtained results are compared with the experimental data. The best approach to reality is observed around \(\alpha=0.80\). One can conclude that fractional model best fit experimental data better than the integer order model.Modeling and stability analysis of the spread of novel coronavirus disease COVID-19https://www.zbmath.org/1483.371112022-05-16T20:40:13.078697Z"Selvam, A. George Maria"https://www.zbmath.org/authors/?q=ai:selvam.a-george-maria"Alzabut, Jehad"https://www.zbmath.org/authors/?q=ai:alzabut.jehad-o"Vianny, D. Abraham"https://www.zbmath.org/authors/?q=ai:vianny.d-abraham"Jacintha, Mary"https://www.zbmath.org/authors/?q=ai:jacintha.mary"Yousef, Fatma Bozkurt"https://www.zbmath.org/authors/?q=ai:yousef.fatma-bozkurtAdaptive chaotic maps and their application to pseudo-random numbers generationhttps://www.zbmath.org/1483.371122022-05-16T20:40:13.078697Z"Tutueva, Aleksandra V."https://www.zbmath.org/authors/?q=ai:tutueva.aleksandra-v"Nepomuceno, Erivelton G."https://www.zbmath.org/authors/?q=ai:nepomuceno.erivelton-geraldo"Karimov, Artur I."https://www.zbmath.org/authors/?q=ai:karimov.artur-i"Andreev, Valery S."https://www.zbmath.org/authors/?q=ai:andreev.valery-s"Butusov, Denis N."https://www.zbmath.org/authors/?q=ai:butusov.denis-nSummary: Chaos-based stream ciphers form a prospective class of data encryption techniques. Usually, in chaos-based encryption schemes, the pseudo-random generators based on chaotic maps are used as a source of randomness. Despite the variety of proposed algorithms, nearly all of them possess many shortcomings. While sequences generated from single-parameter chaotic maps can be easily compromised using the phase space reconstruction method, the employment of multi-parametric maps requires a thorough analysis of the parameter space to establish the areas of chaotic behavior. This complicates the determination of the possible keys for the encryption scheme. Another problem is the degradation of chaotic dynamics in the implementation of the digital chaos generator with finite precision. To avoid the appearance of quasi-chaotic regimes, additional perturbations are usually introduced into the chaotic maps, making the generation scheme more complex and influencing the oscillations regime. In this study, we propose a novel technique utilizing the chaotic maps with adaptive symmetry to create chaos-based encryption schemes with larger parameter space. We compare pseudo-random generators based on the traditional Zaslavsky map and the new adaptive Zaslavsky web map through multi-parametric bifurcation analysis and investigate the parameter spaces of the maps. We explicitly show that pseudo-random sequences generated by the adaptive Zaslavsky map are random, have a weak correlation and possess a larger parameter space. We also present the technique of increasing the period of the chaotic sequence based on the variability of the symmetry coefficient. The speed analysis shows that the proposed encryption algorithm possesses a high encryption speed, being compatible with the best solutions in a field. The obtained results can improve the chaos-based cryptography and inspire further studies of chaotic maps as well as the synthesis of novel discrete models with desirable statistical properties.On Shallit's minimization problemhttps://www.zbmath.org/1483.371132022-05-16T20:40:13.078697Z"Sadov, S. Yu."https://www.zbmath.org/authors/?q=ai:sadov.sergey-yuSummary: In Shallit's problem [\textit{PJ. Shallit}, SIAM Rev., 36, No. 3, 490--491 (1994)], it was proposed to justify a two-term asymptotics of the minimum of a rational function of \(n\) variables defined as the sum of a special form whose number of terms is of order \(n^2\) as \(n\to\infty \). Of particular interest is the second term of this asymptotics (``Shallit's constant''). The solution published in SIAM Review presented an iteration algorithm for calculating this constant, which contained some auxiliary sequences with certain properties of monotonicity. However, a rigorous justification of the properties, necessary to assert the convergence of the iteration process, was replaced by a reference to numerical data. In the present paper, the gaps in the proof are filled on the basis of an analysis of the trajectories of a two-dimensional dynamical system with discrete time corresponding to the minimum points of \(n\)-sums. In addition, a sharp exponential estimate of the remainder in Shallit's asymptotic formula is obtained.A bridge between invariant dynamical structures and uncertainty quantificationhttps://www.zbmath.org/1483.371142022-05-16T20:40:13.078697Z"García-Sánchez, G."https://www.zbmath.org/authors/?q=ai:garcia-sanchez.g"Mancho, A. M."https://www.zbmath.org/authors/?q=ai:mancho.ana-maria|mancho.ana-mariia"Wiggins, S."https://www.zbmath.org/authors/?q=ai:wiggins.stephenSummary: We develop a new quantifier for forward time uncertainty for trajectories that are solutions of models generated from data sets. Our uncertainty quantifier is defined on the phase space in which the trajectories evolve and we show that it has a rich structure that is directly related to phase space structures from dynamical systems theory, such as hyperbolic trajectories and their stable and unstable manifolds. We apply our approach to an ocean data set, as well as standard benchmark models from deterministic dynamical systems theory. A significant application of our results, is that they allow a quantitative comparison of the transport performance described from different ocean data sets. This is particularly interesting nowadays when a wide variety of sources are available since our methodology provides avenues for assessing the effective use of these data sets in a variety of situations.The dynamics of the angular and radial density correlation scaling exponents in fractal to non-fractal morphodynamicshttps://www.zbmath.org/1483.371152022-05-16T20:40:13.078697Z"Nicolás-Carlock, J. R."https://www.zbmath.org/authors/?q=ai:nicolas-carlock.j-r"Solano-Altamirano, J. M."https://www.zbmath.org/authors/?q=ai:solano-altamirano.j-m"Carrillo-Estrada, J. L."https://www.zbmath.org/authors/?q=ai:carrillo-estrada.j-lSummary: Fractal/non-fractal morphological transitions allow for the systematic study of the physics behind fractal morphogenesis in nature. In these systems, the fractal dimension is considered a non-thermal order parameter, commonly and equivalently computed from the scaling of the two-point radial- or angular-density correlations. However, these two quantities lead to discrepancies during the analysis of basic systems, such as in the diffusion-limited aggregation fractal. Hence, the corresponding clarification regarding the limits of the radial/angular scaling equivalence is needed. In this work, considering three fundamental fractal/non-fractal transitions in two dimensions, we show that the unavoidable emergence of growth anisotropies is responsible for the breaking-down of the radial/angular equivalence. Specifically, we show that the angular scaling behaves as a critical power-law, whereas the radial scaling as an exponential that, under the fractal dimension interpretation, resemble first- and second-order transitions, respectively. Remarkably, these and previous results can be unified under a single fractal dimensionality equation.Value distribution of derivatives in polynomial dynamicshttps://www.zbmath.org/1483.371162022-05-16T20:40:13.078697Z"Okuyama, Yûsuke"https://www.zbmath.org/authors/?q=ai:okuyama.yusuke"Vigny, Gabriel"https://www.zbmath.org/authors/?q=ai:vigny.gabrielSummary: For every \(m\in \mathbb{N}\), we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in \(\mathbb{C}\setminus \{0\}\) under the \(m\)th order derivatives of the iterates of a polynomials \(f\in \mathbb{C}[z]\) of degree \(d>1\) towards the harmonic measure of the filled-in Julia set of \(f\) with pole at \(\infty\). We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number field \(k\) for a sequence of effective divisors on \(\mathbb{P}^1(\overline{k})\) having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Hénon-type polynomial automorphism of \(\mathbb{C}^2\) has a given eigenvalue.On the three-dimensional consistency of Hirota's discrete Korteweg-de Vries equationhttps://www.zbmath.org/1483.390092022-05-16T20:40:13.078697Z"Joshi, Nalini"https://www.zbmath.org/authors/?q=ai:joshi.nalini"Nakazono, Nobutaka"https://www.zbmath.org/authors/?q=ai:nakazono.nobutakaSummary: Hirota's discrete Korteweg-de Vries equation (dKdV) is an integrable partial difference equation on \(\mathbb{Z}^2\), which approaches the Korteweg-de Vries equation in a continuum limit. We find new transformations to other equations, including a second-degree second-order partial difference equation, which provide an unusual embedding into a three-dimensional lattice. The consistency of the resulting system extends a property that has been widely used to study partial difference equations on multidimensional lattices.Continuous solutions to two iterative functional equationshttps://www.zbmath.org/1483.390102022-05-16T20:40:13.078697Z"Baron, Karol"https://www.zbmath.org/authors/?q=ai:baron.karolLet \((\Omega,\mathcal{A},P)\) be a probability space and \((X,\rho)\) a separable metric space with the \(\sigma\)-algebra \(\mathcal{B}\) of all its Borel subsets. Let \(f:X\times \Omega \to X\) be a \(\mathcal{B}\otimes \mathcal{A}\) measurable function. The author's aim is to look for continuous solutions \(\varphi: X \to \mathbb{R}\) of the equations
\begin{align*}
\varphi(x)&=F(x)-\int_{\Omega} \varphi(f(x,\omega))P(d\omega), \tag{1} \\
\varphi(x)&=F(x)+\int_{\Omega} \varphi(f(x,\omega))P(d\omega). \tag{2}
\end{align*}
Define
\[
f^0(x,\omega_1,\omega_2,\dots)=x, \qquad f^n(x,\omega_1,\omega_2,\dots)=f(f^{n-1}(x,\omega_1,\omega_2,\dots),\omega_n),
\]
and
\[
\pi_n^f(x,B)=P^{\infty}(f^n(x,\cdot)\in B), \quad n\in \mathbb{N}\cup \{0\}, \ B\in \mathcal{B}.
\]
Under the following conditions
\[
\int_{\Omega} \rho(f(x,\omega),f(z,\omega))P(d\omega)\le \lambda \rho(x,z), \quad x,z \in X, \ \lambda \in (0,1), \tag{3}
\]
and
\[
\int_{\Omega} \rho(f(x,\omega),x)P(d\omega)< \infty,
\]
there exists a probability Borel measure \(\pi^f\) on \(X\) such that for every \(x\in X\) the sequence \((\pi_n^f(x,\cdot))\) converges weakly to \(\pi^f\). Assuming these conditions with a fixed \(\lambda \in (0,1)\), let \(\mathcal{F}(X)\) be defined as the set of all continuous functions \(F:X \to \mathbb{R}\) such that there are a sequence \((F_n)\) of real functions on \(X\) and constants \(\theta \in (0,1)\), \(L\in (0,1/\lambda)\) and \(\alpha, \beta \in (0, \infty)\) such that
\[
|F(x)-F_n(x)|\le \alpha \theta^n, \quad x\in X,\ n\in \mathbb{N},
\]
and
\[
|F_n(x)-F_n(z)|\le \beta L^n\rho(x,z), \quad x,z \in X,\ n\in \mathbb{N}.
\]
The first result can now be stated.
Theorem. Assume the previous conditions. If \(F \in \mathcal{F}(X)\) then
\[
\varphi(x)=F(x)-\frac{1}{2}\int_X F(z)\pi^f(dz)+\sum_{n=1}^{\infty} (-1)^n\Big(\int_X F(z)\pi_n^f(x,dz)-\int_X F(z)\pi^f(dz)\Big), \quad x\in X,
\]
defines a continuous solution of (1). If additionally the condition \(\int_X F(x)\pi^f(dx)=0\) holds true, then the formula
\[
\varphi_0(x)=F(x)+\sum_{n=1}^{\infty} \int_X F(z)\pi_n^f(x,dz), \quad x\in X,
\]
defines a continuous solution \(\varphi_0:X \to \mathbb{R}\) of (2).
