Non-autonomous Discrete Systems Research Articles

Chaos in non-autonomous discrete systems and their induced set-valued systems.

In this paper, the interactions of some given chaotic properties between a non-autonomous discrete system (X,f0,∞) and its induced set-valued system (K(X),f¯0,∞) are obtained. It is proved that the specification property, property P, topological mixing, mild mixing, and topological exactness of (X,f0,∞) are equivalent to those of (K(X),f¯0,∞), respectively. It is shown that Robinson chaos (resp. Kato chaos) between (X,f0,∞) and (K(X),f¯0,∞) are equivalent under certain conditions. Furthermore, Li-Yorke chaos and distributional chaos of (X,f0,∞) imply those of (K(X),f¯0,∞), respectively. Topological equi-conjugacy between two systems is proved to be preserved by their induced set-valued systems. Topological entropy of (X,f0,∞) is guaranteed to be no larger than that of (K(X),f¯0,∞). Two examples are finally provided with computer simulations for illustration.

Induced dynamics in hyperspaces of non-autonomous discrete systems

In this paper, the interrelations of some dynamical properties of a non-autonomous dynamical system (X, f1, ?) and its induced non-autonomous dynamical system (K(X), f1, ?) are studied, where K(X) is the hyperspace of all non-empty compact subsets of X, endowed with Vietoris topology. Various stronger forms of sensitivity and transitivity are considered. Some examples of non-autonomous systems are provided to support the results. A relation between shadowing property of the non-autonomous system (X, f1, ?) and its induced system (K(X), f1, ?) is studied.

On the Iteration Invariance of Distributional Chaos of Type $$2\frac{1}{2}$$ 2 1 2 in Non-autonomous Discrete System

This paper is concerned with invariants for distributional chaos of type $$2\frac{1}{2}$$ in non-autonomous discrete system $$(X, f_{1,\infty })$$ which converges uniformly and proves that $$f_{1,\infty }$$ is $$\text {DC}2\frac{1}{2}$$ if and only if $$f_{1,\infty }^{[k]}$$ is $$\text {DC}2\frac{1}{2}$$ for any $$k\in \mathbb {N}$$ . This result partly answers the questions posed by Wu and Zhu (Appl Math Lett 26:432–436, 2013).

Some weak versions of distributional chaos in non-autonomous discrete systems

This paper is concerned with some weak versions of distributional chaos in a non-autonomous discrete system generated by a given sequence of maps {fn}n=0∞ in a metric space (X, d). It is shown that three versions named DC1, DC2, and DC212 are invariants under iterations when {fn}n=0∞ is equi-continuous in X, which weakens the condition in the literature that {fn}n=0∞ uniformly converges in a compact space X. It is also proved that DC1, DC2, and DC212 are invariants of topological equi-conjugacy. One example is provided with computer simulations for illustration.

Multi-sensitivity and other stronger forms of sensitivity in non-autonomous discrete systems

In this paper we obtain different sufficient conditions for a non-autonomous discrete system to be multi-sensitivite. We study properties of a multi-sensitive non-autonomous system in detail. It is proved that on a compact metric space every finitely generated non-autonomous system which is topologically transitive having dense set of periodic points is thickly syndetically sensitive. We introduce and study the notion of totally sensitive non-autonomous systems. We also provide counter examples to support our results.

Extinction in a Nonautonomous Discrete Lotka-Volterra Competitive System with Time Delay

This paper is concerned with a nonautonomous discrete Lotka-Volterra competitive system with time delay. By using some analytical techniques, we prove that, under certain conditions, one of the species will be driven to extinction while the other one will be globally attractive with any positive solution of a discrete logistic equation.

Lyapunov Exponents, Sensitivity, and Stability for Non-Autonomous Discrete Systems

This paper focuses on the relationships of Lyapunov exponents with sensitivity and stability for non-autonomous discrete systems. Some new concepts are introduced for non-autonomous discrete systems, including Lyapunov exponents, strong sensitivity at a point and in a set, Lyapunov stability, and exponential asymptotical stability. It is shown that the positive Lyapunov exponent at a point implies strong sensitivity for a class of non-autonomous discrete systems. Furthermore, the uniformly positive Lyapunov exponents in a totally invariant set imply strong sensitivity in this set under certain conditions. The negative Lyapunov exponent at a point implies exponential asymptotical stability for a class of non-autonomous discrete systems. The related existing results for autonomous discrete systems are generalized to non-autonomous discrete systems and their conditions are weakened. One example is provided for illustration.

Global Attractivity and Extinction of a Discrete Competitive System with Infinite Delays and Single Feedback Control

A nonautonomous discrete two-species competition system with infinite delays and single feedback control is considered in this paper. Based on the discrete comparison theorem, a set of sufficient conditions which guarantee the permanence of the system is obtained. Then, by constructing some suitable discrete Lyapunov functionals, some sufficient conditions for the global attractivity and extinction of the system are obtained. It is shown that, by choosing some suitable feedback control variable, one of two species will be driven to extinction.

Chaos induced by snap-back repellers in non-autonomous discrete dynamical systems

This paper focuses on chaos induced by snap-back repellers in non-autonomous discrete systems. A new concept of snap-back repeller for non-autonomous discrete systems is introduced and several new criteria of chaos induced by snap-back repellers in non-autonomous discrete systems are established. In addition, it is proved that a regular and nondegenerate snap-back repeller in non-autonomous discrete systems implies chaos in the (strong) sense of Li–Yorke. Two illustrative examples are proved.

