Recent zbMATH articles in MSC 37https://zbmath.org/atom/cc/372021-01-08T12:24:00+00:00WerkzeugHausdorff dimension of chaotic sets caused by a continuous self-map on \(I^n\).https://zbmath.org/1449.280082021-01-08T12:24:00+00:00"Wu, Huaming"https://zbmath.org/authors/?q=ai:wu.huamingSummary: This paper extends the results of Hausdorff dimension of chaotic sets caused by continuous self-maps on \(I\) and \({I^2}\) to the \(n\)-dimensional cube. We prove that there is a residual set \(\mathscr{R}\) in \({C^0} ({I^n})\), if set \(C \subset {I^n}\) is chaotic for any given \(f \in \mathscr{R}\) in the sense of Li-Yorke, then \({\dim_H} (C) \le n-1\). Similarly, the results on high dimensional Cartesian product can be obtained. That is, there are residual sets \({\mathscr{R}_i}\) in \({C^0} ({I^{ni}}, {I^{ni}})\) such that for any \({f_i} \in {\mathscr{R}_i}, i = 1, 2\), if set \({C_i} \subset {I^{ni}}\) is chaotic in the sense of Li-Yorke, then \({\dim_H} ({C_1} \times {C_2}) \le n-1\).Modelling of chaos in smooth piecewise dynamical systems with one discontinuous point.https://zbmath.org/1449.370102021-01-08T12:24:00+00:00"Pourbarat, Mehdi"https://zbmath.org/authors/?q=ai:pourbarat.mehdi"Abbasi, Neda"https://zbmath.org/authors/?q=ai:abbasi.neda"Makrooni, Roya"https://zbmath.org/authors/?q=ai:makrooni.royaSummary: In this paper, we provide conditions on the smooth piecewise dynamical systems that guarantee the existence of Devaney chaos. In fact, we show that if \(f\) is a generalized semi-baker map with two branches and its derivative greater than or equal \(\sqrt{2}\), then the dynamical system related to that is chaotic in the sense of Devaney. Such conditions on the dynamical systems with more than one discontinues point essentially does not satisfy this result.Bi-Hamiltonian structure of three-dimensional Lotka-Volterra system.https://zbmath.org/1449.370372021-01-08T12:24:00+00:00"Xu, Mingxing"https://zbmath.org/authors/?q=ai:xu.mingxing"Zhou, Ran"https://zbmath.org/authors/?q=ai:zhou.ranSummary: We considered two Lotka-Volterra system with three degrees of freedom. Firstly, we introduced the generalized Poisson-bracket and generalized Hamiltonian structures in three-dimensional systems. Secondly, we observed the first integral of the system by constructing local homeomorphism transformation and established the algebraic equation. The Hamiltonian function of the system was obtained by solving the algebraic equation. Finally, we gave the sufficient conditions for the existence of Bi-Hamiltonian structure of Lotka-Volterra system.Optimal and adaptive control for a kind of 3D chaotic and 4D hyper-chaotic systems.https://zbmath.org/1449.370632021-01-08T12:24:00+00:00"Effati, S."https://zbmath.org/authors/?q=ai:effati.sohrab"Saberi-Nadjafi, J."https://zbmath.org/authors/?q=ai:saberi-nadjafi.jafar"Saberi Nik, H."https://zbmath.org/authors/?q=ai:nik.h-saberi|saberi-nik.hassanSummary: This paper is concerned with the chaos control of two autonomous chaotic and hyper-chaotic systems. First, based on the Pontryagin minimum principle (PMP), an optimal control technique is presented. Next, we proposed Lyapunov stability to control of the autonomous chaotic and hyper-chaotic systems with unknown parameters by a feedback control approach. Matlab bvp4c and ode45 have been used for solving the autonomous chaotic systems and the extreme conditions obtained from the PMP. Numerical simulations on the chaotic and hyper-chaotic systems are illustrated to show the effectiveness of the analytical results.Pullback attractors for strongly damped wave equation with delays.https://zbmath.org/1449.351072021-01-08T12:24:00+00:00"Xu, Guigui"https://zbmath.org/authors/?q=ai:xu.guigui"Wang, Libo"https://zbmath.org/authors/?q=ai:wang.libo"Lin, Guoguang"https://zbmath.org/authors/?q=ai:lin.guoguangSummary: In this paper, we study the existence of the pullback attractors for strongly damped wave equation with delay. By means of constructing the energy functional and combining with the idea of contractive function method, we verify the compactness for the process \(\{U (t,\tau)\}_{t \ge \tau}\) in \(C_{V,H}\) generated by the strongly damped wave equation with delay, and then we obtain the existence of the pullback attractors in \(C_{V,H}\) for the process \(\{U (t,\tau)\}_{t \ge \tau}\).Almost uniform and strong convergences in ergodic theorems for symmetric spaces.https://zbmath.org/1449.470242021-01-08T12:24:00+00:00"Chilin, V."https://zbmath.org/authors/?q=ai:chilin.vladimir-ivanovich|chilin.vladmir"Litvinov, S."https://zbmath.org/authors/?q=ai:litvinov.s-v|litvinov.semyon-n|litvinov.sergej|litvinov.s-a|litvinov.sergeyA space $X\subset L^0_\nu$ is fully symmetric on $((0,\infty),\nu)$ if $f\in X$, $g\in L^0_\nu$, and the decreasing rearrangement of $f$ dominates that of $g: g^\ast\leq f^\ast$ pointwise (resp., $\int_0^s g^\ast (t)\, dt\leq \int_0^s f^\ast(t)\, dt$ for all $s>0$) implies that $g\in X$ and $\Vert g\Vert_X\leq \Vert f\Vert_X$.
The first main result extends the Dunford-Schwartz pointwise ergodic theorem in characterizing $\mathcal{R}_\mu=\{f\in L^1+L^\infty: \forall\lambda>0,\, \mu(\vert f\vert >\lambda)<\infty\}$ as a space on which pointwise ergodic limits converge uniformly. To be precise, let $(\Omega,\mathcal{A},\mu)$ be a measure space and $X$ a fully symmetric space on $(\Omega,\mathcal{A},\mu)$ such that the constant $1\notin X$. If $T\in DS$ (that is, $T$ is bounded on both $L^1$ and $L^\infty$) and $f\in X$, then the averages $M_n(T)(f)=\frac{1}{n}\sum_{k=0}^{n-1} T^k(f)$ converge a.u. to some $\hat{f}\in X$. In particular, $M_n(T)(f) =\frac{1}{n}\sum_{k=0}^{n-1} T^k (f)\, \to\hat{f}\in \mathcal{R}_\mu$ a.u. when $f\in \mathcal{R}_\mu$.
The proofs involve a reduction to the $L^1$ case and the Hopf maximal theorem ($\int_{M(T)^\ast(f)>0} f\, d\mu>0$), where $M(T)^\ast(f)(x)=\sup \vert M_n(T)(f)(x)\vert $, and the weak type inequalities $$\mu(M(T)^\ast(\vert f\vert)>\lambda)\leq \bigl(2\frac{\Vert f\Vert_p}{\lambda}\bigr)^p,\quad\lambda>0\, .$$
Theorem 3.4 then states that, if $\mu$ is $\sigma$-finite, then $\mathcal{R}_\mu$ is the largest subspace of $L^1+L^\infty$ on which convergence is almost uniform, that is, if $f\in (L^1+L^\infty)\setminus \mathcal{R}_\mu$, then there is a $T\in DS$ such that the sequence $M_n(T)(f)$ fails to converge almost everywhere. In fact, the maximality of $\mathcal{R}_\mu$ for a subspace $X$ is equivalent to constants not belonging to $X$. Orlicz spaces are used to illustrate the condition that constants are not members. Strong convergence of Cesàro means is also discussed in the context of characterizing validity of mean ergodicity for fully symmetric spaces.
Reviewer: Joseph Lakey (Las Cruces)The strong \(G\)-shadowing property of the inverse limit spaces and the product spaces of group action.https://zbmath.org/1449.370162021-01-08T12:24:00+00:00"Ji, Zhanjiang"https://zbmath.org/authors/?q=ai:ji.zhanjiang"Zhang, Gengrong"https://zbmath.org/authors/?q=ai:zhang.gengrong"Tu, Jingxian"https://zbmath.org/authors/?q=ai:tu.jingxianSummary: The concept of the strong \(G\)-shadowing property is given in the metric spaces under the action of topological group. Then the dynamical properties of the strong \(G\)-shadowing property in the inverse limit spaces and the product spaces under the action of topological group are studied. The following conclusions are obtained. Let the system \( ({X_f}, \bar G, \bar d, \sigma)\) be the inverse limit spaces of the system \( (X, G, D, f)\). Then \(f\) has the \(G\)-shadowing property if and only if \(\sigma\) has the \({\bar G}\)-shadowing property. The product map \({f_1} \times {f_2}\) has the strong \(G\)-shadowing property if and only if the map \({f_1}\) has the strong \({G_1}\)-shadowing property and the map \({f_2}\) has the strong \({G_2}\)-shadowing property. These results enrich the theory of strong \(G\)-shadowing property in the inverse limit spaces and the product spaces under the action of topological group.Periodic solutions for seasonally forced SIRS model with pulse vaccination.https://zbmath.org/1449.341232021-01-08T12:24:00+00:00"Wang, Lin"https://zbmath.org/authors/?q=ai:wang.lin.2|wang.lin.1|wang.lin|wang.lin.3|wang.lin.4"Pang, Yanni"https://zbmath.org/authors/?q=ai:pang.yanni"Li, Wenjin"https://zbmath.org/authors/?q=ai:li.wenjinSummary: Using the coincidence degree theory of Gaines-Mawhin, we prove the existence of periodic solutions for seasonally forced SIRS models with pulse vaccination. The effects of different loss of immunity rates on the infectious disease models are compared with numerical simulations.Asymptotic dynamics of non-autonomous modified Swift-Hohenberg equations with multiplicative noise on unbounded domains.https://zbmath.org/1449.350742021-01-08T12:24:00+00:00"Mohamed, Yagoub Abaker"https://zbmath.org/authors/?q=ai:mohamed.yagoub-abaker"Liu, Tingting"https://zbmath.org/authors/?q=ai:liu.tingting"Ma, Qiaozhen"https://zbmath.org/authors/?q=ai:ma.qiaozhenSummary: We investigate the dynamical behavior of the stochastic non-autonomous modified Swift-Hohenberg equation with time-dependent forcing term and multiplicative noise on \(\mathbb{R}^2\). In order to overcome the difficulty that Sobolev embedding is not compact in the unbounded domain, we first define a continuous cocycle associated with the problem in \({L^2} (\mathbb{R}^2)\), and make some uniform estimates on the tails of solutions for large space variables. With the aid of uniform estimates of solution, we verify the pullback asymptotic compactness of the random dynamical system, and further obtain the existence of random attractors.Existence of periodic solutions of the first-Order non-Autonomous systems at resonance.https://zbmath.org/1449.341172021-01-08T12:24:00+00:00"Chen, Ruipeng"https://zbmath.org/authors/?q=ai:chen.ruipeng"Li, Xiaoya"https://zbmath.org/authors/?q=ai:li.xiaoyaSummary: This paper studies the existence of periodic solutions of the first-order non-autonomous systems at resonance, where nonlinear terms are periodic continuous functions. Several new existence results are established by means of Miranda's theorem and Schauder's fixed point theorem. Our results enrich and complement those available in the literature.Ergodicity of stochastic quasi-geostrophic flows equations with a degenerate multiplicative noise.https://zbmath.org/1449.370062021-01-08T12:24:00+00:00"Wang, Yan"https://zbmath.org/authors/?q=ai:wang.yan.5|wang.yan.1|wang.yan.3|wang.yan.2|wang.yan|wang.yan.4|wang.yan.6"Chen, Guanggan"https://zbmath.org/authors/?q=ai:chen.guanggan"Wang, Pin"https://zbmath.org/authors/?q=ai:wang.pinSummary: This paper is concerned with the stochastic quasi-geostrophic flows equations with a bounded multiplicative degenerate noise. It is a kind of important mathematical model in geophysical fluid mechanics and marine atmospheric science. Due to the perturbation of the bounded multiplicative degenerate noise, the corresponding Malliavin covariance operator is invertible, which causes that the strong Feller property of the probability transition semigroups can not be applied. In this paper, we use the asymptotically strong Feller property to overcome the difficulties caused by the degenerate noise, and finally obtain the ergodicity of the system.The research of periodic shadowing property and equicontinuity in the product \(G\)-space.https://zbmath.org/1449.370142021-01-08T12:24:00+00:00"Ji, Zhanjiang"https://zbmath.org/authors/?q=ai:ji.zhanjiang"Zhang, Gengrong"https://zbmath.org/authors/?q=ai:zhang.gengrong"Tu, Jingxian"https://zbmath.org/authors/?q=ai:tu.jingxianSummary: The concept of \(G\)-periodic shadowing property and \(G\)-equicontinuity is introduced in the product space under the action of topological group. By using the property of the product map, we study the relationship of these dynamical properties between product mapping \(f \times g\) and sub mapping \(f, g\). We obtain the following conclusions. (1) The product map \(f \times g\) has the \(G\)-periodic shadowing property if and only if the map \(f\) has the \({G_1}\)-periodic shadowing property and the map \(g\) has the \({G_2}\)-periodic shadowing property. (2) The product map \(f \times g\) has the \(G\)-equicontinuity if and only if the map \(f\) has the \({G_1}\)-equicontinuity and the map \(g\) has the \({G_2}\)-equicontinuity. These results enrich the theory of \(G\)-periodic shadowing property and \(G\)-equicontinuity in the product space under the action of topological group.Approximation of centre manifolds for multiplicative noise driven stochastic dynamical systems.https://zbmath.org/1449.370512021-01-08T12:24:00+00:00"Li, Qin"https://zbmath.org/authors/?q=ai:li.qin"Chen, Guanggan"https://zbmath.org/authors/?q=ai:chen.guanggan"Yang, Min"https://zbmath.org/authors/?q=ai:yang.minSummary: In this paper, we study the Wong-Zakai type approximation of the centre manifold for a class of stochastic evolution equations driven by multiplicative noise. Based on the convergence of solutions on invariant manifolds, the centre manifold of a stochastic system with smooth noise is used to approximate the centre manifold of the original system, so that the dynamic behavior of the original stochastic system is more clear.Dyons of unit topological charges in gauged Skyrme model.https://zbmath.org/1449.353702021-01-08T12:24:00+00:00"Wu, Zhonglin"https://zbmath.org/authors/?q=ai:wu.zhonglin"Li, Dongya"https://zbmath.org/authors/?q=ai:li.dongyaSummary: Dyons are an important family of topological solitons carrying both electric and magnetic charges and the presence of a nontrivial temporal component of the gauge field essential for the existence of electricity often makes the analysis of the underlying nonlinear equations rather challenging since the governing action functional assumes an indefinite form. In this work, by developing a constrained variational technique, we establish an existence theorem for the dyon solitons in a Skyrme model coupled with \(SO (3)\)-gauge fields. These solutions carry unit monopole and Skyrme charges.Melnikov method of impulsive system and its application to chaos prediction.https://zbmath.org/1449.340492021-01-08T12:24:00+00:00"Niu, Yujun"https://zbmath.org/authors/?q=ai:niu.yujun"Hu, Shuangnian"https://zbmath.org/authors/?q=ai:hu.shuangnianSummary: A necessary condition of chaos appearance in impulsive systems is derived. The Duffing system with impulsive signals is employed to show the efficiency of this method.The long-term dynamic behavior of a class of nonlinear evolution equations.https://zbmath.org/1449.350912021-01-08T12:24:00+00:00"Gao, Qingpei"https://zbmath.org/authors/?q=ai:gao.qingpei"Chai, Yuzhen"https://zbmath.org/authors/?q=ai:chai.yuzhenSummary: The existence of bounded absorption set for coupled beam equations is obtained