Concerning the problem of uniqueness of solution the following is proved.
Theorem. Assume the previous conditions. Let \(F \in \mathcal{F}(X)\).
\begin{itemize}
\item[(i)] If \(\varphi_1, \varphi_2\in \mathcal{F}(X)\) are solutions of (1), then \(\varphi_1=\varphi_2\).
\item[(ii)] If \(\varphi_1, \varphi_2\in \mathcal{F}(X)\) are solutions of (2), then \(\varphi_1-\varphi_2\) is a constant function.
\end{itemize}
The last problem investigated in the paper is about the number of functions \(F\) for which (1) and (2) have continuous solutions.
Assume that \((X,\rho)\) is a compact metric space and that condition (3) holds true. Define
\begin{align*}
\mathcal{F}_1&=\{F\in C(X): \text{Eq. (1) has a continuous solution} \}, \\
\mathcal{F}_2&=\{F\in C_f: \text{Eq. (2) has a continuous solution} \},
\end{align*}
where
\[
C_f=\{F\in C(X): \int_X F(x)\pi^f(dx)=0 \}.
\]
Theorem. Under the previous assumptions, the following holds:
\begin{itemize}
\item[(i)] \(\mathcal{F}_1\) is a Borel and dense subset of \(C(X)\), and if \(\mathcal{F}_1\neq C(X)\), then \(\mathcal{F}_1\) is of first category in \(C(X)\) and a Haar zero subset of \(C(X)\).
\item[(ii)] \(\mathcal{F}_2\) is a Borel and dense subset of \(C_f\), and if \(\mathcal{F}_2\neq C_f\), then \(\mathcal{F}_2\) is of first category in \(C_f\) and a Haar zero subset of \(C_f\).
\end{itemize}
Reviewer: Gian Luigi Forti (Milano)Quantum tomography and the quantum Radon transformhttps://www.zbmath.org/1483.460732022-05-16T20:40:13.078697Z"Ibort, Alberto"https://www.zbmath.org/authors/?q=ai:ibort.alberto"López-Yela, Alberto"https://www.zbmath.org/authors/?q=ai:lopez-yela.albertoSummary: A general framework for the tomographical description of states, that includes, among other tomographical schemes, the classical Radon transform, quantum state tomography and group quantum tomography, in the setting of \(C^\ast\)-algebras is presented. Given a \(C^\ast\)-algebra, the main ingredients for a tomographical description of its states are identified: A generalized sampling theory and a positive transform. A generalization of the notion of dual tomographic pair provides the background for a sampling theory on \(C^\ast\)-algebras and, an extension of Bochner's theorem for functions of positive type, the positive transform.
The abstract theory is realized by using dynamical systems, that is, groups represented on \(C^\ast\)-algebra. Using a fiducial state and the corresponding GNS construction, explicit expressions for tomograms associated with states defined by density operators on the corresponding Hilbert spade are obtained. In particular a general quantum version of the classical definition of the Radon transform is presented. The theory is completed by proving that if the representation of the group is square integrable, the representation itself defines a dual tomographic map and explicit reconstruction formulas are obtained by making a judicious use of the theory of frames. A few significant examples are discussed that illustrate the use and scope of the theory.Fine structure of the dichotomy spectrumhttps://www.zbmath.org/1483.470112022-05-16T20:40:13.078697Z"Pötzsche, Christian"https://www.zbmath.org/authors/?q=ai:potzsche.christianSummary: The dichotomy spectrum is a crucial notion in the theory of dynamical systems, since it contains information on stability and robustness properties. However, recent applications in nonautonomous bifurcation theory showed that a detailed insight into the fine structure of this spectral notion is necessary. On this basis, we explore a helpful connection between the dichotomy spectrum and operator theory. It relates the asymptotic behavior of linear nonautonomous difference equations to the point, surjectivity and Fredholm spectra of weighted shifts. This link yields several dynamically meaningful subsets of the dichotomy spectrum, which not only allows to classify and detect bifurcations, but also simplifies proofs for results on the long term behavior of difference equations with explicitly time-dependent right-hand side.Forward-backward approximation of nonlinear semigroups in finite and infinite horizonhttps://www.zbmath.org/1483.470882022-05-16T20:40:13.078697Z"Contreras, Andrés"https://www.zbmath.org/authors/?q=ai:contreras.andres-a"Peypouquet, Juan"https://www.zbmath.org/authors/?q=ai:peypouquet.juanThe authors consider the problem
\[
\begin{aligned}
-&\dot{u}(t)\in\left( A+B\right) u(t) \text{ for a.e. }t>0,\\
&u(0)=u_{0}\in D(A),
\end{aligned}
\]
in a class of Banach spaces, where \(A\) is \(m\)-accretive and \(B\) is coercive. First, the approximation of solutions is investigated. Solutions are approximated by trajectories constructed by interpolation of sequences generated using forward-backward iteration and these are shown to converge uniformly on a finite time interval, proving existence and uniqueness of solutions. Second, asymptotic equivalence results are given that connect the behaviour of forward-backward iterations as the number of iterations goes to infinity with the behaviour of the solution as time goes to infinity, for step sizes that are sufficiently small. These results are based on a certain inequality which the authors trace back to \textit{E. Hille} [Fysiogr. Sällsk. Lund Förh. 21, No. 14, 130--142 (1951; Zbl 0044.32902)].
Reviewer: Daniel C. Biles (Nashville)Entropic dynamics: from entropy and information geometry to Hamiltonians and quantum mechanicshttps://www.zbmath.org/1483.530202022-05-16T20:40:13.078697Z"Caticha, Ariel"https://www.zbmath.org/authors/?q=ai:caticha.ariel"Bartolomeo, Daniel"https://www.zbmath.org/authors/?q=ai:bartolomeo.daniel"Reginatto, Marcel"https://www.zbmath.org/authors/?q=ai:reginatto.marcelSummary: Entropic Dynamics is a framework in which quantum theory is derived as an application of entropic methods of inference. There is no underlying action principle. Instead, the dynamics is driven by entropy subject to the appropriate constraints. In this paper we show how a Hamiltonian dynamics arises as a type of non-dissipative entropic dynamics. We also show that the particular form of the ``quantum potential'' that leads to the Schrödinger equation follows naturally from information geometry.
For the entire collection see [Zbl 1470.00021].Around Efimov's differential test for homeomorphismhttps://www.zbmath.org/1483.530782022-05-16T20:40:13.078697Z"Alexandrov, Victor"https://www.zbmath.org/authors/?q=ai:alexandrov.victor-aThere is a famous result due to Efimov, more precisely the following Theorem: No surface can be \(C^2\)-immersed in Euclidean 3-space so as to be complete in the induced Riemannian metric, with Gauss curvature \(K \le \) constant \(< 0\).
The paper under review starts with a mini-survey of results related to the previous theorem.
Among other things, Efimov established that the condition \(K \le\) constant \(< 0\) is not the only obstacle for the immersibility of a complete surface of negative curvature; he showed that a rather slow change of Gauss curvature is another obstacle. In all those numerous articles, he used to a large extent one and the same method based on the study of the spherical image of a surface. At that study, an essential role belongs to statements that, under some conditions, a locally homeomorphic mapping \(f : \mathbb{R}^2 \to \mathbb{R}^2\) is a global homeomorphism and \(f(\mathbb{R}^2)\) is a convex domain in \(\mathbb{R}^2\).
Two other theorems of Efimov are recalled in the present paper and the author gives an overview on the analogues of these theorems, their generalizations and applications. The article is devoted to presentation of results motivated by the theory of surfaces, the theory of global inverse function, the Jacobian Conjecture, and the global asymptotic stability of dynamical systems, respectively.
Reviewer: Adela-Gabriela Mihai (Bucureşti)Minimal surfaces under constrained Willmore transformationhttps://www.zbmath.org/1483.530822022-05-16T20:40:13.078697Z"Casinhas Quintino, Áurea"https://www.zbmath.org/authors/?q=ai:quintino.aurea-casinhasSummary: The class of constrained Willmore (CW) surfaces in space-forms constitutes a Möbius invariant class of surfaces with strong links to the theory of integrable systems, with a spectral deformation
[\textit{F. Burstall} et al., Contemp. Math. 308, 39--61 (2002; Zbl 1031.53026)],
defined by the action of a loop of flat metric connections, and Bäcklund transformations
[\textit{F. E. Burstall} and the author, Commun. Anal. Geom. 22, No. 3, 469--518 (2014; Zbl 1306.53051)],
defined by a dressing action by simple factors. Constant mean curvature (CMC) surfaces in 3-dimensional space-forms are
[\textit{J. Richter}, Conformal maps of a Riemannian surface into the space of quaternions. Berlin: TU Berlin, FB Mathematik (1997; Zbl 0896.53005)]
examples of CW surfaces, characterized by the existence of some polynomial conserved quantity
[the author, Constrained Willmore surfaces: symmetries of a Möbius invariant integrable system. University of Bath (PhD Thesis) (2008);
Constrained Willmore surfaces. Symmetries of a Möbius invariant integrable system (to appear). Cambridge: Cambridge University Press (2021; Zbl 07298516);
the author and \textit{S. Duarte Santos}, ``Polynomial conserved quantities for constrained Willmore surfaces'', Preprint, \url{arXiv:1507.01253}].
Both CW spectral deformation and CW Bäcklund transformation preserve the existence of such a conserved quantity, defining, in particular, transformations within the class of CMC surfaces in 3-dimensional space-forms, with, furthermore, preservation of both the space-form and the mean curvature, in the latter case. A classical result by
\textit{G. Thomsen} [Abh. Math. Semin. Univ. Hamb. 3, 31--56 (1923; JFM 49.0530.02)]
characterizes, on the other hand, isothermic Willmore surfaces in 3-space as minimal surfaces in some 3-dimensional space-form. CW transformation preserves the class of Willmore surfaces, as well as the isothermic condition, in the particular case of spectral deformation. We define, in this way, a CW spectral deformation and CW Bäcklund transformations of minimal surfaces in 3-dimensional space-forms into new ones, with preservation of the space-form in the latter case. This paper is dedicated to a reader-friendly overview of the topic.
For the entire collection see [Zbl 1473.53006].Extensions of quasi-morphisms to the symplectomorphism group of the diskhttps://www.zbmath.org/1483.530932022-05-16T20:40:13.078697Z"Maruyama, Shuhei"https://www.zbmath.org/authors/?q=ai:maruyama.shuheiSummary: In this paper, we construct quasi-morphisms on the group of symplectomorphisms of the closed disk \(D\). These quasi-morphisms are extensions of the Ruelle invariant and Gambaudo-Ghys quasi-morphisms. As a corollary, we show that the second bounded cohomology \(H_b^2(\operatorname{Symp}(D))\) is infinite dimensional.Kahan discretizations of skew-symmetric Lotka-Volterra systems and Poisson mapshttps://www.zbmath.org/1483.530992022-05-16T20:40:13.078697Z"Evripidou, C. A."https://www.zbmath.org/authors/?q=ai:evripidou.charalampos-a"Kassotakis, P."https://www.zbmath.org/authors/?q=ai:kassotakis.pavlos-g"Vanhaecke, P."https://www.zbmath.org/authors/?q=ai:vanhaecke.polThe primary result of this paper is a characterization of connected, skew-symmetric graphs \(\Gamma\) with the Kahan-Poisson property. The authors show that a graph as such is a cloning of a graph with vertices \(1,2,\ldots,n\), with an arc \(i \to j\) precisely when \(i<j\), and with all arks having the same value. This characterization helps to understand better the integrability of Lotka-Volterra systems as well as their deformations.