Some Iterative Properties of ( F 1 , F 2 ) -Chaos in Non-Autonomous Discrete Systems

This paper is concerned with invariance -scrambled sets under iterations. The main results are an extension of the compound invariance of Li–Yorke chaos and distributional chaos. New definitions of -scrambled sets in non-autonomous discrete systems are given. For a positive integer k, the properties and of Furstenberg families are introduced. It is shown that, for any positive integer k, for any , Furstenberg family has properties and , where denotes the family of all infinite subsets of whose upper density is not less than s. Then, the following conclusion is obtained. D is an -scrambled set of if and only if D is an -scrambled set of .

On distributional chaos in non-autonomous discrete systems

This paper studies distributional chaos in non-autonomous discrete systems generated by given sequences of maps in metric spaces. In the case that the metric space is compact, it is shown that a system is Li–Yorke δ-chaotic if and only if it is distributionally δ′-chaotic in a sequence; and three criteria of distributional δ-chaos are established, which are caused by topologically weak mixing, asymptotic average shadowing property, and some expanding condition, respectively, where δ and δ′ are positive constants. In a general case, a criterion of distributional chaos in a sequence induced by a Xiong chaotic set is established.

Asymptotic analysis of non-autonomous discrete hungry integrable systems

Asymptotic analysis of non-autonomous discrete hungry integrable systems

Functional envelope of a non-autonomous discrete system

Abstract Let (X, F = {fn}n =0 ∞) be a non-autonomous discrete system by a compact metric space X and continuous maps fn : X → X, n = 0, 1, ....We introduce functional envelope (S(X), G = {Gn}n =0 ∞), of (X, F = {fn}n =0 ∞), where S(X) is the space of all continuous self maps of X and the map Gn : S(X) → S(X) is defined by Gn(ϕ) = Fn ∘ ϕ, Fn = fn ∘ fn-1 ∘ . . . ∘ f1 ∘ f0. The paper mainly deals with the connection between the properties of a system and the properties of its functional envelope.

Extinction for a discrete competition system with feedback controls

We consider a nonautonomous discrete competition system with nonlinear interinhibition terms and feedback controls. By constructing a suitable Lyapunov function, we obtain some criteria about the extinction of one of the two species and the corresponding feedback controls varieties. Our conclusions not only supplement but also improve some existing ones. Numerical simulations are used to illustrate our analytic analysis. We show that feedback control variables play an important role in the extinction property of the system.

On the Relation of Shadowing and Expansivity in Nonautonomous Discrete Systems

On the Relation of Shadowing and Expansivity in Nonautonomous Discrete Systems

Chaos in hyperspaces of nonautonomous discrete systems

We study the interaction of some dynamical properties of a nonautonomous discrete dynamical system (X, f∞) and its induced nonautonomous discrete dynamical system (K(X),f∞¯), where K(X) is the hyperspace of non-empty compact sets in X, endowed with the Vietoris topology. We consider properties like transitivity, weakly mixing, points with dense orbit, density of periodic points, among others. We also present examples of nonautonomous discrete dynamical systems showing that transitivity, density of periodic points and sensitive dependence on initial conditions are independents on the unit interval, i.e., unlike autonomous discrete dynamical systems, in definition of Devaney chaotic there are not redundant conditions for NDS on the interval. Actually, our examples give an even more precise conclusion: the classical result stating that transitivity is a sufficient condition for an autonomous discrete dynamical system on the interval to be Devaney chaotic fails to be true for nonautonomous dynamical systems.

Existence of periodic solutions for nonautonomous second-order discrete Hamiltonian systems

In this paper, we consider the existence of periodic solutions for a class of nonautonomous second-order discrete Hamiltonian systems in case the sum on the time variable of potential is periodic. The tools used in our paper are the direct variational minimizing method and Rabinowitz’s saddle point theorem.

On iteration invariants for (F1,F2) -sensitivity and weak (F1,F2) -sensitivity of non-autonomous discrete systems

On iteration invariants for (F1,F2) -sensitivity and weak (F1,F2) -sensitivity of non-autonomous discrete systems

Chaos induced by weak A-coupled-expansion of non-autonomous discrete dynamical systems

This paper focuses on chaos induced by weak A-coupled-expansion of non-autonomous discrete systems in compact subsets of metric spaces and in bounded and closed subsets of complete metric spaces, separately. A new concept of weak A-coupled-expansion for non-autonomous discrete systems, whose condition is weaker than that of A-coupled-expansion, is introduced, and several new criteria of chaos induced by weak A-coupled-expansion of non-autonomous discrete systems are established. By applying some close relationships between chaotic dynamical behaviours of the original system and its induced systems, two criteria of chaos are established. One example is provided for illustration.

Devaney Chaos in Nonautonomous Discrete Systems

This paper is concerned with Devaney chaos in nonautonomous discrete systems. It is shown that in its definition, the two former conditions, i.e. transitivity and density of periodic points, in a set imply the last one, i.e. sensitivity, in the case that the set is unbounded, while a similar result holds under two additional conditions in the other case that the set is bounded. Some chaotic behavior is studied for a class of nonautonomous discrete systems, each of which is governed by a convergent sequence of continuous maps. In addition, the concepts of some pseudo-orbits and shadowing properties are introduced for nonautonomous discrete systems, and it is shown that some shadowing properties of the system and density of periodic points imply that the system is Devaney chaotic under the condition that the sequence of continuous maps is uniformly convergent in a compact metric space.