by using Galerkin method combined with a priori estimate and some inequality techniques. lt is proved that the solution semigroup \(S (t)\) is asymptotically compact. The global attractor of the equation in space \(H_0^2 (\Omega) \times {L^2} (\Omega) \times {L^2} (\Omega)\) is obtained.The distribution of rational numbers and ergodic theory.https://zbmath.org/1449.370012021-01-08T12:24:00+00:00"Boca, Florin P."https://zbmath.org/authors/?q=ai:boca.florin-petreSummary: Rational numbers, or equivalently roots of unity on the unit circle, are not randomly distributed when they are enumerated by the size of denominators (Farey), or by the sum of digits in their continued fraction expansion (Stern-Brocot). There are four types of measure-preserving transformations (Gauss, Farey, BCZ, Newman), with very different ergodic behaviour, that play a role in gathering information about this distribution. Some of their ergodic properties and applications will be surveyed in this paper.Existence, uniqueness and global asymptotic stability for a class of complex-valued neutral-type neural networks with time delays.https://zbmath.org/1449.370222021-01-08T12:24:00+00:00"Tan, Manchun"https://zbmath.org/authors/?q=ai:tan.manchun"Xu, Desheng"https://zbmath.org/authors/?q=ai:xu.deshengSummary: This paper explores the problem of delay-independent and delay-dependent stability for a class of complex-valued neutral-type neural networks with time delays. Aiming at the neutral-type neural networks, an appropriate function is constructed to derive the existence of equilibrium point. On the basis of homeomorphism theory, Lyapunov functional method and linear matrix inequality techniques, several LMI-based sufficient conditions on the existence, uniqueness and global asymptotic stability of equilibrium point for complex-valued neutral-type neural networks are obtained. Finally, numerical examples are given to illustrate the feasibility and the effectiveness of the proposed theoretical results.Semi-uniform dynamics for non-autonomous Kuramoto-Sivashinsky equations.https://zbmath.org/1449.350972021-01-08T12:24:00+00:00"She, Lianbing"https://zbmath.org/authors/?q=ai:she.lianbing"Zhang, Wenlin"https://zbmath.org/authors/?q=ai:zhang.wenlin"Li, Yangrong"https://zbmath.org/authors/?q=ai:li.yangrongSummary: We introduce a concept of a semi-uniform attractor for an evolution process. A theorem for existence result of semi-uniform attractor is given in this paper. Although the invariance is not involved, but it can induce a semi-uniformly compact pullback attractor. Moreover, under some suitable assumptions, we show that the non-autonomous Kuramoto-Sivashinsky equation has a semi-uniform attractor and a semi-uniformly compact pullback attractor.Attractor family and dimension for a class of high-order nonlinear Kirchhoff equations.https://zbmath.org/1449.350952021-01-08T12:24:00+00:00"Lin, Guoguang"https://zbmath.org/authors/?q=ai:lin.guoguang"Li, Zhuoxi"https://zbmath.org/authors/?q=ai:li.zhuoxiSummary: The initial boundary value problem for a class of high-order Kirchhoff equations with nonlinear nonlocal source terms and strong damping terms is studied. For the nonlinear nonlocal source term and the Kirchhoff stress term, the existence and uniqueness of the global solution of the equation are firstly proved by Galerkin finite element method and a prior estimate. Then the bounded absorption set is obtained by a prior estimate, so the global attractor family of high-order nonlinear Kirchhoff equation is obtained. By linearizing the equation and proving the Frechet differentiability of the solution semigroup, we further prove the decay of the volume element of the linearization problem. Finally, the Hausdorff dimension and fractal dimension of the global attractor family are proved to be finite.A review on stochastic multi-symplectic methods for stochastic Maxwell equations.https://zbmath.org/1449.601112021-01-08T12:24:00+00:00"Zhang, Liying"https://zbmath.org/authors/?q=ai:zhang.liying"Chen, Chuchu"https://zbmath.org/authors/?q=ai:chen.chuchu"Hong, Jialin"https://zbmath.org/authors/?q=ai:hong.jialin"Ji, Lihai"https://zbmath.org/authors/?q=ai:ji.lihaiSummary: Stochastic multi-symplectic methods are a class of numerical methods preserving the discrete stochastic multi-symplectic conservation law. These methods have the remarkable superiority to conventional numerical methods when applied to stochastic Hamiltonian partial differential equations (PDEs), such as long-time behavior, geometric structure preserving, and physical properties preserving. Stochastic Maxwell equations driven by either additive noise or multiplicative noise are a system of stochastic Hamiltonian PDEs intrinsically, which play an important role in fields such as stochastic electromagnetism and statistical radiophysics. Thereby, the construction and the analysis of various numerical methods for stochastic Maxwell equations which inherit the stochastic multi-symplecticity, the evolution laws of energy and divergence of the original system are an important and promising subject. The first stochastic multi-symplectic method is designed and analyzed to stochastic Maxwell equations by \textit{J. Hong} et al. [J. Comput. Phys. 268, 255--268 (2014; Zbl 1349.65536)]. Subsequently, there have been developed various stochastic multi-symplectic methods to solve stochastic Maxwell equations. In this paper, we make a review on these stochastic multi-symplectic methods for solving stochastic Maxwell equations driven by a stochastic process. Meanwhile, the theoretical results of well-posedness and conservation laws of the stochastic Maxwell equations are included.The research of Lipschitz shadowing property and pointwise periodic shadowing property in nonautonomous dynamical systems.https://zbmath.org/1449.370202021-01-08T12:24:00+00:00"Ji, Zhanjiang"https://zbmath.org/authors/?q=ai:ji.zhanjiang"Qin, Guijiang"https://zbmath.org/authors/?q=ai:qin.guijiangSummary: According to the definition of Lipschitz shadowing property and pointwise periodic shadowing property in autonomous dynamical systems, we introduce the concept of Lipschitz shadowing property and pointwise periodic shadowing property in nonautonomous dynamical systems, and study their dynamic properties. The following results are obtained: (1) if \(F = \{f_i\}_{i=0}^\infty\) and \(G = \{g_i\}_{i=0}^\infty\) are topologically conjugate, then \(F\) has Lipschitz shadowing property if and only if \(G\) has Lipschitz shadowing property; (2) if \(F=\{f_i\}_{i=0}^\infty\) and \(G=\{g_i\}_{i=0}^\infty\) are topologically conjugate, then \(F\) has pointwise periodic shadowing property if and only if \(G\) has pointwise periodic shadowing property; (3) the product system \( (X\times Y, F\times G)\) has Lipschitz shadowing property if and only if \( (X,F)\) and \( (Y,G)\) have Lipschitz shadowing property. These conclusions make up for the lack of the theory of Lipschitz shadowing property and pointwise periodic shadowing property in autonomous dynamical systems.Strange nonchaotic attractors in quasiperiodically piecewise logistic system.https://zbmath.org/1449.370282021-01-08T12:24:00+00:00"Shen, Yunzhu"https://zbmath.org/authors/?q=ai:shen.yunzhu"Zhang, Fanhui"https://zbmath.org/authors/?q=ai:zhang.fanhui"Dong, Guangxia"https://zbmath.org/authors/?q=ai:dong.guangxiaSummary: It is not clear that the current understanding of the formation and mechanism of nonsmooth system for strange nonchaotic attractors (SNAs). We focus on the quasiperiodic driven Logistic system, aiming to analyze the various types of strange nonchaotic driven piecewise Logistic attractors by applying the largest Lyapunov exponents and phase-sensitive exponents in this system. It mainly includes the Heagy-Hammel routes, the type-I intermittency routes and the fractalization routes. The results of the study show that these types of strange nonchaotic attractors are abundant in nonsmooth systems and the results provide a theoretical basis for the study of strange nonchaotic attractors in nonsmooth systems.Generating new super dynamical systems in \( (2 + 1)\)-dimensional space.https://zbmath.org/1449.353892021-01-08T12:24:00+00:00"Wei, Hanyu"https://zbmath.org/authors/?q=ai:wei.hanyu"Zhang, Yan"https://zbmath.org/authors/?q=ai:zhang.yan.4|zhang.yan.3|zhang.yan.2"Xia, Tiecheng"https://zbmath.org/authors/?q=ai:xia.tie-chengSummary: In the article, we make use of the binomial-residue-representation (BRR) to generate \( (2+1)\)-dimensional super dynamical systems. By using these systems, a new \( (2+1)\)-dimensional super NLS-MKdV hierarchy is deduced, which can be reduced to super nonlinear Schrödinger equation. Especially, two main results are obtained which have important physical applications. One of them is a set of \( (2+1)\)-dimensional super integrable couplings, the other one is a \( (2+1)\)-dimensional diffusion equation. Furthermore, Super trace identity is used to furnish the super Hamiltonian structures for the new \( (2+1)\)-dimensional super integrable system.On Hausdorff dimension of random attractors for a stochastic wave equation.https://zbmath.org/1449.350902021-01-08T12:24:00+00:00"Ban, Ailing"https://zbmath.org/authors/?q=ai:ban.ailingSummary: This paper mainly examines the upper bound estimation of the Hausdorff dimension of random attractors for strongly damped stochastic wave equations with critical growth exponents. We prove that the obtained upper bound of the Hausdorff dimension of random attractor deceases as the strongly damped coefficient grows and is uniformly bounded when the strongly damped coefficient is sufficiently large.Global attractors for a class of coupled equations with memorizing terms.https://zbmath.org/1449.351122021-01-08T12:24:00+00:00"Zhang, Liyuan"https://zbmath.org/authors/?q=ai:zhang.liyuan"Ren, Yonghua"https://zbmath.org/authors/?q=ai:ren.yonghuaSummary: In this paper, we study the problem of global attractors for a class of coupled equations with memory term. By using Faedo-Galerkin method, we obtain the existence of the solutions of the equations. By proving the existence of the system absorption set and the asymptotic compactness of the semigroup of \(S (t)\), we further prove the existence of the global attractor for the equations.Dynamical behavior of a discrete Leslie-Gower-type food chain model.https://zbmath.org/1449.370592021-01-08T12:24:00+00:00"Su, Qianqian"https://zbmath.org/authors/?q=ai:su.qianqianSummary: In this paper, we study the dynamics behavior of a discrete Leslie-Gower three-dimensional food chain model. By using differential inequality, we get the conclusion that under some conditions, the species \({x_1}\) and \({x_3}\) are permanent and the species \({x_2}\) will be driven to extinction. Then, by constructing a suitable Lyapunov function, sufficient conditions are obtained to ensure the global attractivity of the system, which promotes the results of a literature.Research on complex dynamics of a new 4D hyperchaotic system.https://zbmath.org/1449.370672021-01-08T12:24:00+00:00"Hong, Lingling"https://zbmath.org/authors/?q=ai:hong.lingling"Yang, Qigui"https://zbmath.org/authors/?q=ai:yang.qiguiSummary: In this paper, based on the 3D Lorenz-like chaotic system, a linear feedback controller is designed and a new four-dimensional hyperchaos system with only two times nonlinear terms is proposed. This system has simple algebraic structure, but shows complex dynamic behavior. It is proved theoretically that it is not equivalent to hyperchaotic Li system. In order to study the complex dynamics of the system, the stability of the system at the hyperbolic and non-hyperbolic equilibrium points is discussed in detail, and the Hopf bifurcation is strictly analyzed. The approximate expression and stability of the periodic orbit generated by the Hopf bifurcation are obtained. Furthermore, the Lyapunov exponent spectrum, Poincaré map and bifurcation diagram of the system are obtained by numerical simulation with the help of modern mathematical software, and the existence of the hyperchaotic attractor is verified.Study on chaotic synchronization control of distributed generation system and inductive load network.https://zbmath.org/1449.931122021-01-08T12:24:00+00:00"Li, Jiankang"https://zbmath.org/authors/?q=ai:li.jiankang"Wei, Duqu"https://zbmath.org/authors/?q=ai:wei.duqu"Luo, Xiaoshu"https://zbmath.org/authors/?q=ai:luo.xiaoshu"Qin, Yinghua"https://zbmath.org/authors/?q=ai:qin.yinghuaSummary: In this paper, the permanent magnet synchronous generator (PMSG) is used as the node of distributed generation system, and the permanent magnet synchronous motor (PMSM) is used as the the node of inductive load network. Chaotic oscillation synchronization control between PMSG and PMSM is studied. First, the controller is designed by studying the method of modified function projection synchronization, so that PMSG and PMSM can realize the chaotic synchronization control. Moreover, by changing some parameter values, it is found that the synchronization time can be adjusted. Then the Lyapunov function theory is used to prove the feasibility of this scheme. Finally, the numerical simulation proves the correctness and validity of the theoretical analysis results, which are of great importance to the stable operation of power network.Horseshoe dynamics in Duffing oscillator with fractional damping and multi-frequency excitation.https://zbmath.org/1449.370272021-01-08T12:24:00+00:00"Priyatharsini, S. Valli"https://zbmath.org/authors/?q=ai:priyatharsini.s-valli"Meenakshi, M. V. Sethu"https://zbmath.org/authors/?q=ai:sethu-meenakshi.m-v"Chinnathambi, V."https://zbmath.org/authors/?q=ai:chinnathambi.v"Rajasekar, S."https://zbmath.org/authors/?q=ai:rajasekar.shanmuganathanSummary: The occurrence of horseshoe chaos in Duffing oscillator with fractional damping and multi-frequency excitation is analyzed by using analytical and numerical techniques. Applying Melnikov method, analytical threshold condition for the onset of horseshoe chaos is obtained. The effect of damping exponent and the number of periodic forces on the dynamics of the Duffing oscillator is also analyzed. Due to fractional damping and multi-frequency excitation, suppression of chaos and various nonlinear phenomena are predicted. Analytical predictions are demonstrated through numerical simulations.Some notes on the multiplicative order of \(\alpha+\alpha^{-1}\) in finite fields of characteristic two.https://zbmath.org/1449.111132021-01-08T12:24:00+00:00"Ugolini, Simone"https://zbmath.org/authors/?q=ai:ugolini.simoneThe author proves some results on the possible order of \(\alpha+\alpha^{-1}\) for \(\alpha\) a non-zero element of a finite field of even order. The results are based on his earlier paper, [Contemp. Math. 579, 187--204 (2012; Zbl 1302.37074)].