Reviewer: Iakovos Androulidakis (Athína)Connected components of the space of proper gradient vector fieldshttps://www.zbmath.org/1483.550072022-05-16T20:40:13.078697Z"Starostka, Maciej"https://www.zbmath.org/authors/?q=ai:starostka.maciejSummary: We show that there exist two proper gradient vector fields on \(\mathbb{R}^n\) which are homotopic in the category of proper maps but not homotopic in the category of proper gradient maps.Arithmeticity of hyperbolic \(3\)-manifolds containing infinitely many totally geodesic surfaceshttps://www.zbmath.org/1483.570172022-05-16T20:40:13.078697Z"Mohammadi, Amir"https://www.zbmath.org/authors/?q=ai:mohammadi.amir"Margulis, Gregorii"https://www.zbmath.org/authors/?q=ai:margulis.gregory-aThe main result in this paper is that if a closed hyperbolic 3-manifold \(M\) contains infinitely many totally geodesic surfaces, then \(M\) is arithmetic. The result answers affirmatively an open question asked by Reid and by McMullen, cf. [\textit{D. B. McReynolds} and \textit{A. W. Reid}, Math. Res. Lett. 21, No. 1, 169--185 (2014; Zbl 1301.53039) and \textit{K. Delp} et al., ``Problems In Groups, Geometry, and Three-Manifolds'', Preprint, \url{arXiv:1512.04620}]. The proof of arithmeticity uses a superrigidity theorem. As a consequence, the authors obtain that if \(M = \mathbb{H}^3/\Gamma\) is a closed hyperbolic 3-manifold which contains infinitely many totally geodesic surfaces, the index of \(\Gamma\) in its commensurator group is infinite.
Reviewer: Athanase Papadopoulos (Strasbourg)Polytope Novikov homologyhttps://www.zbmath.org/1483.570382022-05-16T20:40:13.078697Z"Pellegrini, Alessio"https://www.zbmath.org/authors/?q=ai:pellegrini.alessioFor a closed smooth oriented and connected finite dimensional manifold \(M\), Sergey P. Novikov associated a homology with each a cohomology class \(a\in H^1_\mathrm{dR}(M)\), the so-called Novikov homology \(HN_\ast(a)\), cf. [\textit{S. P. Novikov}, Sov. Math., Dokl. 24, 222--226 (1981; Zbl 0505.58011); translation from Dokl. Akad. Nauk SSSR 260, 31--35 (1981), Russ. Math. Surv. 37, No. 5, 1--56 (1982; Zbl 0571.58011); translation from Usp. Mat. Nauk 37, No. 5(227), 3--49 (1982)]. Let \(\Phi_a:\pi_1(M)\to\mathbb{R}\) be the period homomorphism, and let \(\pi:\widetilde{M}_a\to M\) be the minimal regular covering with the group of deck transformations \(\Gamma_a\cong\pi_1(M)/\mathrm{Ker}(\Phi_a)\). Then for any representative \(\alpha\in a\) there exists an \(\tilde{f}_\alpha\in C^\infty(\widetilde{M}_a)\) such that \(\pi^\ast\alpha=d\tilde{f}_\alpha\). For a Riemannian metric \(g\) on \(M\) the pair \((\alpha, g)\) is said to be Morse-Smale if \((\tilde{f}_\alpha, \pi^\ast g)\) satisfies the Morse-Smale condition on \(\widetilde{M}_a\). For each \(i\in\mathbb{N}_0:=\mathbb{N}\cup\{0\}\) let \(\mathrm{Crit}_i(\tilde{f}_\alpha)\) denote the critical points of \(\tilde{f}_\alpha\) with Morse index \(i\). The \(i\)th Novikov chain group \(\mathrm{CN}_i(\alpha)\) consists of all formal sums
\[
\xi=\sum_{\tilde{x}\in \mathrm{Crit}_i(\tilde{f}_\alpha)}\xi_{\tilde{x}}\tilde{x}\in \bigoplus_{\tilde{x}\in \mathrm{Crit}_i(\tilde{f}_\alpha)} \mathbb{Z}\langle\tilde{x}\rangle
\]
such that \(\{\tilde{x}\mid \xi_{\tilde{x}}\in\mathbb{Z}\setminus\{0\}\,\&\, \tilde{f}_\alpha(\tilde{x})>c\}\) is finite for each \(c\in\mathbb{R}\). The boundary operator \(\partial : \mathrm{CN}_i(\alpha) \to \mathrm{CN}_{i-1}(\alpha)\) is defined by
\[
\partial \xi:=\sum_{\tilde{x}, \, \tilde{y}} \, \xi_{\tilde{x}} \cdot \#_{\mathrm{alg}} \, \underline{\mathcal{M}}(\tilde{x},\tilde{y};\tilde{f}_{\alpha}) \, \tilde{y},
\]
where \(\#_{\mathrm{alg}} \, \underline{\mathcal{M}}(\tilde{x},\tilde{y};\tilde{f}_{\alpha})\) counts trajectories of negative gradient of \(\tilde{f}_\alpha\) with respect to \(\tilde{g}:=\pi^\ast g\) with signs from \(\tilde{x}\) to \(\tilde{y}\).
The Novikov ring \(\Lambda_\alpha\) consists of all formal sums
\[
\lambda=\sum_{A\in\Gamma_a}\lambda_A A\in \bigoplus_{A\in\Gamma_a}\mathbb{Z}\langle A\rangle
\]
such that \(\{A\in\Gamma_a\mid \lambda_A\in\mathbb{Z}\setminus\{0\}\,\&\, \Phi_a(A)<c\}\) is finite for each \(c\in\mathbb{R}\). The product is given by the convolution
\[
(\lambda\ast\mu)_A=\sum_{B\in\Gamma_a}\lambda_B\mu_{B^{-1}A}.
\]
According to the obvious action of \(\Lambda_a\) on \(\mathrm{CN}_\ast(\alpha)\), the latter is a finitely generated \(\Lambda_a\)-module. Moreover the boundary operator \(\partial\) is \(\Lambda_a\)-linear, and for each \(i \in \mathbb{N}_0\) the Novikov homology
\[
\mathrm{HN}_i(\alpha,g):=\frac{\ker \partial_i}{\mathrm{im} \, \partial_{i+1}}
\]
carries a \(\Lambda_a\)-module structure. Different choices of cohomologous Morse forms representing \(\alpha\) induce isomorphic Novikov homologies. The isomorphism class is said to be the Novikov homology of pairs \((\alpha, g)\), and denoted by \(\mathrm{HN}_\ast(a)\).
In the paper under review the author generalizes the above Novikov homology and defines polytope Novikov homology. Corresponding to a polytope \(\mathcal{A}=\langle a_0, \dots, a_k \rangle \subset H^1_{\mathrm{dR}}(M)\) with vertices \(a_0,\dots,a_k\), there exists a regular cover \(\pi : \widetilde{M}_{\mathcal{A}} \to M\) with the group of deck transformations
\[ \Gamma_\mathcal{A}\cong {\pi_1(M)}{\bigg /}\bigcap_{l=0}^k \mathrm{Ker}(\Phi_{a_l}), \]
Then for every \(a \in \mathcal{A}\) and for any representative \(\alpha\in a\) there exists a \(\tilde{f}_{\alpha} \in C^{\infty}(\widetilde{M}_{\mathcal{A}})\) such that \(\pi^*\alpha=d\tilde{f}_{\alpha}\). Fix a smooth section \(\theta : \mathcal{A} \longrightarrow \Omega^1(M)\), that is, \(\theta_a\) is a representative of \(a\). For each \(i\in\mathbb{N}_0\) the \(i\)th polytope Novikov chain complex group \(\mathrm{CN}_i(\theta_a,\mathcal{A})\) consists of all formal sums
\[ \xi=\sum_{\tilde{x}\in \mathrm{Crit}_i(\tilde{f}_{\theta_a})}\xi_{\tilde{x}}\tilde{x}\in \bigoplus_{\tilde{x}\in \mathrm{Crit}_i(\tilde{f}_\alpha)} \mathbb{Z}\langle\tilde{x}\rangle \]
such that
\begin{gather*}
\xi=\sum_{\tilde{x} \in \mathrm{Crit}_i\left(\tilde{f}_{\theta_a}\right)} \xi_{\tilde{x}} \, \tilde{x} \in \mathrm{CN}_i(\theta_a, \mathcal{A}) \iff \forall b \in \mathcal{A}, \forall c \in \mathbb{R} : \\ \#\lbrace \tilde{x} \mid \xi_{\tilde{x}} \neq 0, \; \tilde{f}_\beta(\tilde{x})>c \rbrace < +\infty,
\end{gather*}
where \(\beta \in b\) is any representative. The groups \(\mathrm{CN}_\bullet(\theta_a,\mathcal{A})\) may be equipped with boundary operators \(\partial_{\theta_a} : \mathrm{CN}_\ast(\theta_a,\mathcal{A}) \to \mathrm{CN}_{\ast-1}(\theta_a,\mathcal{A})\) given by
\[ \partial_{\theta_a} \xi:= \sum_{\tilde{x}, \tilde{y}} \xi_{\tilde{x}} \cdot \#_{\mathrm{alg}} \, \underline{\mathcal{M}}\left(\tilde{x},\tilde{y};\tilde{f}_{\theta_a}\right) \, \tilde{y}. \]
Let \(\widehat{\mathbb{Z}}[\Gamma_{\mathcal{A}}]^b\) denote the upward completion of the group ring \(\mathbb{Z}[\Gamma_{\mathcal{A}}]\) with respect to the period homomorphism \(\Phi_b : \Gamma_{\mathcal{A}} \to \mathbb{R}\). Define the polytope Novikov ring \(\Lambda_\mathcal{A}=\bigcap_{b \in \mathcal{A}} \widehat{\mathbb{Z}}[\Gamma_{\mathcal{A}}]^b\). The above boundary operator \(\partial_{\theta_a}\) is \(\Lambda_{\mathcal{A}}\)-linear. The homology of the chain complex \(\left(\mathrm{CN}_\ast(\vartheta_a,g_{\vartheta_a},\mathcal{A}),\partial \right)\), denoted by \(\mathrm{HN}_\ast(\vartheta_a,\mathcal{A})\), is called the polytope Novikov homology. It is proved that any two cohomology classes in a prescribed polytope give rise to chain homotopy equivalent polytope Novikov complexes over a Novikov ring associated with said polytope.
An important application is to present a novel approach to the (twisted) Novikov Morse Homology Theorem: For any cohomology class \(a \in H^1_{\mathrm{dR}}(M)\) there exists an isomorphism \(\mathrm{HN}_\ast (a) \cong \mathrm{H}_\ast(M,\Lambda_a)\) of Novikov-modules.
The second application is to prove a new polytope Novikov Principle, which generalizes the ordinary Novikov Principle and a recent result of Pajitnov in the abelian case [\textit{A. Pajitnov}, Eur. J. Math. 6, No. 4, 1303--1341 (2020; Zbl 1470.57050)].
