Chaos of time-varying discrete dynamical systems

This paper is concerned with chaos of time-varying (i.e. non-autonomous) discrete systems in metric spaces. Some basic concepts are introduced for general time-varying systems, including periodic point, coupled-expansion for transitive matrix, uniformly topological equiconjugacy, and three definitions of chaos, i.e. chaos in the sense of Devaney and Wiggins, respectively, and in a strong sense of Li–Yorke. An interesting observation is that a finite-dimensional linear time-varying system can be chaotic in the original sense of Li–Yorke, but cannot have chaos in the strong sense of Li–Yorke, nor in the sense of Devaney in a set containing infinitely many points, and nor in the sense of Wiggins in a set starting from which all the orbits are bounded. A criterion of chaos in the original sense of Li–Yorke is established for finite-dimensional linear time-varying systems. Some basic properties of topological conjugacy are discussed. In particular, it is shown that topological conjugacy alone cannot guarantee two topologically conjugate time-varying systems to have the same topological properties in general. In addition, a criterion of chaos induced by strict coupled-expansion for a certain irreducible transitive matrix is established, under which the corresponding nonlinear system is proved chaotic in the strong sense of Li–Yorke. Two illustrative examples are finally provided with computer simulations for illustration.     

A flutter function reveals the chaos of differential calculus

This paper disproves the widespread opinion that chaos may appear in non-linear connections only. The common differential operator, which assigns the first derivative to each function, is linear and chaotic in the sense of Li and Yorke.     

Study on chaos induced by turbulent maps in noncompact sets

This paper is concerned with chaos induced by strictly turbulent maps in noncompact sets of complete metric spaces. Two criteria of chaos for such types of maps are established, and then a criterion of chaos, characterized by snap-back repellers in complete metric spaces, is obtained. All the maps presented in this paper are proved to be chaotic either in the sense of both Li–Yorke and Wiggins or in the sense of both Li–Yorke and Devaney. The results weaken the assumptions in some existing criteria of chaos. Several illustrative examples are provided with computer simulation.     

Human capital externality and chaotic equilibrium dynamics

This study develops a two-period overlapping generations model in which adults undertake educational investment decisions on behalf of young agents. In addition to educational investment, we argue that the accumulation of human capital is also dependent upon the externality from average human capital within the economy. In a departure from the previous literature in this area, we assume that there is a reduction in the overall productivity of human capital accumulation brought about by human capital externality, and show that complicated dynamics will emerge under this circumstance. In addition to displaying the chaotic dynamics in the sense of Li and Yorke, we also verify the existence of Devaney's chaos and Smale's chaos.     

混沌与拓扑强混合

讨论了拓扑强混合、Li—Yorke混沌和修改的Devaney混沌三者之间的关系,我们得到:Li—Yorke混沌与修改的Devaney混沌无蕴涵关系;Li—Yorke混沌和修改的Devaney混沌均不能蕴涵着拓扑强混合,这解决了文献[1]中提出的两个问题.     

Li–Yorke chaos translation set for linear operators

In order to study Li–Yorke chaos by the scalar perturbation for a given bounded linear operator T on a Banach space X, we introduce the Li–Yorke chaos translation set of T, which is defined by \(S_{LY}(T)=\{\lambda \in {\mathbb {C}};\lambda +T \text { is Li--Yorke chaotic}\}\). In this paper, some operator classes are considered, such as normal operators, compact operators, shift operators, and so on. In particular, we show that the Li–Yorke chaos translation set of the Kalisch operator on the Hilbert space \(\mathcal {L}^2[0,2\pi ]\) is a simple point set \(\{0\}\).     

Li–Yorke chaos in higher dimensions: a review

Li and Yorke not only introduced the term “chaos” along with a mathematically rigorous definition of what they meant by it, but also gave a condition for chaos in scalar difference equations, their equally famous “period three implies chaos” result. Generalizations of the Li and Yorke definition of chaos to difference equations in ? ⁿ are reviewed here as well as higher dimensional conditions ensuring its existence, specifically the “snap-back repeller” condition of Marotto and its counterpart for saddle points. In addition, further generalizations to mappings in Banach spaces and complete metric spaces are considered. These will be illustrated with various simple examples including an application to chaotic dynamics on the metric space (?  ⁿ , D) of fuzzy sets on the base space ? ⁿ .     

Relationships among some chaotic properties of non-autonomous discrete dynamical systems

This paper is concerned with relationships among some chaotic properties of non-autonomous discrete dynamical systems. Some relationships among weak mixing, topologically weak mixing, generic chaos, dense chaos, and sensitivity are investigated. In addition, some equivalent conditions of sensitivity are given and the relationships between sensitivity and Li–Yorke sensitivity are obtained. These results generalize some existing results of autonomous discrete systems, some of which relax the corresponding conditions.     

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.     

Li–Yorke chaos for invertible mappings on compact metric spaces

In this paper, we construct a homeomorphism on the closed unit disk to show that the inverse of a Li–Yorke chaotic mapping on a compact metric space need not be Li–Yorke chaotic.     