Reviewer: Arne Winterhof (Linz)Almost periodic solution of a nonautonomous continuous competitive system.https://zbmath.org/1449.341712021-01-08T12:24:00+00:00"Yu, Shengbin"https://zbmath.org/authors/?q=ai:yu.shengbinSummary: This paper considers the dynamic behaviors of a nonautonomous continuous competitive system with nonlinear inter-inhibition terms. By using the comparison theorem of differential equation and constructing the suitable Lyapunov function, sufficient conditions for the permanence, global attractivity and the existence of a unique positive almost periodic solution of the system are obtained. The results supplement some known ones. Numerical simulations show the feasibility of our results.Phase space homogenization and dynamic characteristics of unimodal chaotic system.https://zbmath.org/1449.370762021-01-08T12:24:00+00:00"Xu, Hui"https://zbmath.org/authors/?q=ai:xu.hui"Tong, Xiaojun"https://zbmath.org/authors/?q=ai:tong.xiaojun"Zhang, Miao"https://zbmath.org/authors/?q=ai:zhang.miao"Liu, Yang"https://zbmath.org/authors/?q=ai:liu.yang.17"Wang, Zhu"https://zbmath.org/authors/?q=ai:wang.zhuSummary: The cryptosystem constructed by classical one-dimensional chaotic mapping has some shortcomings in terms of security such as short-period orbits, small key space and inhomogeneous distribution of phase space. In order to solve the security problem of classical one-dimensional chaotic ciphers, a novel one-dimensional unimodal chaotic system and its improved composite form were proposed. A universal homogenization algorithm was used to transform the chaotic sequence into an equal probability distribution and a probability density mathematical proof was given. The dynamics and random characteristic indicators such as ergodicity, Lyapunov exponents, phase space, bifurcations, information entropy and approximate entropy were calculated and analyzed for the improved unimodal chaotic system. Through comparison with related researches, it can be seen that the improved unimodal chaotic system has stable Lyapunov exponents, extended phase space, uniform probability density and higher approximate entropy. Theoretical derivation and numerical calculation demonstrate that this scheme can meet the security attributes of nonlinear components in cryptosystem.Hidden attractors of a class of Van der Pol-Duffing oscillator.https://zbmath.org/1449.342002021-01-08T12:24:00+00:00"Nie, Jiasheng"https://zbmath.org/authors/?q=ai:nie.jiasheng"Li, Shumin"https://zbmath.org/authors/?q=ai:li.shuminSummary: In order to analyze and study the problem of hidden attractors in the Van der Pol-Duffing system, some new research results are obtained. The Routh-Hurwitz criterion, Hopf bifurcation theory, the harmonic linearization method and the analytical-numerical method are used to study the stability of the equilibrium points and the existence of hidden attractors in the system. There are hidden attractors in the system, and there are phenomena in which hidden attractors coexist with stable equilibrium points, stable periodic orbits, and chaotic attractors.Taming chaos by linear regulation with bound estimation.https://zbmath.org/1449.370732021-01-08T12:24:00+00:00"Wang, Jiqiang"https://zbmath.org/authors/?q=ai:wang.jiqiang"Chen, Weijian"https://zbmath.org/authors/?q=ai:chen.weijianSummary: Chaos control has become an important area of research and consequently many approaches have been proposed to control chaos. This paper proposes a linear regulation method. Different from the existing approaches is that it can provide region of attraction while estimating the bounding behaviour of the norm of the states. The proposed method also possesses design flexibility and can be easily used to cater for special requirement such that control signal should be generated via single input, single state, static feedback and so forth. The applications to the Tigan system, the Genesio chaotic system, the novel chaotic system, and the Lorenz chaotic system justify the above claims.Inverse spectral problem for a pair of self-adjoint Hankel operators.https://zbmath.org/1449.350322021-01-08T12:24:00+00:00"Gérard, Patrick"https://zbmath.org/authors/?q=ai:gerard.patrick"Grellier, Sandrine"https://zbmath.org/authors/?q=ai:grellier.sandrineSummary: We give a precise inverse spectral result for compact self-adjoint Hankel operators. From Megretskii-Peller-Treil, a necessary and sufficient condition on a sequence of non zero real numbers, finite or infinite but tending to zero, to be a sequence of eigenvalues of some self-adjoint and compact Hankel operator is that the multiplicity of an eigenvalue \(\lambda\) should differ from the multiplicity of \(-\lambda\) at most by one. Under this condition, we describe precisely the set of symbols for which the Hankel operator has a given sequence of eigenvalues. This theorem is a consequence of a general inverse spectral result that we proved for non-necessarily self-adjoint Hankel operators. As a by-product, we show how we recover the Megretskii-Peller-Treil condition.
For the entire collection see [Zbl 1404.42002].A fixed point theorem, intermediate value theorem, and nested interval property.https://zbmath.org/1449.550012021-01-08T12:24:00+00:00"Wu, Z."https://zbmath.org/authors/?q=ai:wu.zhiwei|wu.zhaofu|wu.zhiliang|wu.zhijan|wu.zhisong|wu.zhicheng|wu.zhenlong|wu.zhigen|wu.zhaocong|wu.zhoqun|wu.zhehui|wu.zhiqiang|wu.zhaohua|wu.zuyu|wu.zhilu|wu.zuping|wu.zhengjia|wu.zhao|wu.zhengwei|wu.zhishen|wu.zhonghuai|wu.zhengguang|wu.zhaojing|wu.zhenbin|wu.zhiqing|wu.zhuozhuo|wu.zikai|wu.zaixin|wu.zhonglin|wu.zongyu|wu.zhaoxia|wu.zhenguan|wu.zaide|wu.zongze|wu.zhworen|wu.zhangming|wu.ziheng|wu.zhengxiang|wu.zhide|wu.zhongbo|wu.zhanmin|wu.zhaojin|wu.ziwei|wu.zhengpeng|wu.zhongfu|wu.zhirong|wu.zhensen|wu.zhiang|wu.zigao|wu.zhidan|wu.zhonghua|wu.zezhong|wu.zhengwang|wu.ziqin|wu.zehui|wu.zhuanbao|wu.zhenghao|wu.zhixin|wu.ziqiang|wu.zhongcheng|wu.zhengxian|wu.zehao|wu.zongxin|wu.zedong|wu.zhibo|wu.zhengxiao|wu.zhen|wu.zhifeng|wu.zhenwei|wu.zhongtang|wu.zhenggang|wu.zuoren|wu.zhiqiao|wu.zan|wu.zhongtao|wu.zhangzhi|wu.zhemin|wu.zhuwu|wu.zhuangzhi|wu.zhouxiong|wu.zhilei|wu.zhuo|wu.zhengming|wu.zhongke|wu.zhibin|wu.zengbao|wu.zhenkui|wu.zhiquan.1|wu.zhengfei|wu.zizhao|wu.zhanggui|wu.zehu|wu.zongxian|wu.zhihui|wu.zhengrong|wu.zhengren|wu.zongqi|wu.zhengchang|wu.zhizhang|wu.zhenghua|wu.ziku|wu.zhenxiang|wu.zhixiang|wu.zemin|wu.zichen|wu.zhengmao|wu.zijun|wu.zhigang|wu.zhiming|wu.zhipeng|wu.zhimei|wu.zhaoqi|wu.zhixue|wu.zhongze|wu.zhijing|wu.zhenfeng|wu.ziongjian|wu.zhongxiang|wu.zhengzhong|wu.zongliang|wu.zhongchen|wu.zhi|wu.zhang|wu.zhongmin|wu.ziji|wu.zidian|wu.zhilin|wu.zhuang|wu.zhitao|wu.zhenxing|wu.zhuoqun|wu.zhanchun|wu.zhihua|wu.zuobing|wu.zhaoji|wu.zhihai|wu.zhenchao|wu.zhongxi|wu.zhizheng|wu.zhailian|wu.zongmin|wu.zhendong|wu.zili|wu.zengmao|wu.zikaio|wu.zhengtian|wu.zhan|wu.zhouhu|wu.zhenqiang|wu.zhengde|wu.zumin|wu.zheqian|wu.zhe|wu.zihua|wu.zhenming|wu.zhibiao|wu.zuohui|wu.zhenping|wu.zhenke|wu.zhong|wu.zhenglong|wu.zhonghai|wu.zhongming|wu.zizhen|wu.zhijian|wu.zonghua|wu.zhanji|wu.zhaoping|wu.zhansong|wu.zhuzhu|wu.zijing|wu.zhiqin|wu.zaigui|wu.zebin|wu.zheng|wu.zhongdao|wu.zhende|wu.zonglin|wu.zhangjun|wu.zhongchao|wu.zhiren|wu.zhengsheng|wu.zhengda|wu.zhengping|wu.zhifang|wu.zi|wu.zhimin|wu.zhaohui|wu.zhou|wu.zhaotong|wu.zhenhua|wu.zhansheng|wu.zijuan|wu.zhijin|wu.zhenhui|wu.zhaorong|wu.zhiyou|wu.zhaojun|wu.zefang|wu.zhengjiang|wu.zhixiong|wu.ziming|wu.zhixi|wu.zhuoren|wu.zhengang|wu.zhongle|wu.zongtao|wu.zhengguo|wu.zhiquan|wu.zhongshan|wu.zhangqing|wu.zhongqiang|wu.zhihong|wu.zhongru|wu.zhiping|wu.zhongan|wu.zixing|wu.zhongshiang|wu.zhijun.1|wu.ziniu|wu.zongsheng|wu.zhuanhao|wu.zhihuan|wu.zhidong|wu.zeping|wu.ze|wu.zhongwei|wu.zhouting|wu.zhenzhi|wu.zhihao|wu.zhengze|wu.zhaoming|wu.zijie|wu.zhengxing|wu.zhentao|wu.zhengyao|wu.zhiyu|wu.ziyan|wu.zhaoyang|wu.zhaoyan|wu.zhiyuan|wu.zeyan|wu.zunyou|wu.zhenning|wu.zhenying|wu.zhengyun|wu.zhinan|wu.zhengyu|wu.zhaoying|wu.zheyi|wu.zhenye|wu.zeyu|wu.zheyang|wu.zenan|wu.zhengyang|wu.zhenyang|wu.zhengyi|wu.zhenya|wu.ziyu|wu.zhenyong|wu.zhiyong|wu.zhenyu|wu.zhaoyu|wu.zhanyong|wu.zeyang|wu.zhenqun|wu.zhijuanSummary: For a continuous function \(f : [a,b] \to\mathbb{R}\), we prove that \(f\) has a fixed point if and only if the intervals \([a_0,b_0]= [a,b]\) and \([a_n,b_n] = [a_{n-1},b_{n-1}]\cap f([a_{n-1},b_{n-1}])\; (n = 1,2,\ldots)\) are all nonempty. More equivalent statements for the existence of fixed points of \(f\) have also been obtained and used to derive the intermediate value theorem and the nested interval property.Design and synchronization control of a new chaotic circuit.https://zbmath.org/1449.940862021-01-08T12:24:00+00:00"Wei, Qiang"https://zbmath.org/authors/?q=ai:wei.qiang"Bai, Yulong"https://zbmath.org/authors/?q=ai:bai.yulong"Duan, Jikai"https://zbmath.org/authors/?q=ai:duan.jikai"Chang, Mingheng"https://zbmath.org/authors/?q=ai:chang.mingheng"Fan, Manhong"https://zbmath.org/authors/?q=ai:fan.manhongSummary: A new three-dimensional autonomous chaotic system was presented, which consists of four parameter constants and three nonlinear terms. Through theoretical analysis and numerical calculation, the basic dynamic characteristics of the system were analyzed and studied, such as its Lyapunov exponent spectrum, Poincaré cross section and bifurcation diagram, etc. The analog circuit of the chaotic system was designed and simulated on Multisim and the simulation results showed that the chaotic system was not topologically equivalent to the previous chaotic systems. Compared with other chaotic systems, the phase diagram of each phase of the new system was asymmetric, and the motion characteristics were also more complex and disordered. The hardware circuit of the system was set up and the phase diagrams of each phase were obtained, being completely consistent with the theoretical analysis. The synchronization control of two chaotic circuit systems was realized by coupling the feedback synchronization method, thus laying a foundation for the application of the chaotic system.Entropy of infinite systems and transformations.https://zbmath.org/1449.370052021-01-08T12:24:00+00:00"Amini, Massoud"https://zbmath.org/authors/?q=ai:amini.massoudSummary: The Kolmogorov-Sinai entropy is a far reaching dynamical generalization of Shannon entropy of information systems. This entropy works perfectly for probability measure preserving (p.m.p.) transformations. However, it is not useful when there is no finite invariant measure. There are certain successful extensions of the notion of entropy to infinite measure spaces, or transformations with infinite invariant measures. The three main extensions are Parry, Krengel, and Poisson entropies. In this survey, we shortly overview the history of entropy, discuss the pioneering notions of Shannon and later contributions of Kolmogorov and Sinai, and discuss in somewhat more details the extensions to infinite systems. We compare and contrast these entropies with each other and with the entropy on finite systems.Generalized algebraic completely integrable systems.https://zbmath.org/1449.700142021-01-08T12:24:00+00:00"Lesfari, Ahmed"https://zbmath.org/authors/?q=ai:lesfari.ahmedSummary: We tackle in this paper the study of generalized algebraic completely integrable systems. Some interesting cases of integrable systems appear as coverings of algebraic completely integrable systems. The manifolds invariant by the complex flows are coverings of abelian varieties and these systems are called algebraic completely integrable in the generalized sense. The later are completely integrable in the sense of Arnold-Liouville. We shall see how some algebraic completely integrable systems can be constructed from known algebraic completely integrable in the generalized sense. A large class of algebraic completely integrable systems in the generalized sense, are part of new algebraic completely integrable systems. We discuss some interesting and well known examples: a 4-dimensional algebraically integrable system in the generalized sense as part of a 5-dimensiunal algebraically integrable system, the Hénon-Heiles and a 5-dimensional system, the RDG potential and a 5-dimensional system, the Goryachev-Chaplygin top and a 7-dimensional system, the Lagrange top, the (generalized) Yang-Mills system and cyclic covering of abelian varieties.Dual combination function projective synchronization of chaotic systems with disturbances.https://zbmath.org/1449.370642021-01-08T12:24:00+00:00"Fang, Jie"https://zbmath.org/authors/?q=ai:fang.jie"Zhu, Fei"https://zbmath.org/authors/?q=ai:zhu.fei"Lou, Xinjie"https://zbmath.org/authors/?q=ai:lou.xinjie"Liu, Hua"https://zbmath.org/authors/?q=ai:liu.hua"Deng, Wei"https://zbmath.org/authors/?q=ai:deng.weiSummary: Based on dual synchronization and combined synchronization, the dual combination function projective synchronization of four drive chaotic systems and two response chaotic systems is researched in this paper. An adaptive feedback controller is theoretically designed based on Lyapunov stability theory, tracking control thought and adaptive control method, by which the state variables of the response systems are able to track the chaotic trajectory of the drive systems according to the desired function matrices and overcome the effects of the unknown disturbances. When