Reviewer: Guang-Cun Lu (Beijing)The birth of random evolutionshttps://www.zbmath.org/1483.600382022-05-16T20:40:13.078697Z"Hersh, Reuben"https://www.zbmath.org/authors/?q=ai:hersh.reubenFrom the text: The theory of random evolutions was born in Albuquerque in the late 1960s, flourished and matured in the 1970s, sprouted a robust daughter in Kiev in the 1980s, and is today a tool or method, applicable in a variety of ``real-world'' ventures.Random quasi-periodic paths and quasi-periodic measures of stochastic differential equationshttps://www.zbmath.org/1483.600792022-05-16T20:40:13.078697Z"Feng, Chunrong"https://www.zbmath.org/authors/?q=ai:feng.chunrong"Qu, Baoyou"https://www.zbmath.org/authors/?q=ai:qu.baoyou"Zhao, Huaizhong"https://www.zbmath.org/authors/?q=ai:zhao.huaizhongSummary: In this paper, we define random quasi-periodic paths for random dynamical systems and quasi-periodic measures for Markovian semigroups. We give a sufficient condition for the existence and uniqueness of random quasi-periodic paths and quasi-periodic measures for stochastic differential equations and a sufficient condition for the density of the quasi-periodic measure to exist and to satisfy the Fokker-Planck equation. We obtain an invariant measure by considering lifted flow and semigroup on cylinder and the tightness of the average of lifted quasi-periodic measures. We further prove that the invariant measure is unique, and thus ergodic.Spatial ergodicity for SPDEs via Poincaré-type inequalitieshttps://www.zbmath.org/1483.600912022-05-16T20:40:13.078697Z"Chen, Le"https://www.zbmath.org/authors/?q=ai:chen.le"Khoshnevisan, Davar"https://www.zbmath.org/authors/?q=ai:khoshnevisan.davar"Nualart, David"https://www.zbmath.org/authors/?q=ai:nualart.david"Pu, Fei"https://www.zbmath.org/authors/?q=ai:pu.feiSummary: Consider a parabolic stochastic PDE of the form \(\partial_t u=\frac{1}{2}\Delta u+\sigma (u)\eta\), where \(u=u(t,x)\) for \(t\geq 0\) and \(x\in\mathbb{R}^d, \sigma \colon\mathbb{R}\to \mathbb{R}\) is Lipschitz continuous and non random, and \(\eta\) is a centered Gaussian noise that is white in time and colored in space, with a possibly-signed homogeneous spatial correlation \(f\). If, in addition, \(u(0)\equiv 1\), then we prove that, under a mild decay condition on \(f\), the process \(x\mapsto u(t,x)\) is stationary and ergodic at all times \(t>0\). It has been argued that, when coupled with moment estimates, spatial ergodicity of \(u\) teaches us about the intermittent nature of the solution to such SPDEs [\textit{L. Bertini} and \textit{N. Cancrini}, J. Stat. Phys. 78, No. 5--6, 1377--1401 (1995; Zbl 1080.60508); \textit{D. Khoshnevisan}, Analysis of stochastic partial differential equations. Providence, RI: American Mathematical Society (AMS) (2014; Zbl 1304.60005)]. Our results provide rigorous justification of such discussions.
Our methods hinge on novel facts from harmonic analysis and functions of positive type, as well as from Malliavin calculus and Poincaré inequalities. We further showcase the utility of these Poincaré inequalities by: (a) describing conditions that ensure that the random field \(u(t)\) is mixing for every \(t> 0\); and by (b) giving a quick proof of a conjecture of \textit{D. Conus} et al. [Electron. J. Probab. 17, Paper No. 102, 15 p. (2012; Zbl 1296.60165)] about the ``size'' of the intermittency islands of \(u\).
The ergodicity and the mixing results of this paper are sharp, as they include the classical theory of \textit{G. Maruyama} [Mem. Fac. Sci. Kyūsyū Univ., Ser. A 4, 45--106 (1949; Zbl 0045.40602)] (see also [\textit{H. Dym} and \textit{H. P. McKean}, Gaussian processes, function theory, and the inverse spectral problem. Probability and Mathematical Statistics. Vol. 31. New York San Francisco - London: Academic Press, a subsidiary of Harcourt Brace Jovanovich, Publishers. (1976; Zbl 0327.60029)]) in the simple setting where the nonlinear term \(\sigma\) is a constant function.Multiple Markov Gaussian processeshttps://www.zbmath.org/1483.601002022-05-16T20:40:13.078697Z"Kowalski, Zbigniew S."https://www.zbmath.org/authors/?q=ai:kowalski.zbigniew-s.1Summary: We get a necessary and sufficient condition on the density of the spectral measure for stationary Gaussian processes with a discrete set of parameters to be Markov of order \(k\). We introduce a natural definition of the Markov property of order \(r\in \mathbb{R}_+,\) in the case of continuous parameter. Moreover we give an extension of multiple Markov Gaussian processes with discrete parameters to Gaussian semiflows with some weaker property than the multiple Gaussian property.On recurrent properties of Fisher-Wright's diffusion on \((0,1)\) with mutationhttps://www.zbmath.org/1483.601162022-05-16T20:40:13.078697Z"Sineokiy, Roman"https://www.zbmath.org/authors/?q=ai:sineokiy.roman"Veretennikov, Alexander"https://www.zbmath.org/authors/?q=ai:veretennikov.alexander-yuThis short paper proves a refined exponential recurrent bound for a one-dimensional Fisher-Wright diffusion process living on the interval \((0,1)\) subject to mutations. This bound implies an exponential rate of convergence towards the invariant measure.
More precisely, consider the following one-dimensional stochastic differential equation
\[
dX_t =\left[\#1\right]{a(1-X_t) -bX_t}dt + \varepsilon \sqrt{X_t\left(\#1\right){1-X_t}}dW_t
\]
with parameters \(a,b,\varepsilon >0\). The starting point \(X_0=x\in (0,1)\) is deterministic and \((W_t)\) is a one dimensional standard Brownian motion (under \({\mathbb P}_x\)).
Such equations were introduced for the study of population genetics independently by Wright and Fisher and remain a topical area of investigations until now. In the context of the paper, the model is subject to mutations and the parameters \(a\) and \(b\) stand for the mutation rates (for an account on such models see [\textit{L. Chen} and \textit{D. W. Stroock}, SIAM J. Math. Anal. 42, No. 2, 539--567 (2010; Zbl 1221.35013); \textit{C. L. Epstein} and \textit{R. Mazzeo}, SIAM J. Math. Anal. 42, No. 2, 568--608 (2010; Zbl 1221.35063)] in the authors' reference list) and \(\varepsilon\) as a selection parameter.
Well-known classical results on stochastic differential equations ensure that there is a pathwise unique strong solution for this equation and that this solution is a strong Markov process. Since \((0,1)\) is topologically equivalent to \(\mathbb R\), one may apply Feller's test to show that this solution remains in the open interval \((0,1)\) for all times whenever the condition \(\min(a,b)>\varepsilon^2/2\) is satisfied (Feller's condition).
Although one may wish to apply general results ensuring the existence of an invariant measure with an exponential rate of convergence for general one-dimensional models (see for, e.g., [\textit{L. H. Duc} et al., Stochastic Processes Appl. 128, No. 10, 3253--3272 (2018; Zbl 1434.60138)]), the strategy is quite different here and the main result of the paper gives a deeper insight on the recurrent properties of the Fisher-Wright diffusion process with a more detailed explicit result for the recurrence properties of the solution.
More precisely, for any \(\alpha\in (0,1)\) let \(\tau_\alpha = \inf\left(\#1\right){t\geq 0~:~X_t\in [\alpha, 1-\alpha]}\) stand for the time where the solution enters \([\alpha, 1-\alpha]\). Then, assuming that Feller's condition is satisfied, the main result of the paper asserts that for any constant \(c>0\), there exists a point \(\alpha\in (0,1/2)\) and \(m>0\) such that
\[
\mathbb E_x \mathrm{e}^{c\tau_\alpha} \leq C(m)\,c\,\alpha^{m+1}\left(\#1\right){(1-x)^{-m} + x^{-m}} + 1
\]
with \(C(m) = \frac{2}{\min(a,b)m - \varepsilon^2m(m+1)/2}\).
In a nutshell, the methodology of proof is to draw clever consequences of the application of Itô's formula to the function \(v(t,x) = x^{-m}{\exp}(ct)\).
The stated inequality ensures that there are exponential moments for the returns from very small neighborhood of \(0\) or \(1\) to a compact set in \((0,1)\) and moreover that these exponential moments are quantified by an explicit bound derived from the data.
As a by product of their main result, the authors recover the exponential rate of convergence of the distribution of \(X_t\) towards the invariant measure w.r.t the total variation metric (the computations are not written in the text but are announced for a forthcoming research study concerning more general diffusions).
Note also that the paper contains an independent and short proof that the solution remains \({\mathbb P}_x\)-a.s. in \((0,1)\) whenever Feller's condition is satisfied which is inspired by \textit{I. I. Gikhhman} [``A short remark on Feller's square root condition'', Preprint, \url{https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1756450}].
\textbf{NB}: all number references are given by the authors' reference list.
Reviewer: Miguel Martinez (Marne-la-Vallée)Novel mathematics inspired by industrial challengeshttps://www.zbmath.org/1483.650082022-05-16T20:40:13.078697ZPublisher's description: This contributed volume convenes a rich selection of works with a focus on innovative mathematical methods with applications in real-world, industrial problems. Studies included in this book are all motivated by a relevant industrial challenge, and demonstrate that mathematics for industry can be extremely rewarding, leading to new mathematical methods and sometimes even to entirely new fields within mathematics.
The book is organized into two parts: Computational Sciences and Engineering, and Data Analysis and Finance. In every chapter, readers will find a brief description of why such work fits into this volume; an explanation on which industrial challenges have been instrumental for their inspiration; and which methods have been developed as a result. All these contribute to a greater unity of the text, benefiting not only practitioners and professionals seeking information on novel techniques but also graduate students in applied mathematics, engineering, and related fields.