synchronization is achieved, the drive and response systems of the two sets of synchronization systems are able to be combined arbitrarily, and thereby enhancing the flexibility of the synchronization system. MATLAB numerical simulation verifies the correctness and effectiveness of the theoretical analysis.Blaschke product with one Herman ring and two Siegel disks.https://zbmath.org/1449.370332021-01-08T12:24:00+00:00"Chu, Haifeng"https://zbmath.org/authors/?q=ai:chu.haifengSummary: Let \(\alpha\) be an irrational number of bounded type and \(\beta\) be of Brjuno type with \(\alpha \ne 1 - \beta\). Using quasiconformal surgery, we construct a degree 3-Blaschke product \({B_{\alpha, \beta}}\) such that the map \({B_{\alpha, \beta}}\) has one Herman ring and two Siegel disks of rotation number \({\alpha, \beta}\) and \(1 - \beta\), respectively.Random dynamics for stochastic delay lattice systems in \({X_\rho}\) space.https://zbmath.org/1449.370522021-01-08T12:24:00+00:00"Zhang, Yijin"https://zbmath.org/authors/?q=ai:zhang.yijinSummary: The dynamics of a class of stochastic lattice dynamical systems with time delay driven by additive white noise is studied. \({X_\rho}\) space is introduced, basic equalities, Young inequality, Gronwall inequality and Schwarz inequality are applied. The existence, uniqueness and continuous dependence on the initial data of solutions to the stochastic delay lattice equations with additive noise are presented. Then a continuous infinite dimensional random dynamical system generated by the solutions is obtained.Existence of exponential attractors for Boussinesq equations with memory.https://zbmath.org/1449.351012021-01-08T12:24:00+00:00"Wang, Meixia"https://zbmath.org/authors/?q=ai:wang.meixia"Ma, Qiaozhen"https://zbmath.org/authors/?q=ai:ma.qiaozhenSummary: We considered the long-time dynamical behavior of solutions of Boussinesq equations with memory. Firstly, the original equation was transformed into a dynamical system by introducing a new variable. Secondly, the compactness of corresponding solutions semigroups associated with the problem we studied was proved by using the technique of operator decomposition and the compactness theorem on historical spaces. Finally, the exponential attractor of the memory-type Boussinesq equation was obtained by combining the existence of the exponential attractor, and the finite fractal dimension of the global attractor of the problem was also obtained.Random attractors for Berger equation with white noise.https://zbmath.org/1449.351032021-01-08T12:24:00+00:00"Wang, Xuan"https://zbmath.org/authors/?q=ai:wang.xuan"Song, An"https://zbmath.org/authors/?q=ai:song.anSummary: We considered the random asymptotic behaviors of solutions for the Berger equation with white noise. By introducing the equivalent process of isomorphic mapping, and using asymptotic a priori estimation technique and operator decomposition method, we proved the existence of random attractor in \( ({H^2} (U) \bigcap H_0^1 (U)) \times {L^2} (U)\).Propagation of traveling wave solutions for nonlinear evolution equation through the implementation of the extended modified direct algebraic method.https://zbmath.org/1449.351482021-01-08T12:24:00+00:00"Yaro, David"https://zbmath.org/authors/?q=ai:yaro.david"Seadawy, Aly"https://zbmath.org/authors/?q=ai:seadawy.aly-r"Lu, Dian-chen"https://zbmath.org/authors/?q=ai:lu.dianchenSummary: In this work, different kinds of traveling wave solutions and uncategorized soliton wave solutions are obtained in a three dimensional (3-D) nonlinear evolution equations (NEEs) through the implementation of the modified extended direct algebraic method. Bright-singular and dark-singular combo solitons, Jacobi's elliptic functions, Weierstrass elliptic functions, constant wave solutions and so on are attained beside their existing conditions. Physical interpretation of the solutions to the 3-D modified KdV-Zakharov-Kuznetsov equation are also given.Poincáre recurrence theorem in regular uncertain dynamic system.https://zbmath.org/1449.370032021-01-08T12:24:00+00:00"Yao, Xiao"https://zbmath.org/authors/?q=ai:yao.xiao"Ke, Hua"https://zbmath.org/authors/?q=ai:ke.huaSummary: Poincáre recurrence theorem in an uncertain dynamic system is proved in the framework of uncertainty theory, which claims that almost every point of an uncertain event with positive uncertain measure will iterate back to the event for infinite times. This recurrence behaviour can be used to develop new results of uncertain variable in an uncertain dynamic system.Bifurcations of coupled nonlinear oscillators with similar kinematics.https://zbmath.org/1449.370262021-01-08T12:24:00+00:00"Nikitina, N. V."https://zbmath.org/authors/?q=ai:nikitina.nelly-vladimirovna"Talimonova, O. Yu."https://zbmath.org/authors/?q=ai:talimonova.o-yuSummary: The application of the principle of skew symmetry for nonlinear systems that represent a bunch of nonlinear Van der Pol oscillators is analyzed. A bunch of oscillators can (depending on the parameters) form systems of coupled regular limiting cycles and coupled attractors with chaotic or conditionally periodic winding of the trajectory. At a slight change in the parameters of oscillators, the scale of two limiting cycles changes. A strong change in the parameters and the coupling coefficient leads to the appearance of limiting cycles with chaotic winding of the trajectory. When considering three connected limiting cycles, one can reduce them to two ones with a periodic winding and one limiting cycle with a conditionally periodic winding. To clarify the nature of the winding of the trajectories, a topological analysis of the trajectory should be done. In this case, the equations in variations are constructed, and the characteristic indicators of solutions are found.The additivity of pointwise preimage entropy for selfmaps on nilmanifolds.https://zbmath.org/1449.370092021-01-08T12:24:00+00:00"Huang, Baojun"https://zbmath.org/authors/?q=ai:huang.baojunSummary: Pointwise preimage entropy is similar to topological entropy. However, in general, their properties are not completely coincident such as additivity of Cartesian product. In this paper, the additivity of pointwise preimage entropy of the torus maps under Cartesian product is extended to the case of the maps of compact nilmanifolds.Estimation entropy and \(\Delta \)-weakly mixing sets for free semigroup actions.https://zbmath.org/1449.370242021-01-08T12:24:00+00:00"Zhong, Xingfu"https://zbmath.org/authors/?q=ai:zhong.xingfuSummary: We introduce the notion of estimation entropy for semigroup action systems and give some properties. Furthermore, we investigate the concept of \(\Delta \)-weakly mixing sets for semigroup actions and give a characterization of \(\Delta \)-weakly mixing sets by \(\Delta \)-Xiong chaotic sets.The research of strong shadowing property and strong chain recurrent point set.https://zbmath.org/1449.370082021-01-08T12:24:00+00:00"Ji, Zhanjiang"https://zbmath.org/authors/?q=ai:ji.zhanjiang"Zhang, Gengrong"https://zbmath.org/authors/?q=ai:zhang.gengrong"Tu, Jingxian"https://zbmath.org/authors/?q=ai:tu.jingxianSummary: The dynamical properties of strong shadowing property and strong chain recurrent point set are studied in the compact metric spaces and some conclusions are obtained. (1) If the map \(f\) is topologically conjugate to the map \(g\), then \(f\) has the strong shadowing property if and only if \(g\) has the strong shadowing property. (2) The strong chain recurrent point set of the continuous map \(g\) is the image of the strong chain recurrent point set of the continuous map \(f\) under the topological conjugate map \(h\). (3) The strong chain recurrent point set of the the continuous map \({f^n}\) is a subset of the strong chain recurrent point set of the the continuous map \(f\). (4) The strong chain recurrent point set of the shift map \(\sigma\) is a subset of the inverse limit space of the continuous map \(f\) in its strong chain recurrent point set. These conclusions generalize and improve the results of strong shadowing property and strong chain recurrent point in the existing literature.The reducibility of a class of quasi-periodic nonlinear Hamiltonian systems.https://zbmath.org/1449.370402021-01-08T12:24:00+00:00"Li, Jia"https://zbmath.org/authors/?q=ai:li.jia.2|li.jia|li.jia.1|li.jia.3"Zhu, Chunpeng"https://zbmath.org/authors/?q=ai:zhu.chunpengSummary: In the paper, we considered the reducibility of a class of quasi-periodic nonlinear Hamiltonian systems with multiple eigenvalues. Under suitable hypothesis of non-resonance conditions and non-degeneracy conditions, by a quasi-periodic symplectic transformation, we proved that for most sufficiently small \(\varepsilon\), the Hamiltonian system is reducible.The research of asymptotically periodic point and pointwise shadowing property under strongly uniform convergence.https://zbmath.org/1449.370192021-01-08T12:24:00+00:00"Ji, Zhanjiang"https://zbmath.org/authors/?q=ai:ji.zhanjiang"Zhang, Gengrong"https://zbmath.org/authors/?q=ai:zhang.gengrong"Tu, Jingxian"https://zbmath.org/authors/?q=ai:tu.jingxianSummary: The relationship of the asymptotically periodic property and pointwise shadowing property between the sequence map and the limit map under strongly uniform convergence was studied. By using the properties of the strong uniform convergence and equicontinuity, the following results were obtained: (1)Suppose that the sequence map \(\{{f_n}\}\) converged strongly uniformly to the equicontinuous map \(f\) and the sequence of points \(\{{x_k}\}\) be the asymptotically periodic point of every map \({f_n}\), if \(\underset{k \to \infty}{\lim}{x_k} = x\), then the point \(x\) was the asymptotically periodic point of the map \(f\); (2) If the sequence map \(\{{f_n}\}\) converged strongly uniformly to the equicontinuous map \(f\), then limsup\(A\)Per\( ({f_n}) \subset A{\mathrm{Per}} (f)\); (3) Suppose that the sequence map \(\{{f_n}\}\) converged strongly uniformly to the map \(f\), if \({f_n}\) had the fine pointwise shadowing property, then \(f\) had pointwise shadowing property.The assymmetric Lorenz attractor as an example of a new pseudohyperbolic attractor of three-dimensional systems.https://zbmath.org/1449.340452021-01-08T12:24:00+00:00"Kazakov, A. O."https://zbmath.org/authors/?q=ai:kazakov.alexey-o"Kozlov, A. D."https://zbmath.org/authors/?q=ai:kozlov.a-dThe authors consider a class of three-dimensional systems of the form \[\begin{cases} \dot{x}=y+g_1(x,y,z),\\ \dot{y}=z+g_2(x,y,z),\\ \dot{z}=Ax+By+Cz+g_3(x,y,z), \end{cases}\] where \(A\), \(B\) and \(C\) are parameters of the system; \(g_i\), \(i=1,2,3\) are nonlinear terms, satisfying the relations \(g_i(0,0,0)=\dfrac{\partial g_i}{\partial x}(0,0,0)=\dfrac{\partial g_i}{\partial y}(0,0,0)=\dfrac{\partial g_i}{\partial z}(0,0,0)\), \(i=1,2,3\). A new method for constructing three-dimensional systems with various, including pseudohyperbolic, attractors is presented. Using the proposed method, the authors construct an example of a three-dimensional system with a pseudohyperbolic attractor of a new type, an asymmetric Lorenz attractor. It differs from the classical one in the absence of symmetry in any of the coordinates. To search for the asymmetric Lorenz attractor in the class of three-dimensional systems, the authors use the saddle chart method, and to check its pseudohyperbolicity, they use the LMP-method.
Reviewer: Artyom Andronov (Saransk)Stability analysis of the zero solution for time-periodic linear system with small perturbations.https://zbmath.org/1449.341952021-01-08T12:24:00+00:00"Li, Zhuo"https://zbmath.org/authors/?q=ai:li.zhuo"Li, Xia"https://zbmath.org/authors/?q=ai:li.xiaSummary: In this paper, we study the stability of the zero solution for time-periodic linear perturbed systems. After investigating the properties of periodic linear systems, we apply the method of finding nonsingular and differentiable periodic matrices to the study of the properties of linear systems with constant coefficients. Then we extend the zero solution's stability of the constant coefficient linear system with small perturbation to the periodic linear system with small perturbations.Representation of spaciously situated perfect attractors of diffeomorphisms by geodesic laminations.https://zbmath.org/1449.370212021-01-08T12:24:00+00:00"Grines, V. Z."https://zbmath.org/authors/?q=ai:grines.vyacheslav-z"Kurenkov, E. D."https://zbmath.org/authors/?q=ai:kurenkov.evgenii-dmitrievichFor each one-dimensional spatially located perfect attractor \(\Lambda\) of the \(A\)-diffeomorphism \(f\) a unique geodesic lamination \(\mathcal L\), the complement of which consists of a finite number of areas homeomorphic to the disk, is constructed. In the case when the attractor does not contain connectives of degree 2, the existence of a homeomorphism of a surface homotopic to the identical one that maps unstable manifolds of points of the attractor \(\Lambda\) to the layers of the constructed geodesic lamination \(\mathcal L\) is established. Moreover, the authors prove that if the nonwandering sets of homotopic \(A\)-diffeomorphisms \(f\) and \(f'\) have spatially located perfect attractors \(\Lambda\) and \(\Lambda'\), then the geodesic laminations \(\mathcal L\) and \(\mathcal L'\) corresponding to these attractors coincide.