The articles of this volume will be reviewed individually.Krylov subspace methods for estimating operator-vector multiplications in Hilbert spaceshttps://www.zbmath.org/1483.650802022-05-16T20:40:13.078697Z"Hashimoto, Yuka"https://www.zbmath.org/authors/?q=ai:hashimoto.yuka"Nodera, Takashi"https://www.zbmath.org/authors/?q=ai:nodera.takashiSummary: The Krylov subspace method has been investigated and refined for approximating the behaviors of finite or infinite dimensional linear operators. It has been used for approximating eigenvalues, solutions of linear equations, and operator functions acting on vectors. Recently, for time-series data analysis, much attention is being paid to the Krylov subspace method as a viable method for estimating the multiplications of a vector by an unknown linear operator referred to as a transfer operator. In this paper, we investigate a convergence analysis for Krylov subspace methods for estimating operator-vector multiplications.Feedback integrators for nonholonomic mechanical systemshttps://www.zbmath.org/1483.652022022-05-16T20:40:13.078697Z"Chang, Dong Eui"https://www.zbmath.org/authors/?q=ai:chang.dong-eui"Perlmutter, Matthew"https://www.zbmath.org/authors/?q=ai:perlmutter.matthewSummary: The theory of feedback integrators is extended to handle mechanical systems with nonholonomic constraints with or without symmetry, so as to produce numerical integrators that preserve the nonholonomic constraints as well as other conserved quantities. To extend the feedback integrators, we develop a suitable extension theory for nonholonomic systems and also a corresponding reduction theory for systems with symmetry. It is then applied to various nonholonomic systems such as the Suslov problem on \(\mathrm{SO}(3)\), the knife edge, the Chaplygin sleigh, the vertical rolling disk, the roller racer, the Heisenberg system, and the nonholonomic oscillator.An accurate numerical method and algorithm for constructing solutions of chaotic systemshttps://www.zbmath.org/1483.652042022-05-16T20:40:13.078697Z"Pchelintsev, Alexander N."https://www.zbmath.org/authors/?q=ai:pchelintsev.alexander-nSummary: In various fields of natural science, the chaotic systems of differential equations are considered more than 50 years. The correct prediction of the behaviour of solutions of dynamical model equations is important in understanding of evolution process and reduce uncertainty. However, often used numerical methods are unable to do it on large time segments. In this article, the author considers the modern numerical method and algorithm for constructing solutions of chaotic systems on the example of tumor growth model. Also a modification of Benettin's algorithm presents for calculation of Lyapunov exponents.Scientific visualization for the ODE-based simulator BedSim of LKABhttps://www.zbmath.org/1483.652372022-05-16T20:40:13.078697Z"Dechevsky, Lubomir"https://www.zbmath.org/authors/?q=ai:dechevsky.lubomir-t"Gundersen, Joakim"https://www.zbmath.org/authors/?q=ai:gundersen.joakimSummary: This work describes an applied-mathematic and computer-graphics project for scientific visualization of the industrial processes simulated in the testbed BedSim of the Swedish company LKAB. The various computational, geometric and simulation aspects discussed in the paper are common for the visualization of the output of any simulators of large and complex dynamical systems. One main conclusion (in Section 6) is the necessity to upgrade from ODE-system-based model to PDE-based one. The follow-up papers to this one, see Parts I--IV [Zbl 1483.65238; Zbl 1483.65239; Zbl 1483.65240; Zbl 1483.65241] describe some challenges and solutions on the way to achieving such an upgrade.Minimal multiset grammars for recurrent dynamicshttps://www.zbmath.org/1483.681152022-05-16T20:40:13.078697Z"Farinelli, Alessandro"https://www.zbmath.org/authors/?q=ai:farinelli.alessandro"Franco, Giuditta"https://www.zbmath.org/authors/?q=ai:franco.giuditta"Rizzi, Romeo"https://www.zbmath.org/authors/?q=ai:rizzi.romeoSummary: A biochemical network modeled by a multiset grammar may be investigated from a dynamical viewpoint by a linear recurrence system. This interesting connection between computation by a multiset grammar and a (network) recurrent dynamics poses a minimization problem, which turns out to be NP-hard.
For the entire collection see [Zbl 1358.68015].Complexity of fixed point counting problems in Boolean networkshttps://www.zbmath.org/1483.682402022-05-16T20:40:13.078697Z"Bridoux, Florian"https://www.zbmath.org/authors/?q=ai:bridoux.florian"Durbec, Amélia"https://www.zbmath.org/authors/?q=ai:durbec.amelia"Perrot, Kevin"https://www.zbmath.org/authors/?q=ai:perrot.kevin"Richard, Adrien"https://www.zbmath.org/authors/?q=ai:richard.adrienSummary: A \textit{Boolean network} (BN) with \(n\) components is a discrete dynamical system described by the successive iterations of a function \(f : \{ 0,1\}^n \to \{ 0,1\}^n\). This model finds applications in biology, where fixed points play a central role. For example, in genetic regulations, they correspond to cell phenotypes. In this context, experiments reveal the existence of positive or negative influences among components. The digraph of influences is called \textit{signed interaction digraph} (SID), and one SID may correspond to a large number of BNs. The present work opens a new perspective on the well-established study of fixed points in BNs. When biologists discover the SID of a BN they do not know, they may ask: given that SID, can it correspond to a BN having at least/at most \(k\) fixed points? Depending on the input, we prove that these problems are in \(\mathsf{P}\) or complete for \(\mathsf{NP}\), \(\mathsf{NP}^{\mathsf{NP}}\), \(\mathsf{NP}^{\#\mathsf{P}}\) or \(\mathsf{NEXPTIME}\).Dynamics of a spherical robot with variable moments of inertia and a displaced center of masshttps://www.zbmath.org/1483.700032022-05-16T20:40:13.078697Z"Artemova, Elizaveta M."https://www.zbmath.org/authors/?q=ai:artemova.elizaveta-m"Karavaev, Yury L."https://www.zbmath.org/authors/?q=ai:karavaev.yuri-leonidovich"Mamaev, Ivan S."https://www.zbmath.org/authors/?q=ai:mamaev.ivan-s"Vetchanin, Evgeny V."https://www.zbmath.org/authors/?q=ai:vetchanin.evgenii-vladimirovichSummary: The motion of a spherical robot with periodically changing moments of inertia, internal rotors and a displaced center of mass is considered. It is shown that, under some restrictions on the displacement of the center of mass, the system of interest features chaotic dynamics due to separatrix splitting. A stability analysis is made of the upper equilibrium point of the ball and of the periodic solution arising in its neighborhood, in the case of periodic rotation of the rotors. It is shown that the lower equilibrium point can become unstable in the case of fixed rotors and periodically changing moments of inertia.Basins of convergence of equilibrium points in the restricted three-body problem with modified gravitational potentialhttps://www.zbmath.org/1483.700322022-05-16T20:40:13.078697Z"Zotos, Euaggelos E."https://www.zbmath.org/authors/?q=ai:zotos.euaggelos-e"Chen, Wei"https://www.zbmath.org/authors/?q=ai:chen.wei.1|chen.wei.2|chen.wei.4|chen.wei|chen.wei.3"Abouelmagd, Elbaz I."https://www.zbmath.org/authors/?q=ai:abouelmagd.elbaz-i"Han, Huiting"https://www.zbmath.org/authors/?q=ai:han.huitingSummary: This article aims to investigate the points of equilibrium and the associated convergence basins in the restricted problem with two primaries, with a modified gravitational potential. In particular, for one of the primary bodies, we add an external gravitational term of the form \(1/r^3\), which is very common in general relativity and represents a gravitational field much stronger than the classical Newtonian one. Using the well-known Newton-Raphson iterator we numerically locate the position of the points of equilibrium, while we also obtain their linear stability. Furthermore, for the location of the points of equilibrium, we obtain semi-analytical functions of both the mass parameter and the transition parameter. Finally, we demonstrate how these two variable parameters affect the convergence dynamics of the system as well as the fractal degree of the basin diagrams. The fractal degree is derived by computing the (boundary) basin entropy.Kowalewski top and complex Lie algebrashttps://www.zbmath.org/1483.700372022-05-16T20:40:13.078697Z"Jurdjevic, V."https://www.zbmath.org/authors/?q=ai:jurdjevic.velimirSummary: This paper identifies a natural Hamiltonian on a ten dimensional complex Lie algebra that unravels the mysteries encountered in Kowalewski's famous paper on the motions of a rigid body around its fixed point under the influence of gravity. This system reveals that the enigmatic conditions of Kowalewski, namely, two principal moments of inertia equal to each other and twice the value of the remaining moment of inertia, and the centre of gravity in the plane spanned by the directions corresponding to the equal moments of inertia, are both necessary and sufficient for the existence of an isospectral representation \(\frac{dL(\lambda)}{dt}=[M(\lambda), L(\lambda)]\) with a spectral parameter \(\lambda \). This representation then yields a crucial spectral invariant that naturally accounts for all the integrals of motion, known as Kowalewski type integrals in the literature of the top. This result is fundamentally dependent on a preliminary discovery that the equality of two principal moments of inertia and the placement of the centre of mass in the plane spanned by the corresponding directions is intimately tied to the existence of another integral of motion on whose zero level surface the above spectral representation resides. The link between mechanical tops and Hamiltonian systems on Lie algebras is provided by an earlier result in which it is shown that the equations of mechanical tops with a linear potential, (heavy tops, in particular) can be represented on certain coadjoint orbits in the semi-direct product \(\mathfrak{g}=\mathfrak{p}\rtimes\mathfrak{k}\) induced by a closed subgroup \(K\) of the underlying group \(G\). The passage to complex Lie algebras is motivated by Kowalewski's mysterious use of complex variables. It is shown that the complex variables in her paper are naturally identified with complex quaternions and the representation of \(\mathfrak{so}(4,\mathbb{C})\) as the product \(\mathfrak{sl}(2,\mathbb{C})\times \mathfrak{sl}(2,\mathbb{C})\). The paper also shows that all the equations of Kowalewski type can be solved by a uniform integration procedure over the Jacobian of a hyperelliptic curve, as in the original paper of Kowalewski.Non-commutative integrability, exact solvability and the Hamilton-Jacobi theoryhttps://www.zbmath.org/1483.700382022-05-16T20:40:13.078697Z"Grillo, Sergio"https://www.zbmath.org/authors/?q=ai:grillo.sergio-danielFor a given Hamiltonian system, the simplest property that ensures the exact solvability of their corresponding equations of motion is given by the commutative-integrability property, which means that a complete system of known independent first integrals Poisson commute. When such a system, of first integrals, do not commute, one have to use the non-commutative integrability property (NCI). This property requires two extra-conditions: \textit{the isotropy condition} regarding the rank of the matrix formed with the Poisson brackets and \textit{the closure condition} with respect to the Poisson brackets. In this paper, the author constructs two methods that allow to integrate the equation of motion for a given Hamiltonian system from a known set of isotropic first integrals, without using the closure condition.
Reviewer: Ioan Bucataru (Iaşi)Dynamics of a tourism sustainability model with distributed delayhttps://www.zbmath.org/1483.760102022-05-16T20:40:13.078697Z"Kaslik, Eva"https://www.zbmath.org/authors/?q=ai:kaslik.eva"Neamţu, Mihaela"https://www.zbmath.org/authors/?q=ai:neamtu.mihaelaSummary: This paper generalizes the existing minimal mathematical model of a given generic touristic site by including a distributed time-delay to reflect the whole past history of the number of tourists in their influence on the environment and capital flow. A stability and bifurcation analysis is carried out on the coexisting equilibria of the model, with special emphasis on the positive equilibrium. Considering general delay kernels and choosing the average time-delay as bifurcation parameter, a Hopf bifurcation analysis is undertaken in the neighborhood of the positive equilibrium. This leads to the theoretical characterization of the critical values of the average time delay which are responsible for the occurrence of oscillatory behavior in the system. Extensive numerical simulations are also presented, where the influence of the investment rate and competition parameter on the qualitative behavior of the system in a neighborhood of the positive equilibrium is also discussed.Occurrence of vibrational resonance in an oscillator with an asymmetric Toda potentialhttps://www.zbmath.org/1483.780042022-05-16T20:40:13.078697Z"Kolebaje, Olusola"https://www.zbmath.org/authors/?q=ai:kolebaje.olusola"Popoola, O. O."https://www.zbmath.org/authors/?q=ai:popoola.oyebola-o"Vincent, U. E."https://www.zbmath.org/authors/?q=ai:vincent.uchechukwu-eSummary: Vibrational resonance (VR) is a phenomenon wherein the response of a nonlinear oscillator driven by biharmonic forces with two different frequencies, \(\omega\) and \(\varOmega\), such that \(\varOmega \gg \omega\), is enhanced by optimizing the parameters of high-frequency driving force. In this paper, an counterintuitive scenario in which a biharmonically driven nonlinear oscillator does not vibrate under the well known VR conditions is reported. This behaviour was observed in a system with an integrable and asymmetric Toda potential driven by biharmonic forces in the usual VR configuration. It is shown that with constant dissipation and in the presence of biharmonic forces, VR does not take place, whereas with nonlinear displacement-dependent periodic dissipation multiple VR can be induced at certain values of high-frequency force parameters. Theoretical analysis are validated using numerical computation and Simulink implementation in MATLAB. Finally, the regime in parameter space of the dissipation for optimum occurrence of multiple VR in the Toda oscillator was estimated. This result would be relevant for experimental applications of dual-frequency driven laser models where the Toda potential is extensively employed.Rational vector rogue waves for the \(n\)-component Hirota equation with non-zero backgroundshttps://www.zbmath.org/1483.780052022-05-16T20:40:13.078697Z"Weng, Weifang"https://www.zbmath.org/authors/?q=ai:weng.weifang"Zhang, Guoqiang"https://www.zbmath.org/authors/?q=ai:zhang.guoqiang"Wang, Li"https://www.zbmath.org/authors/?q=ai:wang.li.4|wang.li.1|wang.li.6|wang.li|wang.li.3|wang.li.5|wang.li.2"Zhang, Minghe"https://www.zbmath.org/authors/?q=ai:zhang.minghe"Yan, Zhenya"https://www.zbmath.org/authors/?q=ai:yan.zhenyaSummary: In this paper, the dimensionless \(n\)-component Hirota (alias the \(n\)-Hirota) equation is investigated, which describes the wave propagations of \(n\) ultrashort optical fields in a fiber. Starting from the modified Darboux transform and its Lax pair with initial non-zero plane-wave conditions, we find the novel multi-parametric families of rational vector rogue wave (RW) solutions for the \(n\)-Hirota equation. Furthermore, some weak and strong interactions of rational vector RWs are exhibited for the \(n\)-Hirota equation with \(n = 2, 3, 4, 5, 6\) in detail. In particular, we also deduce the rational vector \(W\)-shaped dark and bright solitons of the \(n\)-component complex mKdV equation, whose representative wave structures are illustrated for \(n = 2, 3, 4, 5\). Finally, the effect of a small non-integrable deformation of the 3-Hirota equation is explored numerically on the excitation of vector RWs in terms of the Fourier spectral method. These obtained rational vector RW and W-shaped soliton solutions will be useful to further explore the related nonlinear wave phenomena in the sense of multi-component physical systems with non-zero backgrounds.Bohr-Sommerfeld levels for quantum completely integrable systemshttps://www.zbmath.org/1483.810702022-05-16T20:40:13.078697Z"Guillemin, Victor"https://www.zbmath.org/authors/?q=ai:guillemin.victor-w"Wang, Zuo Qin"https://www.zbmath.org/authors/?q=ai:wang.zuoqinSummary: In this paper we will show how the Bohr-Sommerfeld levels of a quantum completely integrable system can be computed modulo \(O(\hbar^{\infty})\) by an inductive procedure starting at stage zero with the Bohr-Sommerfeld levels of the corresponding classical completely integrable system.Autocorrelation functions for quantum particles in supersymmetric Pöschl-Teller potentialshttps://www.zbmath.org/1483.810802022-05-16T20:40:13.078697Z"Cellarosi, Francesco"https://www.zbmath.org/authors/?q=ai:cellarosi.francescoSummary: We consider autocorrelation functions for supersymmetric quantum mechanical systems (consisting of a fermion and a boson) confined in trigonometric Pöschl-Teller partner potentials. We study the limit of rescaled autocorrelation functions (at random time) as the localization of the initial state goes to infinity. The limiting distribution can be described using pairs of Jacobi theta functions on a suitably defined homogeneous space, as a corollary of the work of Cellarosi and Marklof. A construction by Contreras-Astorga and Fernández provides large classes of Pöschl-Teller partner potentials to which our analysis applies.Multi-component supersymmetric D type Drinfeld-Sokolov hierarchy and its Virasoro symmetryhttps://www.zbmath.org/1483.810842022-05-16T20:40:13.078697Z"Li, Chuanzhong"https://www.zbmath.org/authors/?q=ai:li.chuanzhong.1|li.chuanzhongSummary: In this paper, we define a multi-component supersymmetric B type 2KP(MS2BKP) hierarchy. Under a reduction, we derive a multi-component supersymmetric D type Drinfeld-Sokolov hierarchy which has a multi super Virasoro algebraic structure.Correlations between quark mass and flavor mixing hierarchieshttps://www.zbmath.org/1483.811462022-05-16T20:40:13.078697Z"Fritzsch, Harald"https://www.zbmath.org/authors/?q=ai:fritzsch.harald"Xing, Zhi-zhong"https://www.zbmath.org/authors/?q=ai:xing.zhizhong"Zhang, Di"https://www.zbmath.org/authors/?q=ai:zhang.diSummary: We calculate the quark flavor mixing matrix \(V\) based on the Hermitian quark mass matrices \(M_{\mathrm{u}}\) and \(M_{\mathrm{d}}\) with vanishing \((1, 1)\), \((1, 3)\) and \((3, 1)\) entries. The popular leading-order prediction \(|V_{ub}/V_{cb} | \simeq \sqrt{m_u/m_c}\) is significantly modified, and the result agrees with the current experimental value. We find that behind the strong \textit{mass} hierarchy of up- or down-type quarks is the weak \textit{texture} hierarchy of \(M_{\mathrm{u}}\) or \(M_{\mathrm{d}}\) characterized by an approximate seesaw-like relation among its \((2, 2)\), \((2, 3)\) and \((3, 3)\) elements.Confronting the inverse seesaw mechanism with the recent muon g-2 resulthttps://www.zbmath.org/1483.811552022-05-16T20:40:13.078697Z"Pinheiro, João Paulo"https://www.zbmath.org/authors/?q=ai:pinheiro.joao-paulo"de S. Pires, C. A."https://www.zbmath.org/authors/?q=ai:de-s-pires.c-a"Queiroz, Farinaldo S."https://www.zbmath.org/authors/?q=ai:queiroz.farinaldo-s"Villamizar, Yoxara S."https://www.zbmath.org/authors/?q=ai:villamizar.yoxara-sSummary: Since the heavy neutrinos of the inverse seesaw mechanism mix largely with the standard ones, the charged currents formed with them and the muons have the potential of generating robust and positive contribution to the anomalous magnetic moment of the muon. However, bounds from the non-unitary in the leptonic mixing matrix may restrict so severely the parameters of the mechanism that, depending on the framework under which the mechanism is implemented, may render it unable to explain the recent muon g-2 result. In this paper we show that this happens when we implement the mechanism into the standard model and into two versions of the 3-3-1 models.Introduction to the algebraic Bethe ansatzhttps://www.zbmath.org/1483.820032022-05-16T20:40:13.078697Z"Slavnov, N. A."https://www.zbmath.org/authors/?q=ai:slavnov.nikita-aThis note is a short introduction to the Algebraic Bethe Ansatz that is one of the essential achievements of the Quantum Inverse Scattering Method. This note is based on the lecture given in the Scientific and Educational Center of Steklov Mathematical Institute in Moscow. The symmetry is limited to \(\mathfrak{sl}_2\) in this note. In Section 2 the \(R\)-matrix \(R(u,v)\) and the monodromy matrix \({T}(u)\) are introduced by the Yang-Baxter relation and the \(RTT\)-relation. The transfer matrix \(\mathcal{T}(u)\) is introduced as the sum of diagonal elements of the monodromy matrix \(T(u)=\left(\begin{smallmatrix} A(u)&B(u)\\ C(u)&D(u) \end{smallmatrix}\right)\). The transfer matrix \(\mathcal{T}(u)\) commutes with the Hamiltonian, hence constructing eigenfunctions of the transfer matrix determines those of the Hamiltonian. In Section 3 the XXX Heisenberg magnet is introduced as an example. In Section 4 the monodromy matrix of the XXZ Heisenberg magnet is introduced. In Section 5 is devoted to construction of the eigenfunctions of the transfer matrix \(\mathcal{T}(u)\). The \(RTT\) relation and the requirements of the spectral parameters called the Bethe equations allow construction of the eigenfunctions of the transfer matrix \(\mathcal{T}(u)\). The spectrums and the Bethe equations of the XXX Heisenberg magnet are given as an example.
For the entire collection see [Zbl 1472.53006].