Reviewer: Artyom Andronov (Saransk)Study on geometric evolution properties of planar closed curve flow.https://zbmath.org/1449.530522021-01-08T12:24:00+00:00"Ding, Danping"https://zbmath.org/authors/?q=ai:ding.danping"Cheng, Yongting"https://zbmath.org/authors/?q=ai:cheng.yongtingSummary: In this paper, the governing equation of curve geometric variables is used to discuss the geometric characteristics of plane curve flow, and the description and characterization of the evolution properties of correlation variables are obtained. The global evolution law and characteristics of planar closed curve flow are characterized by the distance from the outer point of the curve to the curve, and the global evolution speed of plane simple closed curve is found to be limited.Existence and uniqueness of mild solutions for nonlinear fractional integro-differential evolution equations.https://zbmath.org/1449.370492021-01-08T12:24:00+00:00"Hou, Mimi"https://zbmath.org/authors/?q=ai:hou.mimi"Xi, Xuanxuan"https://zbmath.org/authors/?q=ai:xi.xuanxuan"Zhou, Xianfeng"https://zbmath.org/authors/?q=ai:zhou.xianfengSummary: In this paper, we study a class of nonlinear fractional integro-differential evolution equations in a Banach space \(X\). We use the fractional power of operators and the theory of analytic semigroups to prove the existence and uniqueness of the solution for the given problem. Furthermore, we give the Hölder continuity of the obtained mild solution.A hyper-chaotic system with four-wing attractor.https://zbmath.org/1449.370292021-01-08T12:24:00+00:00"Zhang, Li"https://zbmath.org/authors/?q=ai:zhang.li.3|zhang.li.10|zhang.li.1|zhang.li.5|zhang.li.2|zhang.li.7|zhang.li|zhang.li.6|zhang.li.8|zhang.li.4|zhang.li.12|zhang.li.9|zhang.li.11"Xie, Yue"https://zbmath.org/authors/?q=ai:xie.yue"An, Xinlei"https://zbmath.org/authors/?q=ai:an.xinleiSummary: A nonlinear system with four-wing hyperchaotic attractors was built, and its hyperchaotic characteristics by the analysis of Poincaré sections, the power spectrum and Lyapunov exponents were verified in this paper. We studied the stability of equilibrium points of the system, and found that the system has an unstable saddle point, six unstable focuses and two stable focuses. The impact of parameter on system dynamics was discussed. Numerical simulation results show that there is a typical double period bifurcation phenomenon, which results in chaos eventually. The proposed hyper-chaotic system with four wings has potential applications in the fields of secret communication and information safety.Asymptotic average and Lipschitz shadowing property of the product map under group action.https://zbmath.org/1449.370152021-01-08T12:24:00+00:00"Ji, Zhanjiang"https://zbmath.org/authors/?q=ai:ji.zhanjiang"Zhang, Gengrong"https://zbmath.org/authors/?q=ai:zhang.gengrong"Tu, Jingxian"https://zbmath.org/authors/?q=ai:tu.jingxianSummary: The shadowing property is significant both in theory and application. In this paper, we introduce the concept of \(G\)-asymptotic average shadowing property and \(G\) Lipschitz shadowing property in the product space under the action of a topological group. By means of properties of the product map and zero density sets, we study the relationship of these shadowing propertes between product mapping \(f \times g\) and sub mapping \(f, g\). We obtain the following result: (1) the product map \(f \times g\) has the \(G\)-asymptotic average shadowing property if and only if the map \(f\) has the \({G_1}\)-asymptotic average shadowing property and the map \(g\) has the \({G_2}\)-asymptotic average shadowing property; (2) the product map \(f \times g\) has the \(G\)-Lipschitz shadowing property if and only if the map \(f\) has the \({G_1}\)-Lipschitz shadowing property and the map \(g\) has the \({G_2}\)-Lipschitz shadowing property. These results enrich the theory of asymptotic average shadowing property and Lipschitz shadowing property in the product space under the action of topological group.Existence of periodic solutions for a class of second order Hamiltonian systems with perturbation.https://zbmath.org/1449.370442021-01-08T12:24:00+00:00"Zheng, Lingling"https://zbmath.org/authors/?q=ai:zheng.lingling"Chen, Xingfan"https://zbmath.org/authors/?q=ai:chen.xingfanSummary: The existence result of periodic solution for a class of second order Hamiltonian systems with perturbation is obtained by the saddle point theorem. Previous research results are improved and an example is given.The integrability of the KdV-shallow water waves equation.https://zbmath.org/1449.353802021-01-08T12:24:00+00:00"Hao, Xiaohong"https://zbmath.org/authors/?q=ai:hao.xiaohong"Cheng, Zhilong"https://zbmath.org/authors/?q=ai:cheng.zhilongSummary: In this paper, the binary Bell polynomials are used to construct bilinear forma, bilinear Bäcklund transformation, Lax pair of the KdV-shallow water waves equation. Through bilinear Bäcklund transformation, some soliton solutions are presented. Moreover, the infinite conservation laws are also derived by Bell polynomials. All conserved densities and fluxes are given with explicit recursion formulas.Noether's theorems based on El-Nabulsi extended exponentially quasi-fractional models in event space.https://zbmath.org/1449.370452021-01-08T12:24:00+00:00"Wang, Ze"https://zbmath.org/authors/?q=ai:wang.ze"Zhang, Yi"https://zbmath.org/authors/?q=ai:zhang.yi.1|zhang.yi.8|zhang.yi.11|zhang.yi.5|zhang.yi.7|zhang.yi.12|zhang.yi|zhang.yi.3|zhang.yi.10|zhang.yi.9|zhang.yi.2|zhang.yi.4Summary: In order to further study the dynamic behavior of non-conservative systems and reveal the relationship between the symmetries and conserved quantities of dynamic systems, we proposed and investigated the Noether theorems based on El-Nabulsi extended exponentially quasi fractional models in event space. Firstly, we put forward the El-Nabulsi quasi fractional variational problem based on the extended exponentially fractional integral, and established the differential equations of motion for holonomic systems and nonholonomic systems. Secondly, we provided the definition and criterion of Noether symmetric transformation and Noether quasi-symmetric transformation based on the invariance of the action functional under the infinitesimal transformations. Finally, we presented and proved the Noether theorems based on El-Nabulsi extended exponentially quasi fractional models in event space. Two examples were given to illustrate the application of the results.Duplex selections, equilibrium points, and viability tubes.https://zbmath.org/1449.340572021-01-08T12:24:00+00:00"Kánnai, Zoltán"https://zbmath.org/authors/?q=ai:kannai.zoltanSummary: Existence of viable trajectories to nonautonomous differential inclusions are proven for time-dependent viability tubes. In the convex case we prove a double-selection theorem and a new Scorza-Dragoni type lemma. Our result also provides a new and palpable proof for the equilibrium form of Kakutani's fixed point theorem.Traveling waves for a diffusive SIR-B epidemic model with multiple transmission pathways.https://zbmath.org/1449.350802021-01-08T12:24:00+00:00"Song, Haifeng"https://zbmath.org/authors/?q=ai:song.haifeng"Zhang, Yuxiang"https://zbmath.org/authors/?q=ai:zhang.yuxiangSummary: In this work, we consider a diffusive SIR-B epidemic model with multiple transmission pathways and saturating incidence rates. We first present the explicit formula of the basic reproduction number \(\mathcal{R}_0\). Then we show that if \(\mathcal{R}_0>1\), there exists a constant \(c^*>0\) such that the system admits traveling wave solutions connecting the disease-free equilibrium and endemic equilibrium with speed \(c\) if and only if \(c\geq c^*\). Since the system does not admit the comparison principle, we appeal to the standard Schauder's fixed point theorem to prove the existence of traveling waves. Moreover, a suitable Lyapunov function is constructed to prove the upward convergence of traveling waves.Threshold dynamics of discrete HIV virus model with therapy.https://zbmath.org/1449.370582021-01-08T12:24:00+00:00"Ma, Xia"https://zbmath.org/authors/?q=ai:ma.xia"Cao, Hui"https://zbmath.org/authors/?q=ai:cao.huiSummary: In the paper, we mainly investigate the permanence and the global stability of the discrete HIV model with therapy. By defining the basic reproduction number \({R_0}\), we conclude that the uninfected state \({P^0}\) is globally stable when \({R_0} < 1\), therefore the virus will be extinct. However, the virus will be persistent when \({R_0} > 1\). we also obtain that the infected state \(P^*\) is globally stable when \(1 < {R_0} < N\) by constructing Lyapunov function. The threshold dynamical behavior is in agreement with the continuous differential model.Stable manifolds for non-instantaneous impulsive nonautonomous differential equations.https://zbmath.org/1449.341282021-01-08T12:24:00+00:00"Li, Mengmeng"https://zbmath.org/authors/?q=ai:li.mengmeng"Wang, JinRong"https://zbmath.org/authors/?q=ai:wang.jinrong"O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donalSummary: In this paper, we study stable invariant manifolds for a class of nonautonomous non-instantaneous impulsive equations where the homogeneous part has a nonuniform exponential dichotomy. We establish a stable invariant manifold result for sufficiently small perturbations by constructing stable and unstable invariant manifolds and we also show that the stable invariant manifolds are of class \(C^{1}\) outside the jumping times using the continuous Fiber contraction principle technique.Remarks on criteria of ergodicity for birth-death processes.https://zbmath.org/1449.370022021-01-08T12:24:00+00:00"Wu, Bingyao"https://zbmath.org/authors/?q=ai:wu.bingyao"Wang, Jian"https://zbmath.org/authors/?q=ai:wang.jian.7|wang.jian.1|wang.jian.4|wang.jian.2|wang.jian.3|wang.jian.5|wang.jian.9Summary: In order to study the relations among various ergodic properties of birth-death processes on \(\textbf{Z}_+\), we establish sufficient conditions under which ergodic property directly yields exponential ergodicity, the existence of discrete spectral and strong ergodicity as well as other stronger properties. Some examples are given which satisfy the ergodicity but the sufficient conditions and conclusions are not valid.The topological pressure on an arbitrary topological space.https://zbmath.org/1449.370122021-01-08T12:24:00+00:00"Wang, Wei"https://zbmath.org/authors/?q=ai:wang.wei.30"Cao, Jie"https://zbmath.org/authors/?q=ai:cao.jieSummary: The purpose of this paper is to enrich the theory of topological pressure and to gain a wider range of topological structure. Using the compact method this paper deals with general topological space, and makes the spatial structure more concise. A new topological pressure is defined in new space. The properties of the new topological pressure are discussed and proved.Integrating factors and conserved quantities for Birkhoffian systems on time scales.https://zbmath.org/1449.370352021-01-08T12:24:00+00:00"Yang, Lixia"https://zbmath.org/authors/?q=ai:yang.lixia"Zhang, Yi"https://zbmath.org/authors/?q=ai:zhang.yi.8|zhang.yi.11|zhang.yi.10|zhang.yi.2|zhang.yi.9|zhang.yi.12|zhang.yi.5|zhang.yi.3|zhang.yi|zhang.yi.7|zhang.yi.4|zhang.yi.1Summary: This paper studied the conserved quantities of Birkhoffian systems on time scales. The method of integrating factors was proposed to find the conserved quantities of Birkhoffian systems on time scales. And the energy equation of Birkhoff's equations was established on time scales. The integrating factors and conservation theorems for Birkhoffian systems on time scales were investigated. The integrating factors and conservation theorems for Hamiltonian systems and Lagrangian systems on time scales were special cases of Birkhoffian systems on time scales. Finally, an example was given to illustrate the application of the results.The structural model of tense deformated condition of solids which take into account intermolecular interaction.https://zbmath.org/1449.740122021-01-08T12:24:00+00:00"Anisimov, V. N."https://zbmath.org/authors/?q=ai:anisimov.valerii-nikolaevich|anisimov.vladimir-nSummary: The structural model to describe tense deformated condition of solids from the position of intermolecular interaction is offered. Nonlinear correlations between strains and tensions in the elastic sphere with the help of this model are received.Synchronization of chaos in simultaneous time-frequency domain.https://zbmath.org/1449.370682021-01-08T12:24:00+00:00"Liu, Meng-Kun"https://zbmath.org/authors/?q=ai:liu.mengkun"Suh, C. Steve"https://zbmath.org/authors/?q=ai:suh.c-steveSummary: Synchronization of chaos presents many challenges for controller design. The novel notion of exerting concurrent control in the joint time-frequency domain is applied to formulate a chaos synchronization scheme that requires no linearization or heuristic trial-and-errors for nonlinear controller design. The concept is conceived through recognizing the basic attributes inherent of all chaotic systems, including the simultaneous deterioration of dynamics in both the time and frequency domains when bifurcates, nonstationarity, and sensitivity to initial conditions. Having its philosophical bases established in simultaneous time-frequency control, on-line system identification, and adaptive control, the chaos synchronization scheme incorporates multiresolution analysis, adaptive filters, and filtered-x Least Mean Square algorithm as its physical features. Without \textit{A priori} knowledge of the driven system parameters, synchronization is invariably achieved regardless of the initial and forcing conditions the response system is subjected to. In addition, driving and driven trajectories are seen robustly synchronized with negligible errors in spite of the infliction of high frequency noise.Inverse scattering by obstacles and Santalo's formula.https://zbmath.org/1449.370232021-01-08T12:24:00+00:00"Stoyanov, Luchezar"https://zbmath.org/authors/?q=ai:stoyanov.luchezar-nThe author considers a general inverse problem to obtain information about obstacles of scattering trajectories
either from measurements of traveling times or from the scattering length spectrum. The stability property of the trapping set is discussed by applying Santalo's formula. Some other applications of this formula to scattering problems are presented.