Reviewer: Takeo Kojima (Yonezawa)Solution of the Kadanoff-Baym equations by iterated expansionshttps://www.zbmath.org/1483.820052022-05-16T20:40:13.078697Z"Özel, Cenap"https://www.zbmath.org/authors/?q=ai:ozel.cenap"Linker, Patrick"https://www.zbmath.org/authors/?q=ai:linker.patrick"Nauman, Syed Khalid"https://www.zbmath.org/authors/?q=ai:nauman.syed-khalidSummary: In this paper, we will show how the general nonlinear Kadanoff-Baym equations can be solved with iterated series expansion. In this regard, we will neglect vertex corrections. Further, we will obtain a formal solution in terms of colored tree graphs. The iteration procedure will show that after proper discretization, well-structured matrix equations would be obtained which are easy to implement in numerical simulation.Complete characterization of flocking versus nonflocking of Cucker-Smale model with nonlinear velocity couplingshttps://www.zbmath.org/1483.820062022-05-16T20:40:13.078697Z"Kim, Jong-Ho"https://www.zbmath.org/authors/?q=ai:kim.jongho"Park, Jea-Hyun"https://www.zbmath.org/authors/?q=ai:park.jea-hyunSummary: We consider the Cucker-Smale model with a regular communication rate and nonlinear velocity couplings, which can be understood as the parabolic equations for the discrete \(p\)-Laplacian \((p\geq 1)\) with nonlinear weights involving a parameter \(\beta(>0)\). For this model, we study the initial data and the ranges of \(p\) and \(\beta\) to characterize when flocking and nonflocking occur. Specifically, we analyze the nonflocking case, subdividing it into \textit{semi-nonflocking} (only velocity alignment holds) and \textit{full nonflocking} (group formation and velocity alignment do not hold). More precisely, we show that if \(\beta\in(0, 1]\), \(p\in[1,3)\), then flocking occurs for any initial data. If \(\beta\in(0,1]\), \(p\in[3,\infty)\), then semi-nonflocking occurs for any initial data. If \(\beta\in(1,\infty)\), \(p\in[1,3)\), then flocking occurs for some initial data. In the case \(\beta\in(1,\infty)\) and \(p\in[3,\infty)\), we observe alternative states. Finally, we have numerically verified the conclusions obtained by analytical calculations.Helicity and spin conservation in linearized gravityhttps://www.zbmath.org/1483.830142022-05-16T20:40:13.078697Z"Aghapour, Sajad"https://www.zbmath.org/authors/?q=ai:aghapour.sajad"Andersson, Lars"https://www.zbmath.org/authors/?q=ai:andersson.lars-erik|andersson.lars-ake|andersson.lars-l"Bhattacharyya, Reebhu"https://www.zbmath.org/authors/?q=ai:bhattacharyya.reebhuSummary: The duality-symmetric, Maxwell-like, formulation of linearized gravity introduced by \textit{S. M. Barnett} [New J. Phys. 16, No. 2, Article ID 023027, 23 p. (2014; Zbl 1451.83014)] is used to generalize the conservation laws for helicity, the spin part of angular momentum, and spin-flux, to the case of linearized gravity. These conservation laws have been shown to follow from the conservation property of the helicity array, an analog of Lipkin's zilch tensor. The analog of the helicity array for linearized gravity is constructed and is shown to be conserved.Modifications to the signal from a gravitational wave event due to a surrounding shell of matterhttps://www.zbmath.org/1483.830162022-05-16T20:40:13.078697Z"Naidoo, Monos"https://www.zbmath.org/authors/?q=ai:naidoo.monos"Bishop, Nigel T."https://www.zbmath.org/authors/?q=ai:bishop.nigel-t"van der Walt, Petrus J."https://www.zbmath.org/authors/?q=ai:van-der-walt.petrus-jSummary: In previous work, we established theoretical results concerning the effect of matter shells surrounding a gravitational wave (GW) source, and we now apply these results to astrophysical scenarios. Firstly, it is shown that GW echoes that are claimed to be present in LIGO data of certain events, could not have been caused by a matter shell. However, it is also shown that there are scenarios in which matter shells could make modifications of order a few percent to a GW signal; these scenarios include binary black hole mergers, binary neutron star mergers, and core collapse supernovae.A note on size-momentum correspondence and chaoshttps://www.zbmath.org/1483.830552022-05-16T20:40:13.078697Z"Mahish, Sandip"https://www.zbmath.org/authors/?q=ai:mahish.sandip"Mohapatra, Shrohan"https://www.zbmath.org/authors/?q=ai:mohapatra.shrohan"Sil, Karunava"https://www.zbmath.org/authors/?q=ai:sil.karunava"Bhamidipati, Chandrasekhar"https://www.zbmath.org/authors/?q=ai:bhamidipati.chandrasekharSummary: The aim of this note is to explore \textit{L. Susskind}'s proposal [``Why do things fall?'', Preprint, \url{arXiv:1802.01198}] on the connection between operator size in chaotic theories and the bulk momentum of a particle falling into black holes (see also
[\textit{A. R. Brown} et al., ``Falling toward charged black holes'', Phys. Rev. D (3) 98, No. 12, Article ID 126016, 8 p. (2018, \url{doi:10.1103/PhysRevD.98.126016});
\textit{D. S. Ageev} and \textit{I. Ya. Aref'eva}, J. High Energy Phys. 2019, No. 1, Paper No. 100, 9 p. (2019; Zbl 1409.81101);
\textit{L. Susskind}, ``Complexity and Newton's laws'', Preprint, \url{arXiv:1904.12819};
\textit{J. L. F. Barbón} et al., J. High Energy Phys. 2020, No. 7, Paper No. 169, 23 p. (2020; Zbl 1451.83034);
\textit{L. Susskind} and \textit{Y. Zhao}, J. High Energy Phys. 2021, No. 3, Paper No. 239, 13 p. (2021; Zbl 1461.83021)]
for more recent generalizations), in a broad class of models involving Gauss-Bonnet(GB) and Lifshitz-Hyperscaling violating theories in AdS. For Gauss-Bonnet black holes, the operator size is seen to be suppressed as the coupling constant \(\lambda\) is increased. For the Lifshitz-hyperscaling violating theories characterised by the parameters \(z\) and \(\theta \), the operator size is higher as compared to case \(z = 1\), \(\theta = 0\) (Reissner-Nordstrom AdS black holes). In the case of operators with global charge corresponding to charged particles falling into black holes, suppression of chaos is seen in general theories of gravity, in conformity with the original proposal [Susskind, arXiv:1802.01198, loc. cit.] and earlier findings [Ageev and Aref'eva, loc. cit.].Evolution of fractional-order chaotic economic systems based on non-degenerate equilibrium pointshttps://www.zbmath.org/1483.860082022-05-16T20:40:13.078697Z"Zhang, Guoxing"https://www.zbmath.org/authors/?q=ai:zhang.guoxing"Qian, Pengxiao"https://www.zbmath.org/authors/?q=ai:qian.pengxiao"Su, Zhaoxian"https://www.zbmath.org/authors/?q=ai:su.zhaoxianSummary: The economic system is an irreversible entropy increase process which is constructed by many elements and is far away from the equilibrium point; and affected by various parameters change, it is quite common that its motion state appears chaotic phenomenon due to instability. The extremely complex and not completely random aperiodic motion form of chaotic phenomenon is strongly sensitive to initial conditions. The development of nonlinear science, especially the emergence and development of chaos and fractal theory, has gradually become a powerful tool for economists to study the complexity, uncertainty and nonlinearity of social economic systems; and some visionary economists began to apply the research results of nonlinear science to economics, which has produced nonlinear economics. On the basis of summarizing and analyzing previous research works, this paper first obtains the non-degenerate equilibrium point of some typical fractional-order chaotic economic systems and transforms the equilibrium points of those systems to the origin through coordinate transformation, and then analyzes the Jacobi matrixes of new systems obtained through coordinate translation, and the parameter conditions of bifurcation in the economic systems are finally given and the numerical simulation of the fractional-order chaotic economic system evolution is carried out through bifurcation diagram, phase diagram and time series diagram. The study results of this paper provide a reference for the further study of the evolution of fractional-order chaotic economic systems with non-degenerate equilibrium points.Complexity evolution of chaotic financial systems based on fractional calculushttps://www.zbmath.org/1483.912252022-05-16T20:40:13.078697Z"Wen, Chunhui"https://www.zbmath.org/authors/?q=ai:wen.chunhui"Yang, Jinhai"https://www.zbmath.org/authors/?q=ai:yang.jinhaiSummary: Economics and finance are extremely complex nonlinear systems involving human subjects with many subjective factors. There are numerous attribute properties that cannot be described by the theory of integer-order calculus; so it is necessary to theoretically study the internal complexity of the economic and financial system using the method of bifurcation and chaos of fractional nonlinear dynamics. Fractional calculus can more accurately describe the existence characteristics of complex physical, financial or medical systems, and can truly reflect the actual state properties of these systems; therefore the application of fractional order in chaotic systems has great significance to study the mathematical analysis of nonlinear dynamic systems, and the use of fractional calculus theory to model the complexity evolution of fractional chaotic financial systems has attracted more and more scholars' attention. On the basis of summarizing and analyzing previous studies, this paper qualitatively analyzes the stability of equilibrium solution of fractional-order chaotic financial system, and explores the complexity evolution law of the financial system near the equilibrium point and the occurring conditions of asymptotic chaotic state near this equilibrium point, and simulate the complexity evolution of chaotic financial systems using the Admas-Bashforth-Moulton finite difference method for mapping, phase diagram and time series graph. The research results of this paper provide a reference for government to formulate relevant economic policies, decision-making or further research on the complexity evolution of fractional-order chaotic financial systems.Delay-induced synchronization in two coupled chaotic memristive Hopfield neural networkshttps://www.zbmath.org/1483.920252022-05-16T20:40:13.078697Z"Wang, Zhen"https://www.zbmath.org/authors/?q=ai:wang.zhen.2"Parastesh, Fatemeh"https://www.zbmath.org/authors/?q=ai:parastesh.fatemeh"Rajagopal, Karthikeyan"https://www.zbmath.org/authors/?q=ai:rajagopal.karthikeyan"Hamarash, Ibrahim Ismael"https://www.zbmath.org/authors/?q=ai:hamarash.ibrahim-ismael"Hussain, Iqtadar"https://www.zbmath.org/authors/?q=ai:hussain.iqtadarSummary: This paper is concerned with the synchronization of two coupled hyperbolic-type Hopfield neural networks with a memristive synaptic connection. The results show that the coupling with no time-delay cannot lead to complete synchronization and increasing the coupling strength causes anti-phase synchronization. While adding time-delays to the coupling term not only changes the dynamical behavior of the network but also facilitates reaching the full synchronization manifold. The network is investigated in two different cases of single and multiple time-delays. The calculated average synchronization error indicates that using two time-delays has a better effect on the synchronization of systems. However, the synchronization region is lessened by increasing the time-delay values.Uniqueness of weakly reversible and deficiency zero realizations of dynamical systemshttps://www.zbmath.org/1483.920732022-05-16T20:40:13.078697Z"Craciun, Gheorghe"https://www.zbmath.org/authors/?q=ai:craciun.gheorghe"Jin, Jiaxin"https://www.zbmath.org/authors/?q=ai:jin.jiaxin"Yu, Polly Y."https://www.zbmath.org/authors/?q=ai:yu.polly-yThe authors address the relationship between chemical reaction networks obeying mass-action kinetics and the topological dynamics of corresponding systems of differential equations. A reaction network together with rate constants gives rise to a specific differential equation. The question is in how far a differential equation determines a relevant reaction network. In general, the differential equation does not select a particular network. The main result tells us that this is nevertheless the case for the simplest and most important class of networks, consisting of those that are weakly reversible and deficiency zero (Theorem 3.12). If different networks show the same dynamics then at most one belongs to this special class. The paper presents a thorough introduction to the pertinent theory together with a description of its development, equipped with numerous references to the literature.