Reviewer: Angela Slavova (Sofia)Counting closed orbits for the Dyck shift.https://zbmath.org/1449.370172021-01-08T12:24:00+00:00"Alsharari, Fahad"https://zbmath.org/authors/?q=ai:alsharari.fahad"Noorani, Mohd Salmi Md."https://zbmath.org/authors/?q=ai:noorani.mohd-salmi-mohd"Akhadkulov, Habibulla"https://zbmath.org/authors/?q=ai:akhadkulov.habibullaSummary: The prime orbit theorem and Mertens' theorem are proved for a shift dynamical system of infinite type called the Dyck shift. Different and more direct methods are used in the proof without any complicated theoretical discussion.Forwards and pullback behaviour of a non-autonomous predator-prey system with the Beddington-DeAngelis functional response.https://zbmath.org/1449.341562021-01-08T12:24:00+00:00"Shen, Yixin"https://zbmath.org/authors/?q=ai:shen.yixin"Pu, Zhilin"https://zbmath.org/authors/?q=ai:pu.zhilin"Hu, Huashu"https://zbmath.org/authors/?q=ai:hu.huashuSummary: For the non-autonomous predator-prey system with the Beddington-DeAngelis response, in this paper, we use the sub-super solution, logistic function and comparison principle to explore the asymptotic behaviour of the solution. The time asymptotic behaviour as \(t \to \infty\) is considered, including forward permanence and predator extinction. Then, we explore the asymptotic behaviour as \(s \to -\infty\), including the existence of the pullback attractor and pullback permanence.Stochastic fractional non-autonomous Ginzburg-Landau equations with multiplicative noise in weighted space.https://zbmath.org/1449.354662021-01-08T12:24:00+00:00"Wang, Yunxiao"https://zbmath.org/authors/?q=ai:wang.yunxiao"Shu, Ji"https://zbmath.org/authors/?q=ai:shu.ji"Yang, Yuan"https://zbmath.org/authors/?q=ai:yang.yuan"Li, Qian"https://zbmath.org/authors/?q=ai:li.qian"Wang, Chunjiang"https://zbmath.org/authors/?q=ai:wang.chunjiangSummary: In this paper, we consider the asymptotic dynamic for random attractors of stochastic fractional non-autonomous Ginzburg-Landau equations with multiplicative noise in \(L_\rho^2 (\text bf{R}^n)\). Firstly, we transform the partial differential equation into the random equation that only induces the random parameters. Then, using estimates for far-field values of solutions and a cut-off technique, asymptotic compactness is proved. At last, the existence of a random attractor in \(L_\rho^2(\text bf{R}^n)\) for the random dynamical system is established.Complex evolution of a multi-particle system.https://zbmath.org/1449.370072021-01-08T12:24:00+00:00"Machado, J. A. Tenreiro"https://zbmath.org/authors/?q=ai:machado.jose-antonio-tenreiroSummary: This paper studies a discrete dynamical system of interacting particles that evolve by interacting among them. The computational model is an abstraction of the natural world, and real systems can range from the huge cosmological scale down to the scale of biological cell, or even molecules. Different conditions for the system evolution are tested. The emerging patterns are analysed by means of fractal dimension and entropy measures. It is observed that the population of particles evolves towards geometrical objects with a fractal nature. Moreover, the time signature of the entropy can be interpreted at the light of complex dynamical systems.The effective reducibility of a class of quasi-periodic linear Hamiltonian systems.https://zbmath.org/1449.370412021-01-08T12:24:00+00:00"Li, Jia"https://zbmath.org/authors/?q=ai:li.jia|li.jia.1|li.jia.2|li.jia.3"Zhu, Chunpeng"https://zbmath.org/authors/?q=ai:zhu.chunpengSummary: In this paper, we consider the effective reducibility of a class of quasi-periodic linear Hamiltonian systems with multiple eigenvalues. Under only non-resonance conditions but no non-degeneracy condition, by a quasi-periodic symplectic transformation, we prove that for all parameter \(\varepsilon\), the Hamiltonian system can be effective reducible.Reconstruction of chaotic dynamic systems using non-linear filters.https://zbmath.org/1449.370602021-01-08T12:24:00+00:00"Sánchez, Luis"https://zbmath.org/authors/?q=ai:sanchez.luis-m|sanchez.luis|sanchez.luis-f|sanchez.luis-gonzalez|sanchez.luis-angel|sanchez.luis-r"Infante, Saba"https://zbmath.org/authors/?q=ai:infante.sabaSummary: This article proposes a methodology based on sequential Monte Carlo techniques that permits state estimate of chaotic dynamic systems with Gaussian errors and non-linear dynamics in real time. Such systems arise naturally in many varied applications. We illustrate the methodology through the reconstruction of the states of the chaotic maps of Henon, Ikeda, Tinkerbell and Lorenz, using four different algorithms, namely, generic particle filter (GPF), particle filter with re-sampling (PFR), unscented Kalman filter (UKF) and an unscented particle filter (UPF). The performance of the filters was evaluated in terms of the empirical standard deviation and the computation times showing little variance among the estimated errors and a rapid execution of the algorithms.The semidirect sum of Lie algebras and its applications to C-KdV hierarchy.https://zbmath.org/1449.370482021-01-08T12:24:00+00:00"Dong, Xia"https://zbmath.org/authors/?q=ai:dong.xia"Xia, Tiecheng"https://zbmath.org/authors/?q=ai:xia.tie-cheng"Li, Desheng"https://zbmath.org/authors/?q=ai:li.deshengSummary: By use of the loop algebra \(\overset\simeq G\), integrable coupling of C-KdV hierarchy and its bi-Hamiltonian structures are obtained by \textit{G. Tu} scheme [J. Math. Phys. 30, No. 2, 330--338 (1989; Zbl 0678.70015)]
and the quadratic-form identity. The method can be used to produce the integrable coupling and its Hamiltonian structures to the other integrable systems.Parameter characteristics of the hidden attractor in an improved Sprott-A system.https://zbmath.org/1449.370662021-01-08T12:24:00+00:00"Hao, Jianhong"https://zbmath.org/authors/?q=ai:hao.jianhong"Liu, Bolun"https://zbmath.org/authors/?q=ai:liu.bolunSummary: In this paper, a nonlinear function is introduced into the system to obtain an improved Sprott-A model. By selecting the nonlinear function as the sine function, we can get the multi-scroll hidden attractor. The dynamic characteristics of the system are analyzed by calculating the phase orbit, time evolution and Lyapunov index, and the influence of system parameters on the change of scroll-hiding attractor is studied. Finally, an electronic circuit of the system is implemented in Multisim, and the results of electronic circuit are consistent with that of the corresponding phase orbit.Pullback attractor of \(p\)-Laplacian equation with time-dependent parameters on the entire space.https://zbmath.org/1449.351042021-01-08T12:24:00+00:00"Wang, Yi"https://zbmath.org/authors/?q=ai:wang.yi.5|wang.yi.7|wang.yi.8|wang.yi.1|wang.yi.10|wang.yi.2|wang.yi.4|wang.yi.3|wang.yi.9|wang.yi.6"Ma, Qiaozhen"https://zbmath.org/authors/?q=ai:ma.qiaozhenSummary: Long time dynamical behavior of the \(p\)-Laplacian equation \({u_t} - {\mathrm{div}} (\varepsilon (t)|\nabla u|^{p-2}\nabla u) + f (x,u) = g (x,t)\) is considered. We prove that the process associated with the equation is asymptotically compact under the condition that the forcing term satisfies certain integral condition. By using the tail estimates of solution, the existence of pullback attractor is proved as well.Uniform exponential trisplitting -- a new criterion for discrete skew-product semiflows.https://zbmath.org/1449.370252021-01-08T12:24:00+00:00"Biris, Larisa Elena"https://zbmath.org/authors/?q=ai:biris.larisa-elena"Ceausu, Traian"https://zbmath.org/authors/?q=ai:ceausu.traian"Mihit, Claudia Luminita"https://zbmath.org/authors/?q=ai:mihit.claudia-luminita"Popa, Ioan-Lucian"https://zbmath.org/authors/?q=ai:popa.ioan-lucianSummary: In the present paper the concept of uniform exponential trisplitting for skew-product flows in Banach space is considered. This concept is a generalisation of the well-known concept of uniform exponential trichotomy. Connections between these concepts are presented and some illustrating examples prove that these are distinct. Also, we present necessary and sufficient conditions for the uniform exponential trisplitting concept with invariant and strongly invariant projectors. Finally, a characterisation in terms of Lyapunov sequences is given.Identification of 4D Lü hyper-chaotic system using identical systems synchronization and fractional adaptation law.https://zbmath.org/1449.370612021-01-08T12:24:00+00:00"Abedini, Mohammad"https://zbmath.org/authors/?q=ai:abedini.mohammad-javad"Gomroki, Mehdi"https://zbmath.org/authors/?q=ai:gomroki.mehdi"Salarieh, Hassan"https://zbmath.org/authors/?q=ai:salarieh.hassan"Meghdari, Ali"https://zbmath.org/authors/?q=ai:meghdari.aliSummary: In this paper, the parameters of a 4D Lü hyper-chaotic system are identified via synchronization of two identical systems. Unknown parameters of the drive system are identified by an adaptive method. Stability of the closed-loop system with one state feedback controller is studied by using the Lyapunov theorem. Also the convergence of the parameters to their true values is proved. Then a fractional adaptation law is applied to reduce the time of parameter convergence. Finally the results of both integer and fractional methods are compared.A central limit theorem for random dynamical systems.https://zbmath.org/1449.600402021-01-08T12:24:00+00:00"Lv, Kening"https://zbmath.org/authors/?q=ai:lv.kening"Zheng, Yan"https://zbmath.org/authors/?q=ai:zheng.yanSummary: In this article, we establish a central limit theorem for random dynamical systems, which is a supplement to the ergodic theory of random dynamical system. We can apply the theorem to a certain hyperbolic system to investigate the distribution of stochastic orbits, and further discuss the stochastic stability.Rolling regime in the Higgs model with friction.https://zbmath.org/1449.830042021-01-08T12:24:00+00:00"Piskovskiĭ, Evgeniĭ Viktorovich"https://zbmath.org/authors/?q=ai:piskovskii.evgenii-viktorovichSummary: The Higgs model with friction is considered. The hyperbolic analog of the Krylov-Bogoliubov averaging method is used to obtain an approximate solution. The obtained solution is compared to a numerical solution of the considered equation.Stronger forms of sensitivity on product dynamical system via Furstenberg families.https://zbmath.org/1449.370112021-01-08T12:24:00+00:00"Thakur, Rahul"https://zbmath.org/authors/?q=ai:thakur.rahul"Das, Ruchi"https://zbmath.org/authors/?q=ai:das.ruchiSeveral theorems about variants of Furstenberg families related to sensitivity on a countably infinite product of dynamical systems and its induced hyperspace and vice versa are established. For related papers, see, e.g., the works of \textit{R. Li} [Chaos Solitons Fractals 45, No. 6, 753--758 (2012; Zbl 1263.37022)], \textit{R. Li} et al. [J. Nonlinear Sci. Appl. 10, No. 9, 4940--4953 (2017; Zbl 1412.54041)] and \textit{X. Wu} et al. [J. Math. Anal. Appl. 429, No. 1, 16--26 (2015; Zbl 1370.37022)].
Reviewer: Salvatore Sessa (Napoli)Existence and multiplicity of periodic solutions for the non-autonomous second-order Hamiltonian systems.https://zbmath.org/1449.370422021-01-08T12:24:00+00:00"Ceng, Yusong"https://zbmath.org/authors/?q=ai:ceng.yusong"Chang, Hejie"https://zbmath.org/authors/?q=ai:chang.hejieSummary: In this paper, we study the existence and multiplicity of periodic solutions of the non-autonomous second-order Hamiltonian systems \[\begin{cases}\ddot u (t) = \nabla F (t,u (t))\;\;\; {\mathrm{a.e.}}\; t \in [0,T], \\ u (0)-u (T) = \dot u (0)- \dot u (T) = 0, \end{cases}\] where \(T > 0\). Under suitable assumptions on \(F\), some new existence and multiplicity theorems are obtained by using the least action principle and minimax methods in critical point theory.Hidden attractor with two memristors and Hamilton energy control.https://zbmath.org/1449.370702021-01-08T12:24:00+00:00"Mu, Nana"https://zbmath.org/authors/?q=ai:mu.nana"An, Xinlei"https://zbmath.org/authors/?q=ai:an.xinlei"Xu, Haonan"https://zbmath.org/authors/?q=ai:xu.haonanSummary: Based on the Kirchhoff's law, a nonlinear chaotic circuit model is designed which includes a flux-controlled memristor and a charge-controlled memristor. The hidden attractor is found through analyzing characteristics of the equilibrium points. And then, simulation and comparison studies of bifurcation diagram and phase trajectory and Poincaré map of parameter are carried out by simulation software. The Hamiltonian energy function of the memristor system is calculated according to Helmholtz theorem, and the relationship between the current and the energy is discussed. Based on this, the new Hamilton energy control method is proposed to control the memristor system to the desired states by choosing different control parameter.Consecutive Rosochatius deformations of the Garnier system and the Hénon-Heiles system.https://zbmath.org/1449.370392021-01-08T12:24:00+00:00"Xia, Baoqiang"https://zbmath.org/authors/?q=ai:xia.baoqiang"Zhou, Ruguang"https://zbmath.org/authors/?q=ai:zhou.ruguangSummary: An algorithm of constructing infinitely many symplectic realizations of generalized sl(2) Gaudin magnet is proposed. Based on this algorithm, the consecutive Rosochatius deformations of integrable Hamiltonian systems are presented. As examples, the consecutive Rosochatius deformations of the Garnier system and the Hénon-Heiles system as well as their Lax representations, are obtained.The Hamiltonian structures and algebro-geometric solution of the generalized Kaup-Newell soliton equations.https://zbmath.org/1449.353692021-01-08T12:24:00+00:00"Wei, Hanyu"https://zbmath.org/authors/?q=ai:wei.hanyu"Pi, Guomei"https://zbmath.org/authors/?q=ai:pi.guomeiSummary: Starting from a new spectral problem, a hierarchy of the generalized Kaup-Newell soliton equations is derived. By employing the trace identity, their Hamiltonian structures are also generated. Then, the generalized Kaup-Newell soliton equations are decomposed into two systems of ordinary differential equations. The Abel-Jacobi coordinates are introduced to straighten the flows, from which the algebro-geometric solutions of the generalized Kaup-Newell soliton equations are obtained in terms of the Riemann theta functions.The optimal strategies of \(SI\) pest control models with impulsive intervention.https://zbmath.org/1449.370562021-01-08T12:24:00+00:00"Chen, Miaomiao"https://zbmath.org/authors/?q=ai:chen.miaomiao"Pei, Yongzhen"https://zbmath.org/authors/?q=ai:pei.yongzhen"Liang, Xiyin"https://zbmath.org/authors/?q=ai:liang.xiyin"Lv, Yunfei"https://zbmath.org/authors/?q=ai:lv.yunfeiSummary: In view of the side effects, the technique relying on diseased pest releases as a valuable non-chemical tool is getting much more essentiality in pest management. Inspired the results in literature, the present thesis firstly focuses on a susceptible and infected pest model for pest management, which possesses multiple dynamic behaviors but does not eradicate susceptible individuals. For eliminating the pests, human impulsive interventions are embroiled in this model. Then the sufficient conditions for the global asymptotic stability of the susceptible pest-eradication periodic solution are established by unlimited pulse interventions. However, the strategy driving susceptible pests to extinction is unadvisable from ecological and economical aspects since the appropriate amount of pests in the field is beneficial for conservation of natural enemies and maintaining the crop overcompensation after pest injury. Hence, three different optimal problems involving different pest control tactics are deliberated in order to diminish the susceptible population at the terminal time and keep this in balance with the cost of the intervention (control). Subsequently, by time scaling and translation transformation techniques, the gradients for the cost functional on durations, fractions of susceptible pests killed due to chemical sprays as well as the number of infected pest released at each impulsive intervention moment are computed, which are vital to capture the optimal control strategy for pest regulation. Finally, on the basis of simulations, the strategy of alternative integrated control at unfixed time is validated to be the most effective compared with the other two policies. In addition, by comparing our optimal strategy with pest-extinction one, it is revealed that our strategy is more desirable.Adaptive unidirectional correlation method is used to control the new hyperchaotic system.https://zbmath.org/1449.931292021-01-08T12:24:00+00:00"Chen, Lu"https://zbmath.org/authors/?q=ai:chen.lu"Wang, Tao"https://zbmath.org/authors/?q=ai:wang.tao.8|wang.tao.3|wang.tao.5|wang.tao.6|wang.tao.1|wang.tao.2|wang.tao.4|wang.tao.9|wang.tao.7|wang.tao"Xu, Rongjin"https://zbmath.org/authors/?q=ai:xu.rongjin"Li, Muzi"https://zbmath.org/authors/?q=ai:li.muzi"Yue, Lijuan"https://zbmath.org/authors/?q=ai:yue.lijuanSummary: The unidirectional correlation method is used to control the new hyperchaotic system. An adaptive unidirectional correlation method is proposed by simplifying the rule in the unidirectional correlation method and adding adaptive strategies. In the case of small adaptive speed and simple rule, the new hyperchaotic system is controlled to periodic and stable point. The control time is shortened. Numerical simulation results show that the method is effective.Phase transitions on \(C^*\)-algebras arising from number fields and the generalized Furstenberg conjecture.https://zbmath.org/1449.460572021-01-08T12:24:00+00:00"Laca, Marcelo"https://zbmath.org/authors/?q=ai:laca.marcelo"Warren, Jacqueline M."https://zbmath.org/authors/?q=ai:warren.jacqueline-mFor an algebraic number field \(K\) with ring of integers \(O_K\), the (multiplicative) monoid \(O_K^{\times}\) of non-zero integers action on the (additive) group \(O_K\) gives rise to the semi-direct product \(O_K\rtimes O_K^{\times}\), called here the ``affine'' or ``\(ax+b\)'' monoid of algebraic integers in \(K\). The Toeplitz-like \(C^*\)-algebra generated by the left regular representation of the \(ax+b\) monoid acting by isometries on \(\ell^2(O_K\rtimes O_K^{\times})\) was studied by \textit{J. Cuntz} et al. [Math. Ann. 355, No. 4, 1383--1423 (2013; Zbl 1273.22008)], who analysed the equilibrium states of the time evolution on this \(C^*\)-algebra determined by the absolute norm, and characterized the simplex of KMS equilibrium states of this dynamical system for any inverse temperature \(\beta\in(0,\infty]\).
In the paper under review, the low-temperature range of the classification of KMS equilibrium states is studied, using the parametrization in terms of tracial states of direct sums of group \(C^*\)-algebras. Because of the action of units arising here, a higher-dimensional version of Furstenberg's seminal conjecture on rigidity for probability measures on the circle invariant under the multiplicative action of a non-lacunary semigroup of integers [\textit{H. Furstenberg}, Math. Syst. Theory 1, 1--49 (1967; Zbl 0146.28502)] enters the picture. The main results classify the behaviours arising in terms of the ideal class group, the degree, and the unit rank of \(K\), and an explicit description of the primitive ideal space of the associated transformation group \(C^*\)-algebra for number fields of unit rank at least \(2\) that are not complex multiplication fields.