Reviewer: Dieter Erle (Dortmund)Bifurcation, multistability in the dynamics of tumor growth and electronic simulations by the use of Pspicehttps://www.zbmath.org/1483.920802022-05-16T20:40:13.078697Z"Kemwoue, Florent Feudjio"https://www.zbmath.org/authors/?q=ai:kemwoue.florent-feudjio"Dongo, Jean Marie"https://www.zbmath.org/authors/?q=ai:dongo.jean-marie"Mballa, Rose Ngono"https://www.zbmath.org/authors/?q=ai:mballa.rose-ngono"Gninzanlong, Carlos Lawrence"https://www.zbmath.org/authors/?q=ai:gninzanlong.carlos-lawrence"Kemayou, Marcel Wouapi"https://www.zbmath.org/authors/?q=ai:kemayou.marcel-wouapi"Mokhtari, Bouchra"https://www.zbmath.org/authors/?q=ai:mokhtari.bouchra"Biya-Motto, Frederick"https://www.zbmath.org/authors/?q=ai:biya-motto.frederick"Atangana, Jacques"https://www.zbmath.org/authors/?q=ai:atangana.jacquesSummary: This contribution throws more light on the nonlinear dynamics of a cancer HET model proposed by \textit{L. G. de Pillis} and \textit{A. Radunskaya} [Math. Comput. Modelling 37, No. 11, 1221--1244 (2003; Zbl 1043.92018)] and adjusted by \textit{M. Itik} and \textit{S. P. Banks} [Int. J. Bifurcation Chaos Appl. Sci. Eng. 20, No. 1, 71--79 (2010; Zbl 1183.34064)]. Research shows that the number of equilibrium points in the model fluctuates between 3 and 8 depending on whether the biological parameters of the system vary. For some well precise parameters, the system has 5 equilibrium points, 3 of which are always unstable, 1 always quasi-stable and the last, on the other hand, the only point where all cell populations change stability according to the values of the rate of growth parameter of the host cells. We show that the Hopf bifurcation occurs in this system, when the growth rate of the host cells varies and reaches a critical value. Applying the theory of the normal bifurcation form, we describe the formulae for determining the direction and stability of the periodic solutions of the Hopf bifurcation. Numerical simulations are performed to illustrate its theoretical analysis. Numerical simulations, performed in terms of bifurcation diagrams, Lyapunov exponent graph and phase portraits, permits to highlight the rich and complex phenomena presented by the model. The exploitation of these numerical results reveals that the system degenerates a transition to chaos intermittently through the saddle-node bifurcation. We find that the system presents the diversity of bifurcations such as the chaos, periodic window, saddle-node bifurcations, internal crises, when mentioned parameters are varied in small steps. In addition, we find that the model also has multiple attractors for a precise set of parameters. The basins of attraction of various coexisting attractors present extremely complex structures, thus justifying jumps between coexisting attractors. Finally, in addition to mathematical analysis developed in the work and digital processing, we propose an electronic implementation of the three-dimensional model of cancer capable of imitating the model of mathematical evolution. We determine a circuit equivalent to the ODE model, we determine the electrical components equivalent to the ODE parameters of the mathematical model and the equivalent electronic circuit is simulated numerically with the Orcad-Pspice software. The numerical results obtained from the proposed Orcad-Pspice electronic circuit exhibit the same behavior as that obtained by the numerical simulations and the comparison results provide proof of the reliability and precision of the proposed circuit.Dynamic behavior of a fractional order prey-predator model with group defensehttps://www.zbmath.org/1483.921072022-05-16T20:40:13.078697Z"Alidousti, Javad"https://www.zbmath.org/authors/?q=ai:alidousti.javad"Ghafari, Elham"https://www.zbmath.org/authors/?q=ai:ghafari.elhamSummary: In this paper, we consider a fractional order prey predator model with a prey and two predator species with the group defense capability. In this model, we use the Holling-IV functional response, called Monod-Haldane function, for interactions between prey and predator species. Boundedness of the solution will be proved. Local stability of system's equilibrium points will be investigated analytically and the required conditions for existence of Hopf bifurcation will be obtained. Finally, by using numerical methods, the validity of the obtained results and more dynamical behaviors of system, such as chaotic and periodic solutions will be assessed.Vaccination and vector control effect on dengue virus transmission dynamics: modelling and simulationhttps://www.zbmath.org/1483.921182022-05-16T20:40:13.078697Z"Abidemi, A."https://www.zbmath.org/authors/?q=ai:abidemi.afeez"Abd Aziz, M. I."https://www.zbmath.org/authors/?q=ai:abd-aziz.m-i"Ahmad, R."https://www.zbmath.org/authors/?q=ai:ahmad.reyaz|ahmad.riyaz|ahmad.robiah|ahmad.rauf|ahmad.riaz|ahmad.r-badlishah|ahmad.rehan|ahmad.rana-zeeshan|ahmad.rodina|ahmad.rubi|ahmad.rashdi-shah|ahmad.rashid.1|ahmad.rokiah-rozita|ahmad.rana-tariq-mehmood|ahmad.rohanin|ahmad.rais|ahmad.raheelSummary: This paper presents a two-strain compartmental dengue model with variable humans and mosquitoes populations sizes. The model incorporates two control measures: \textit{Dengvaxia} vaccine and insecticide (adulticide) to forecast the transmission and effective control strategy for dengue in Madeira Island if there is a new outbreak with a different virus serotype after the first outbreak in 2012. The basic reproduction number, \(\mathcal{R}_0=\max\{\sqrt{\mathcal{R}_{01}}, \sqrt{\mathcal{R}_{0j}}\}\), associated with the model is computed using the next generation matrix operator. The disease-free equilibrium is found to be locally asymptotically stable when both \(\mathcal{R}_{01},\mathcal{R}_{0j}<1\), but unstable otherwise. The global asymptotic stability of the model is derived using the comparison theorem. Sensitivity analysis is carried out on the model parameters. The results of the analysis show that mosquito biting and death rates are the most sensitive parameters. Three strategies: the use of \textit{Dengvaxia} vaccine only, the use of adulticide only, and the combination of \textit{Dengvaxia} vaccine and adulticide, are considered for the control implementation under two scenarios (less and more aggressive cases). The numerical results show that a strategy which is based on \textit{Dengvaxia} vaccine and adulticide is the most effective strategy for controlling dengue disease transmission in both scenarios among the considered strategies.A new zoonotic visceral leishmaniasis dynamic transmission model with age-structurehttps://www.zbmath.org/1483.921222022-05-16T20:40:13.078697Z"Bi, Kaiming"https://www.zbmath.org/authors/?q=ai:bi.kaiming"Chen, Yuyang"https://www.zbmath.org/authors/?q=ai:chen.yuyang"Zhao, Songnian"https://www.zbmath.org/authors/?q=ai:zhao.songnian"Ben-Arieh, David"https://www.zbmath.org/authors/?q=ai:ben-arieh.david"Wu, Chih-Hang (John)"https://www.zbmath.org/authors/?q=ai:wu.chih-hangSummary: Visceral leishmaniasis (VL) is a fatal, neglected tropical disease primarily caused by \textit{Leishmania donovani} (\textit{L. donovani}) and \textit{Leishmania infantum} (\textit{L. infantum}). According to VL infectious data reports from severely affected countries, children and teenagers (ages 0--20) have a significantly higher vulnerability to VL infection than other populations. This paper utilizes an infected function (by age) established from epidemic prevalence data to propose a new partial differential equation (PDE) model for infection transmission patterns for various age groups. This new PDE model can be used to study VL epidemics in time and age dimensions. Disease-free and endemic equilibriums are discussed in relation to theoretical stability of the PDE system. This paper also proposes system output adjustment using historical VL data from the World Health Organization. Statistical methods such as the moving average and the autoregressive methods are used to calibrate estimated prevalence trends, potentially minimizing differences between stochastic stimulation results and reported real-world data. Results from simulation experiments using the PDE model were used to predict the worldwide VL severity of the epidemic in the next four years (from 2017 to 2020).Analysis of a stochastic distributed delay epidemic model with relapse and Gamma distribution kernelhttps://www.zbmath.org/1483.921252022-05-16T20:40:13.078697Z"Caraballo, Tomás"https://www.zbmath.org/authors/?q=ai:caraballo.tomas"El Fatini, Mohamed"https://www.zbmath.org/authors/?q=ai:el-fatini.mohamed"El Khalifi, Mohamed"https://www.zbmath.org/authors/?q=ai:el-khalifi.mohamed"Gerlach, Richard"https://www.zbmath.org/authors/?q=ai:gerlach.richard-h"Pettersson, Roger"https://www.zbmath.org/authors/?q=ai:pettersson.rogerSummary: In this work, we investigate a stochastic epidemic model with relapse and distributed delay. First, we prove that our model possesses and unique global positive solution. Next, by means of the Lyapunov method, we determine some sufficient criteria for the extinction of the disease and its persistence. In addition, we establish the existence of a unique stationary distribution to our model. Finally, we provide some numerical simulations for the stochastic model to assist and show the applicability and efficiency of our results.Global stability analysis of a two-strain epidemic model with non-monotone incidence rateshttps://www.zbmath.org/1483.921442022-05-16T20:40:13.078697Z"Meskaf, Adil"https://www.zbmath.org/authors/?q=ai:meskaf.adil"Khyar, Omar"https://www.zbmath.org/authors/?q=ai:khyar.omar"Danane, Jaouad"https://www.zbmath.org/authors/?q=ai:danane.jaouad"Allali, Karam"https://www.zbmath.org/authors/?q=ai:allali.karamSummary: In this paper, we study an epidemic model describing two strains with non-monotone incidence rates. The model consists of six ordinary differential equations illustrating the interaction between the susceptible, the exposed, the infected and the removed individuals. The system of equations has four equilibrium points, disease-free equilibrium, endemic equilibrium with respect to strain 1, endemic equilibrium with respect to strain 2, and the last endemic equilibrium with respect to both strains. The global stability analysis of the equilibrium points was carried out through the use of suitable Lyapunov functions. Two basic reproduction numbers \(R_0^1\) and \(R_0^2\) are found; we have shown that if both are less than one, the disease dies out. It was established that the global stability of each endemic equilibrium depends on both basic reproduction numbers and also on the strain inhibitory effect reproduction number(s) \(R_m\) and/or \(R_k\). It was also shown that any strain with highest basic reproduction number will automatically dominate the other strain. Numerical simulations were carried out to support the analytic results and to show the effect of different problem parameters on the infection spread.Synchronization patterns with strong memory adaptive control in networks of coupled neurons with chimera states dynamicshttps://www.zbmath.org/1483.933082022-05-16T20:40:13.078697Z"Vázquez-Guerrero, P."https://www.zbmath.org/authors/?q=ai:vazquez-guerrero.p"Gómez-Aguilar, J. F."https://www.zbmath.org/authors/?q=ai:gomez-aguilar.jose-francisco"Santamaria, F."https://www.zbmath.org/authors/?q=ai:santamaria.fidel"Escobar-Jiménez, R. F."https://www.zbmath.org/authors/?q=ai:escobar-jimenez.ricardo-fabricioSummary: This work presents the Hindmarsh-Rose fractional model of three-state using the Atangana-Baleanu-Caputo fractional derivative with strong memory. The model allows simulating the chimera states in a neural network. To achieve the synchronization was developed a fractional adaptive controller which is based on the uncertainty of the coupling parameters. The synchronization was studied using different fractional-orders and for 15, 40, 65 and 90 neurons. We consider fractional derivatives with nonlocal and non-singular Mittag-Leffler law. The simulations results show that the neurons synchronization is reached using the proposed method. We believe that the application of fractional operators to synchronization of chimera states open a new direction of research in the near future.A novel image encryption algorithm based on chaotic billiardshttps://www.zbmath.org/1483.940382022-05-16T20:40:13.078697Z"Charif, Khalid"https://www.zbmath.org/authors/?q=ai:charif.khalid"Guennoun, Zine El Abidine"https://www.zbmath.org/authors/?q=ai:el-abidine-guennoun.zineSummary: In this paper, a novel symmetric image encryption algorithm is proposed on the basis of the chaotic billiard (Sinai billiard). These chaotic systems, characterized by the highest degree of chaos, have been recently introduced in cryptography. The design of the algorithm relies on the random walk of three-point particles that move freely in the billiard. It includes several rounds of the confusion and diffusion processes. In each ciphering round, the permutation of the image pixels is performed according to the random numbers obtained from the output of a particle system, by using a new calculation technique. Thus, the pixel values are hidden by the pseudo-random sequences generated by the systems of the other two particles. The simulation results demonstrate that the algorithm is simple to implement and is highly secure. It has successfully passed all security tests and has shown the desirable properties in a good secure cryptosystem.A new approach to fuzzy sets: application to the design of nonlinear time series, symmetry-breaking patterns, and non-sinusoidal limit-cycle oscillationshttps://www.zbmath.org/1483.940782022-05-16T20:40:13.078697Z"García-Morales, Vladimir"https://www.zbmath.org/authors/?q=ai:garcia-morales.vladimirSummary: It is shown that characteristic functions of sets can be made fuzzy by means of the \(\mathcal{B}_\kappa\)-function, recently introduced by the author, where the fuzziness parameter \(\kappa\in\mathbb{R}\) controls how much a fuzzy set deviates from the crisp set obtained in the limit \(\kappa\rightarrow 0\). As applications, we present first a general expression for a switching function that may be of interest in electrical engineering and in the design of nonlinear time series. We then introduce another general expression that allows wallpaper and frieze patterns for every possible planar symmetry group (besides patterns typical of quasicrystals) to be designed. We show how the fuzziness parameter \(\kappa\) plays an analogous role to temperature in physical applications and may be used to break the symmetry of spatial patterns. As a further, important application, we establish a theorem on the shaping of limit cycle oscillations far from bifurcations in smooth deterministic nonlinear dynamical systems governed by differential equations. Following this application, we briefly discuss a generalization of the Stuart-Landau equation to non-sinusoidal oscillators.Motifs for processes on networkshttps://www.zbmath.org/1483.940802022-05-16T20:40:13.078697Z"Schwarze, Alice C."https://www.zbmath.org/authors/?q=ai:schwarze.alice-c"Porter, Mason A."https://www.zbmath.org/authors/?q=ai:porter.mason-a