Reviewer: Thomas B. Ward (Leeds)Directional preimage entropy for \({\mathbb{Z}}_+^k\)-actions.https://zbmath.org/1449.370312021-01-08T12:24:00+00:00"Gao, Yanan"https://zbmath.org/authors/?q=ai:gao.yanan"Zhang, Ziyao"https://zbmath.org/authors/?q=ai:zhang.ziyaoSummary: In this paper, a new type of entropy, directional preimage entropy including topological and measure theoretic versions for \({\mathbb{Z}}_+^k\)-actions, is introduced. Some of their properties including relationships and the invariance are obtained. Moreover, several systems including \({\mathbb{Z}}_+^k\)-actions generated by the expanding maps, \({\mathbb{Z}}_+^k\)-actions defined on nine graphs and some infinite graphs with zero directional preimage branch entropy are studied.The radial distribution of Julia sets of some entire functions with infinite lower order.https://zbmath.org/1449.370322021-01-08T12:24:00+00:00"Qiu, Ling"https://zbmath.org/authors/?q=ai:qiu.ling"Xuan, Zuxing"https://zbmath.org/authors/?q=ai:xuan.zuxing"Zhao, Yan"https://zbmath.org/authors/?q=ai:zhao.yanSummary: This article investigates the radial distribution of Julia sets of some entire functions with infinite lower order which are solutions, the polynomial or differential polynomial of solutions of the equation \(f'' (z) + A (z)f' (z) + B (z)f (z) = 0\).Time-dependent pullback attractors for nonclassical diffusion equations with time delays.https://zbmath.org/1449.351002021-01-08T12:24:00+00:00"Wang, Fangping"https://zbmath.org/authors/?q=ai:wang.fangping"Ma, Qiaozhen"https://zbmath.org/authors/?q=ai:ma.qiaozhenSummary: We first proved the well-posedness of weak solutions for the nonclassical diffusion equations with time delays by using the Faedo-Galerkin method, and then we gave the pullback \(\mathcal{D}\)-asymptotic compactness by using the contraction function method, which proved the existence of time-dependent pullback attractor.Initial value randomization of nonlinear evolution equations.https://zbmath.org/1449.370502021-01-08T12:24:00+00:00"Huang, Jianhua"https://zbmath.org/authors/?q=ai:huang.jianhua"Yan, Wei"https://zbmath.org/authors/?q=ai:yan.weiSummary: This paper aims to introduce some nonlinear evolution equations. Firstly, we present Schrödinger equations with random data, KdV equation with random data, and wave equation with random data. Then we give the harmonic analysis tools which are used to solve random data problem. At last, some unsolved problems related to random data are presented.Existence of uniform random attractor for non-autonomous stochastic strongly damped wave equation on unbounded domains.https://zbmath.org/1449.351112021-01-08T12:24:00+00:00"Zhang, Jie"https://zbmath.org/authors/?q=ai:zhang.jie|zhang.jie.5|zhang.jie.2|zhang.jie.4|zhang.jie.1|zhang.jie.3"Li, Xiaojun"https://zbmath.org/authors/?q=ai:li.xiaojunSummary: In this paper, we study the existence of uniform attractors for a class of nonautonomous stochastic strongly damped wave equations with additive white noise on unbounded domains. Firstly, by using the uniform estimates of solutions of transformed system, we prove that the stochastic dynamical system corresponding to the original equation has a uniformly pullback absorbing set. Second, by asymptotic tail estimation, we obtain that the solution is uniformly pullback asymptotically compact. The existence of uniform random attractor of the original system is obtained.Existence of periodic solution for a kind of \( (m,n)\)-order generalized neutral differential equation.https://zbmath.org/1449.342362021-01-08T12:24:00+00:00"Yao, Shaowen"https://zbmath.org/authors/?q=ai:yao.shaowenSummary: In this paper, we consider the following higher-order \(p\)-Laplacian generalized neutral differential equation with variable parameter \[ (\varphi_p(x (t)-c (t)x (t-\sigma))^{(n)})^{(m)}+g (t,x (t),x (t-\tau (t)), x' (t),\cdots, x^{(m)} (t)) = e (t).\] By the coincidence degree theory, sufficient conditions for the existence of periodic solutions are established.Convergence of attractors and invariant measures for a \(p\)-Laplace equation in \(\mathbb{R}^n\).https://zbmath.org/1449.350962021-01-08T12:24:00+00:00"Miao, Fahe"https://zbmath.org/authors/?q=ai:miao.fahe"Liu, Hui"https://zbmath.org/authors/?q=ai:liu.hui.2|liu.hui.4|liu.hui.1|liu.hui.3"Xin, Jie"https://zbmath.org/authors/?q=ai:xin.jieSummary: Using the conditions of uniform boundedness about the pullback attractor, the convergence of attractors for a \(p\)-Laplace equation in the whole space \(\mathbb{R}^n\) is studied. Then, the existence of a unique family of Borel invariant probability measures for the pullback attractor is established.Liouville correspondence between the short-wave model of Novikov hierarchy and the Sawada-Kotera hierarchy.https://zbmath.org/1449.353832021-01-08T12:24:00+00:00"Kang, Ting"https://zbmath.org/authors/?q=ai:kang.ting"Guo, Xu"https://zbmath.org/authors/?q=ai:guo.xu"Guo, Mingyue"https://zbmath.org/authors/?q=ai:guo.mingyue"Shi, Zhenhua"https://zbmath.org/authors/?q=ai:shi.zhen-huaSummary: In this paper, we study an explicit correspondence between the integrable short-wave model of Novikov hierarchy and the Sawada-Kotera hierarchy. A Liouville transformation between the isospectral problems of short-wave model of Novikov and Sawada-Kotera equation is used to link their respective recursion operators, and thus the one-to-one correspondence between each pair of integrable equations and each pair of Hamiltonian conservation laws is established.A class of quasi-fractional Noether's theorems for nonconservative systems in event space.https://zbmath.org/1449.370462021-01-08T12:24:00+00:00"Wang, Ze"https://zbmath.org/authors/?q=ai:wang.ze"Zhang, Yi"https://zbmath.org/authors/?q=ai:zhang.yi.10|zhang.yi.3|zhang.yi.11|zhang.yi|zhang.yi.8|zhang.yi.2|zhang.yi.7|zhang.yi.12|zhang.yi.5|zhang.yi.1|zhang.yi.9|zhang.yi.4Summary: To study the symmetry and conserved quantity of fractional non-conservative dynamic systems, the Noether theorem based on El-Nabulsi periodic law quasi-fractional model in event space is proposed and studied. Firstly, the fractional order variational problem based on the El-Nabulsi periodic law quasi-fractional model is established in the event space, and the differential equations of the holonomic nonconservative system and the nonholonomic nonconservative system are derived. Secondly, based on the invariance of the action functional under the infinitesimal transformation, the definition and criterion of the Noether symmetric transform and the Noether quasi-symmetric transformation are given. Finally, the Noether theorem based on the El-Nabulsi periodic law quasi-fractional model in the event space is proposed and proved. Two examples are given to illustrate the application of the results.Approximation of smooth stable invariant manifolds for stochastic partial differential equations.https://zbmath.org/1449.370342021-01-08T12:24:00+00:00"Guo, Zhongkai"https://zbmath.org/authors/?q=ai:guo.zhongkai"Yan, Xingjie"https://zbmath.org/authors/?q=ai:yan.xingjie"Yang, Xinguang"https://zbmath.org/authors/?q=ai:yang.xinguangSummary: Invariant manifolds are complicated random sets used for describing and understanding the qualitative behavior of nonlinear dynamical systems. The purpose of the present paper is to try to approximate smooth stable invariant manifolds for a type of stochastic partial differential equations with multiplicative white noise near the fixed point. Two examples are given to illustrate our results.Hypothesis testing for ergodicity of inhomogeneous diffusions.https://zbmath.org/1449.621042021-01-08T12:24:00+00:00"Shao, Jin"https://zbmath.org/authors/?q=ai:shao.jin"Jiang, Hui"https://zbmath.org/authors/?q=ai:jiang.huiSummary: For a class of time inhomogeneous diffusions, to test their ergodicity, we construct a suitable statistic. Then the unbiasedness and consistency of our test could be proved. Moreover, we also apply our main results to \(\alpha\)-Wiener bridge.Modeling of the extracellular information field influence in dynamics of the formation and development risks of a cancer tumor.https://zbmath.org/1449.370542021-01-08T12:24:00+00:00"Artemova, Ol'ga Igor'evna"https://zbmath.org/authors/?q=ai:artemova.olga-igorevna"Krevchik, Vladimir Dmitrievich"https://zbmath.org/authors/?q=ai:krevchik.vladimir-dmitrievich"Semenov, Mikhaĭl Borisovich"https://zbmath.org/authors/?q=ai:semenov.mikhail-borisovichSummary: The dynamic nonlinear 2D model of the extracellular information field influence in the dynamics of risks of the cancer tumor formation and development has been considered. Physical properties of the extracellular matrix, availability of nutrients, oxygen concentration, pH of the extracellular matrix, interaction with stromal cells, and etc. are considered as the main external parameters forming the informational metabolic potential. Within the framework of the constructed 2D analytical model, it has been shown that microinteraction through the extracellular matrix of emerging cancer cells through a dynamic informational metabolic profile significantly influences the risk dynamics of the formation and development of a cancer tumor. It is shown that, depending on the structure of the 2D informational metabolic profile, a number of characteristic nonlinear features such as 2D bifurcations, beats, chaos, imposed on integral dynamic curves resembling by the Gompertz function, describing the probable risks of the formation and development of a cancerous tumor, are appeared. A comparison of the results of our analytical model under consideration with the results of the modeling of other authors on the consideration of chaotic and bifurcation dynamics in the ``tumor-immune cluster-virus'' system has been made. As a result of the quantitative estimations carried out within framework of the proposed theoretical model, we can formulate a method for assessing the risks of developing malignant neoplasms, characterized in that subfebrile temperature, caspase level, colposcopic Raid index, which determine the threshold for the formation of malignant neoplasms, and identified as the risk factors.Adaptive synchronization of dynamical networks via states of several nodes as target orbit.https://zbmath.org/1449.370752021-01-08T12:24:00+00:00"Xiao, Yuzhu"https://zbmath.org/authors/?q=ai:xiao.yuzhu"Tang, Sufang"https://zbmath.org/authors/?q=ai:tang.sufang"Yang, Xiaoli"https://zbmath.org/authors/?q=ai:yang.xiaoliSummary: In this paper, based on the invariance principle of differential equation, a simple adaptive control method is proposed to synchronize the dynamical networks with the general coupling functions. Comparing with other adaptive control methods, the weighted average of a few nodes' states is used as target orbit to design controller. To show the effectiveness of proposed method, some numerical simulations are performed.Generation and suppression of a new hyperchaotic nonlinear model with complex variables.https://zbmath.org/1449.370692021-01-08T12:24:00+00:00"Mahmoud, Emad E."https://zbmath.org/authors/?q=ai:mahmoud.emad-eSummary: In this paper, we introduce a new hyperchaotic complex Chen model. This hyperchaotic complex system is constructed by adding a complex nonlinear term to the third equation of the chaotic complex Chen system with consideration it's all variables are complex. The new system is a 6-dimensional continuous real autonomous hyperchaotic system. The properties of this system including invariance, dissipation, equilibria and their stability, Lyapunov exponents, Lyapunov dimension, bifurcation diagrams and hyperchaotic behavior are studied. Different forms of hyperchaotic complex Chen systems are constructed. We suppress the hyperchaotic behavior of our system via passive control method by using one complex controller. The hyperchaotic attractors of the new system are converted to its unstable trivial fixed point and tracked to its unstable non trivial fixed points and periodic orbits. Block diagrams of our system are designed by using Matlab/Simulink after and before the suppression process to ensure the validity of the analytical results.Some criteria for the global finite-time synchronization of two Lorenz-stenflo systems coupled by a new controller.https://zbmath.org/1449.370622021-01-08T12:24:00+00:00"Chen, Yun"https://zbmath.org/authors/?q=ai:chen.yun"Shi, Zhangsong"https://zbmath.org/authors/?q=ai:shi.zhangsong"Lin, Chunsheng"https://zbmath.org/authors/?q=ai:lin.chunshengSummary: This paper investigates the global finite-time synchronization of two chaotic Lorenz-Stenflo systems coupled by a new controller called the generalized variable substitution controller. First of all, the generalized variable substitution controller is designed to establish the master-slave finite-time synchronization scheme for the Lorenz-Stenflo systems. And then, based on the finite-time stability theory, a sufficient criterion on the finite-time synchronization of this scheme is rigorously verified in the form of matrix and the corresponding estimation for the synchronization time is analytically given. Applying this criterion, some sufficient finite-time synchronization criteria under various generalized variable substitution controllers are further derived in the algebraic form. Finally, some numerical examples are introduced to compare the results proposed in this paper with those proposed in the existing literature, verifying the effectiveness of the criteria obtained.Dynamical behavior analysis of a generalized single-species population model in a polluted environment.https://zbmath.org/1449.370552021-01-08T12:24:00+00:00"Cao, Ming"https://zbmath.org/authors/?q=ai:cao.ming"Wang, Xia"https://zbmath.org/authors/?q=ai:wang.xia"Tang, Sanyi"https://zbmath.org/authors/?q=ai:tang.sanyiSummary: The environmental pollution problem caused by the rapid development of modern industry and agriculture and other production activities has become one of the most important ecological problems. In this paper, a new generalized single-species population model in polluted environment is established to study the influence of environmental toxins and food chain toxins on the existence of the species. By using the average integral method, sufficient conditions for uniform persistence, non-persistence in the mean, weak persistence in the mean and extinction, and the threshold between weak persistence and extinction of the population are obtained. These results provide reliable scientific basis for the conservation of species and the management of the environment.Dynamics of a ratio-dependent Lotka-Volterra cooperative system with delays.https://zbmath.org/1449.342832021-01-08T12:24:00+00:00"Ahmadjan, Muhammadhaji"https://zbmath.org/authors/?q=ai:ahmadjan.muhammadhajiSummary: This paper studies a class of nonautonomous two-species ratio-dependent Lotka-Volterra cooperative system with delays. Some sufficient conditions on the boundedness, permanence, existence of periodic solution and global attractivity of the system are established by means of the comparison method and Lyapunov function method.Measures of maximal entropy.https://zbmath.org/1449.370042021-01-08T12:24:00+00:00"Amini, M."https://zbmath.org/authors/?q=ai:amini.mostafa|amini.morteza.1|amini.massih-reza|amini.morteza|amini.mohammad-m|amini-dehak.mohammad|amini.marzei|amini.massoud|amini.mahraz|amini.m-hadi|amini.m-rSummary: We extend the results of \textit{P. Walters} [Trans. Am. Math. Soc. 236, 127--153 (1978; Zbl 0375.28009); Proc. Lond. Math. Soc. (3) 28, 500--516 (1974; Zbl 0319.28011)] on the uniqueness of invariant measures with maximal entropy on compact groups to an arbitrary locally compact group. We show that the maximal entropy is attained at the left Haar measure and the measure of maximal entropy is unique.Analysis, linear feedback synchronization and circuit realization of a hyperchaotic Tang's system.https://zbmath.org/1449.370652021-01-08T12:24:00+00:00"Gao, Zhizhong"https://zbmath.org/authors/?q=ai:gao.zhizhongSummary: In order to generate complex hyperchaotic attractors, a new four-dimensional Tang hyperchaotic system based on Tang system is built. The phase diagram, bifurcation diagram and Lyapunov exponents spectrum diagram of the system are analyzed by means of numerical simulations. Numerical simulations show that the new system's dynamics behavior can be periodic, quasi-periodic, chaotic and hyperchaotic as the parameter varies. Compared to the previous hyperchaos, the system possesses the large change range with new parameters \(k\), and the system changing with \(k\) and \(p\) shows the same dynamic behavior and a certain proportion. Linear controller is designed to realize synchronization of the hyperchaotic system. Results demonstrate that the method is correct and effective. Finally, a corresponding experimental circuit is designed. The hyperchaotic dynamical behavior of the circuit system and synchronization results of the driver system as well as response system are observed by an oscilloscope. The results are basically consistent with those of numerical simulation.Topological entropy of free semigroup actions.https://zbmath.org/1449.370132021-01-08T12:24:00+00:00"Zhang, Wenda"https://zbmath.org/authors/?q=ai:zhang.wenda"Xue, Licui"https://zbmath.org/authors/?q=ai:xue.licuiSummary: In this paper, we define the entropy and preimage entropy of free semigroup actions in a new method. Based on these definitions, we get some relations between topological entropy and measure entropy, and the relations among kinds of preimage entropies. The main results of this paper are as follows: (1) the topological entropy is invariant under equi-conjugacy; (2) the power rule for the measure-theoretic entropy holds.On some superintegrable Hamiltonian systems in curved spaces.https://zbmath.org/1449.370382021-01-08T12:24:00+00:00"Yan, Mengjiao"https://zbmath.org/authors/?q=ai:yan.mengjiao"Huang, Qing"https://zbmath.org/authors/?q=ai:huang.qingSummary: Some 2-dimensional superintegrable Hamiltonian systems in curved spaces are constructed in this paper. The second-order integrals of the superintegrable systems are built on the Killing vectors to the kinetic energy of the systems. In addition, the Poisson algebra for each superintegrable system is exhibited and the polynomial algebraic dependence relations of the first integrals are given explicitly.Pullback \(D\)-attractor of coupled rod equations with nonlinear moving heat source.https://zbmath.org/1449.350992021-01-08T12:24:00+00:00"Wang, Danxia"https://zbmath.org/authors/?q=ai:wang.danxia"Zhang, Jianwen"https://zbmath.org/authors/?q=ai:zhang.jianwen"Wang, Yinzhu"https://zbmath.org/authors/?q=ai:wang.yinzhuSummary: We consider the pullback \(D\)-attractor for the nonautonomous nonlinear equations of thermoelastic coupled rod with a nonlinear moving heat source. By Galerkin method, the existence and uniqueness of global solutions are proved under homogeneous boundary conditions and initial conditions. By prior estimates combined with some inequality skills, the existence of the pullback \(D\)-absorbing set is obtained. By proving the properties of compactness about the nonlinear operator \(g_1(\cdot), g_2(\cdot)\), and then proving the pullback \(D\)-condition (C), the existence of the pullback \(D\)-attractor of the equations previously mentioned is given.The existence of triple classical solutions to impulsive problems with small non-autonomous perturbations.https://zbmath.org/1449.341002021-01-08T12:24:00+00:00"Liu, Jian"https://zbmath.org/authors/?q=ai:liu.jian.1"Zhao, Zengqin"https://zbmath.org/authors/?q=ai:zhao.zengqin"Yu, Wenguang"https://zbmath.org/authors/?q=ai:yu.wenguangSummary: We study the existence of solutions to nonlinear impulsive boundary value problems with small non-autonomous perturbations on the half-line. We show the existence of at least three distinct classical solutions by using variational methods and a three critical points theorem.Weighted upper metric mean dimension for subadditive potentials.https://zbmath.org/1449.370182021-01-08T12:24:00+00:00"Ding, Zhihui"https://zbmath.org/authors/?q=ai:ding.zhihui"Li, Zhiming"https://zbmath.org/authors/?q=ai:li.zhimingSummary: In this paper we introduce the notion of subadditive weighted upper metric mean dimension and weighted upper measure-theoretic mean dimension which are analogs of pressures. A variational principle of subadditive weighted upper mean dimensions is presented.Dynamical analysis of the Lorenz-84 atmospheric circulation model.https://zbmath.org/1449.370532021-01-08T12:24:00+00:00"Wang, Hu"https://zbmath.org/authors/?q=ai:wang.hu"Yu, Yongguang"https://zbmath.org/authors/?q=ai:yu.yongguang"Wen, Guoguang"https://zbmath.org/authors/?q=ai:wen.guoguangSummary: The dynamical behaviors of the Lorenz-84 atmospheric circulation model are investigated based on qualitative theory and numerical simulations. The stability and local bifurcation conditions of the Lorenz-84 atmospheric circulation model are obtained. It is also shown that when the bifurcation parameter exceeds a critical value, the Hopf bifurcation occurs in this model. Then, the conditions of the supercritical and subcritical bifurcation are derived through the normal form theory. Finally, the chaotic behavior of the model is also discussed, the bifurcation diagrams and Lyapunov exponents spectrum for the corresponding parameter are obtained, and the parameter interval ranges of limit cycle and chaotic attractor are calculated in further. Especially, a computer-assisted proof of the chaoticity of the model is presented by a topological horseshoe theory.Entropy of dynamical systems on weights of a graph.https://zbmath.org/1449.370302021-01-08T12:24:00+00:00"Ebrahimzadeh, A."https://zbmath.org/authors/?q=ai:ebrahimzadeh.ataollah|ebrahimzadeh.asyieh|ebrahimzadeh.abolfazl|ebrahimzadeh.ataollh"Ebrahimi, M."https://zbmath.org/authors/?q=ai:ebrahimi.masoumeh|ebrahimi.mansour|ebrahimi.mehran|ebrahimi.m-j|ebrahimi.mohammad-ali|ebrahimi.masoud|ebrahimi.mohammad-mahdi|ebrahimi.morteza|ebrahimi.mahnaz|ebrahimi.m-r|ebrahimi.mohamad|ebrahimi.mohammad-mehdi|ebrahimi.mahdiSummary: Let \(G\) be a finite simple graph whose vertices and edges are weighted by two functions. In this paper, we define and calculate the entropy of a dynamical system on weights of the graph \(G\) by using the weights of vertices and edges of \(G\). We examine the conditions under which entropy of the dynamical system is zero, positive or \(+\infty\). At the end, it is shown that, for \(r\in [0,+\infty]\), there exists an order preserving transformation with entropy \(r\).Asymptotic metric behavior of random Cayley graphs of finite abelian groups.https://zbmath.org/1449.051362021-01-08T12:24:00+00:00"Shapira, Uri"https://zbmath.org/authors/?q=ai:shapira.uri"Zuck, Reut"https://zbmath.org/authors/?q=ai:zuck.reutSummary: Using methods of \textit{J. Marklof} and \textit{A. Strömbergsson} [ibid. 33, No. 4, 429--466 (2013; Zbl 1340.05063)] we establish several limit laws for metric parameters of random Cayley graphs of finite abelian groups with respect to a randomly chosen set of generators of a fixed size. Doing so we settle a conjecture of \textit{G. Amir} and \textit{O. Gurel-Gurevich} [Groups Complex. Cryptol. 2, No. 1, 59--65 (2010; Zbl 1194.05054)].Existence of solutions of implicit integral equations via \(Z\)-contraction.https://zbmath.org/1449.540892021-01-08T12:24:00+00:00"Patle, Pradip R."https://zbmath.org/authors/?q=ai:patle.pradip-ramesh"Patel, Deepesh Kumar"https://zbmath.org/authors/?q=ai:patel.deepesh-kumarSummary: The main focus of this work is to assure that the sum of a compact operator with a \(Z\)-contraction admits a fixed point. The concept of condensing mapping (in the sense of Hausdorff non-compactness measure) is used to establish the concerned result which generalizes some of the existing state-of-art in the literature. Presented result is used to verify the actuality of solutions of implicit integral equations.Chaos control of fractional order Rabinovich-Fabrikant system and synchronization between chaotic and chaos controlled fractional order Rabinovich-Fabrikant system.https://zbmath.org/1449.370722021-01-08T12:24:00+00:00"Srivastava, M."https://zbmath.org/authors/?q=ai:srivastava.manjari-k|srivastava.muni-shanker|srivastava.manoj-kumar|srivastava.meera|srivastava.mani-b|srivastava.madhurima|srivastava.m-m|srivastava.milan|srivastava.mukesh|srivastava.mahesh-c|srivastava.manindra-kumar|srivastava.meenakshi|srivastava.mayank|srivastava.marindra-kumar|srivastava.mohit-kumar|srivastava.m-p|srivastava.madhu"Agrawal, S. K."https://zbmath.org/authors/?q=ai:agrawal.suresh-kumar|agrawal.sunil-kumar|agrawal.saurabh-k"Vishal, K."https://zbmath.org/authors/?q=ai:vishal.k"Das, S."https://zbmath.org/authors/?q=ai:das.subir-kSummary: In this article the local stability of the Rabinovich-Fabrikant (R-F) chaotic system with fractional order time derivative is analyzed using fractional Routh-Hurwitz stability criterion. Feedback control method is used to control chaos in the considered fractional order system and after controlling the chaos the authors have introduced the synchronization between fractional order non-chaotic R-F system and the chaotic R-F system at various equilibrium points. The fractional derivative is described in the Caputo sense. Numerical simulation results which are carried out using Adams-Boshforth-Moulton method show that the method is effective and reliable for synchronizing the systems.Bargmann type systems for the generalization of Toda lattices.https://zbmath.org/1449.370362021-01-08T12:24:00+00:00"Li, Fang"https://zbmath.org/authors/?q=ai:li.fang.3|li.fang.2|li.fang.6|li.fang.5|li.fang.1|li.fang|li.fang.4"Lu, Liping"https://zbmath.org/authors/?q=ai:lu.lipingSummary: Under a constraint between the potentials and eigenfunctions, the nonlinearization of the Lax pairs associated with the discrete hierarchy of a generalization of the Toda lattice equation is proposed, which leads to a new symplectic map and a class of finite-dimensional Hamiltonian systems. The generating function of the integrals of motion is presented, by which the symplectic map and these finite-dimensional Hamiltonian systems are further proved to be completely integrable in the Liouville sense. Finally, the representation of solutions for a lattice equation in the discrete hierarchy is obtained.A novel hierarchy of differential equations, conservation laws and Darboux transformation.https://zbmath.org/1449.353052021-01-08T12:24:00+00:00"He, Guoliang"https://zbmath.org/authors/?q=ai:he.guoliang"Zheng, Zhenzhen"https://zbmath.org/authors/?q=ai:zheng.zhenzhenSummary: With the aid of the zero-curvature equation, a novel integrable hierarchy of nonlinear evolution equations associated with a \(3 \times 3\) matrix spectral problem was proposed. Based on two linear spectral problems, the infinite many conservation laws of the first two members and explicit solutions constructed from the Darboux transformation of the first member in the hierarchy were obtained.Multiplicity of periodic solutions for a class of non-autonomous second-order Hamiltonian systems.https://zbmath.org/1449.370432021-01-08T12:24:00+00:00"Huang, Delong"https://zbmath.org/authors/?q=ai:huang.delong"Guo, Fei"https://zbmath.org/authors/?q=ai:guo.feiSummary: The multiplicity of periodic solutions of Hamiltonian systems is obtained based on the existing results of the existence of periodic solutions of Hamiltonian systems via the critical point theorem.On the fractional-order extended Kalman filter and its application to chaotic cryptography in noisy environment.https://zbmath.org/1449.370712021-01-08T12:24:00+00:00"Sadeghian, Hoda"https://zbmath.org/authors/?q=ai:sadeghian.hoda"Salarieh, Hassan"https://zbmath.org/authors/?q=ai:salarieh.hassan"Alasty, Aria"https://zbmath.org/authors/?q=ai:alasty.aria"Meghdari, Ali"https://zbmath.org/authors/?q=ai:meghdari.aliSummary: In this paper via a novel method of discretized continuous-time Kalman filter, the problem of synchronization and cryptography in fractional-order systems has been investigated in presence of noisy environment for process and output signals. The fractional-order Kalman filter equation, applicable for linear systems, and its extension called the extended Kalman filter, which can be used for nonlinear systems, are derived. The result is utilized for chaos synchronization with the aim of cryptography while the transmitter system is fractional-order, and both the transmitter and transmission channel are noisy. The fractional-order stochastic chaotic Chen system is then presented to apply the proposed method for chaotic signal cryptography. The results show the effectiveness of the proposed method.Phase-response synchronization in coupled neuronal oscillator population with time delay.https://zbmath.org/1449.342622021-01-08T12:24:00+00:00"Zhang, Zhaokun"https://zbmath.org/authors/?q=ai:zhang.zhaokun"Jiao, Xianfa"https://zbmath.org/authors/?q=ai:jiao.xianfaSummary: In the nervous system, there is a time delay in the information transmission from the neuronal of pre-synapse to the neuronal of post-synapse. In this paper, a dynamical evolution model of the population of coupled neuronal oscillators with time delay in the presence of external periodic stimulus and noise is proposed. The average number density is used to describe the synchronous activity of the population of neuronal oscillators, and the evolution equation of average number density is derived. Numerical simulations indicate that the time delay influences the synchronous activity of the neuronal oscillator population. As the time delay increases, the neuronal population shows stable periodic synchronous oscillation and the degree of synchronization decreased. As the stimulus intensity increases, the synchronization degree of the neuronal population increases. As the stimulus frequency increases, the synchronization degree of the neuronal population decreases rapidly. As the stimulus frequency is close to the population characteristic frequency, the synchronization degree of the neuronal population decreases. As the noise intensity increases, the synchronization degree of the neuronal population increases.Construction of general solution of degenerating Pohlmeyer-Lund-Regge system.https://zbmath.org/1449.370472021-01-08T12:24:00+00:00"Gur'eva, A. M."https://zbmath.org/authors/?q=ai:gureva.a-mSummary: It is demonstrated that degenerating Pohlmeyer-Lund-Regge system is a Liouville-type system, formulas are obtained for \(x\)- and \(y\)-integrals at the first and the second orders. It is demonstrated how they can be used in order to construct a general solution on this equation system.Dynamic analysis and energy feedback control of chaotic systems without equilibrium point.https://zbmath.org/1449.370742021-01-08T12:24:00+00:00"Wang, Wenjing"https://zbmath.org/authors/?q=ai:wang.wenjing"An, Xinlei"https://zbmath.org/authors/?q=ai:an.xinlei"Yu, Huanhuan"https://zbmath.org/authors/?q=ai:yu.huanhuanSummary: A new kind of chaotic autonomous system is proposed by introducing new parameters to the chaotic system with infinitely many equilibria. By using theoretical analysis, Lyapunov exponent, bifurcation graph and other nonlinear system analysis, the hidden dynamic behaviors of the new system, such as one period, two periods and chaos, are studied when the new parameters change. In addition, when initial conditions change, limit cycles and chaotic attractors coexist in the new system. Finally, the Hamiltonian energy of the system is calculated and an energy feedback controller is designed to eliminate chaos in a certain period of time.Stationary distribution and extinction of a stochastic population model with Allee effect and nonlinear perturbation.https://zbmath.org/1449.370572021-01-08T12:24:00+00:00"Chen, Xianli"https://zbmath.org/authors/?q=ai:chen.xianliSummary: This paper investigates the stationary distribution and extinction of a stochastic population model with Allee effect and nonlinear perturbation. The author first proves the existence of global positive solution of the model. Then by constructing a suitable stochastic Lyapunov function, the author establishes sufficient conditions for the existence of an ergodic stationary distribution of the model. Then the author obtains sufficient conditions for extinction of the population in two cases. One is how Allee effect affects extermination of the population and the other is how noises affect extermination of the population. At last, some examples together with numerical simulations are provided to illustrate the analytical results.Critical states of a damaged non-linearly deformed medium during high-speed collision with an obstacle.https://zbmath.org/1449.741462021-01-08T12:24:00+00:00"Petushkov, V. A."https://zbmath.org/authors/?q=ai:petushkov.v-aSummary: Generalized model of the nonlinear interconnected deformation and fracture of a damaged polycrystalline media is presented at high-speed shock influence conditions. Geometrical non-linearity caused by finite non-linear deformations depending on speed, behavior of materials with variable micro structure, anisotropic hardening and Baushinger-effect are considered. Particular attention is paid to problems of damage localization, progression and final fracture of non-linearity deformed bodies. Justification of the proposed model is implemented and nonlinear wave processes in a thin-walled shell are studied at high-speed collision condition with an obstacle.