The Marotto Theorem on planar monotone or competitive maps

In 1978, Marotto generalized Li–Yorke’s results on the criterion for chaos from one-dimensional discrete dynamical systems to n-dimensional discrete dynamical systems, showing that the existence of a non-degenerate snap-back repeller implies chaos in the sense of Li–Yorke. This theorem is very useful in predicting and analyzing discrete chaos in multi-dimensional dynamical systems. Yet, besides it is well known that there exists an error in the conditions of the original Marotto Theorem, and several authors had tried to correct it in different way, Chen, Hsu and Zhou pointed out that the verification of “non-degeneracy” of a snap-back repeller is the most difficult in general and expected, “almost beyond reasonable doubt,” that the existence of only degenerate snap-back repeller still implies chaotic, which was posed as a conjecture by them. In this paper, we shall give necessary and sufficient conditions of chaos in the sense of Li–Yorke for planar monotone or competitive discrete dynamical systems and solve Chen–Hsu–Zhou Conjecture for such kinds of systems.     

Entropy,Chaos, and Weak Horseshoe for Infinite‐Dimensional Random Dynamical Systems

In this paper, we study the complicated dynamics of infinite‐dimensional random dynamical systems that include deterministic dynamical systems as their special cases in a Polish space. Without assuming any hyperbolicity, we prove if a continuous random map has a positive topological entropy, then it contains a topological horseshoe. We also show that the positive topological entropy implies the chaos in the sense of Li‐Yorke. The complicated behavior exhibited here is induced by the positive entropy but not the randomness of the system.© 2017 Wiley Periodicals, Inc.     

Li-Yorke chaos in the system with impacts

The analogue of Li-Yorke chaos [T.Y. Li, J. Yorke, Period three implies chaos, Amer. Math. Monthly 87 (1975) 985-992] for a special initial value problem of a non-autonomous impulsive differential equation is developed.     

Period Three Implications for Expansive Maps in R N

A multidimensional version of the Li–Yorke cycle coexisting theorem [Li, T.-Y. and Yorke, J.A. “Period three implies chaos”, Am. Math. Monthly, 82, 985–992] is established for certain (e.g. expansive) maps. The related fixed- and periodic-point theorems are developed in R ⁿ . Implications of 3-orbits are discussed.     

Structure stability of maps with snap-back repellers in Banach spaces

In this paper, we study small perturbations of a class of chaotic discrete systems in Banach spaces induced by snap-back repellers. If a map has a regular and non-degenerate snap-back repeller, then it still has a regular and non-degenerate snap-back repeller under a sufficiently small perturbation. Consequently, the perturbed system is still chaotic in the sense of both Devaney and Li–Yorke as the original one. Furthermore, in order to study structural stability of maps with regular and non-degenerate snap-back repellers, we first discuss structural stability of strictly A-coupled-expanding maps in Banach spaces. Applying this result, we show that a map with a regular and non-degenerate snap-back repeller in a Banach space is C ¹ structurally stable on its chaotic invariant set.     

DC3 and Li–Yorke chaos

There are three versions of distributional chaos, namely DC1, DC2 and DC3. By using an example of constant-length substitution system, we show that DC3 need not imply Li–Yorke chaos. (In this paper, chaos means the existence of an uncountable scrambled set of the corresponding type, while the existing example only deals with a single pair of points.)     

A note on distributional chaos with respect to a sequence

The aim of this note is to use methods developed by Kuratowski and Mycielski to prove that some more common notions in topological dynamics imply distributional chaos with respect to a sequence. In particular, we show that the notion of distributional chaos with respect to a sequence is only slightly stronger than the definition of chaos due to Li and Yorke. Namely, positive topological entropy and weak mixing both imply distributional chaos with respect to a sequence, which is not the case for distributional chaos as introduced by Schweizer and Smítal.     

Li–Yorke chaos and synchronous chaos in a globally nonlocal coupled map lattice

This paper investigates a globally nonlocal coupled map lattice. A rigorous proof to the existence of chaos in the scene of Li–Yorke in that system is presented in terms of the Marotto theorem. Analytical sufficient conditions under which the system is chaotic, and has synchronous behaviors are determined, respectively. The wider regions associated with chaos and synchronous behaviors are shown by simulations. Spatiotemporal chaos, synchronous chaos and some other synchronous behaviors such as fixed points, 2-cycles and 2²-cycles are also shown by simulations for some values of the parameters.     

Distributional chaos in a sequence

In this paper we prove a sufficient condition for the continuous map of a compact metric space for being distributively chaotic in a sequence. As an application, it is proved that a continuous map of an interval is chaotic in the Li–Yorke sense if and only if it is distributively chaotic in a sequence.     

A weakly mixing dynamical system with the whole space being a transitive extremal distributionally scrambled set

It is known that the whole space can be a Li–Yorke scrambled set in a compact dynamical system, but this does not hold for distributional chaos. In this paper we construct a noncompact weekly mixing dynamical system, and prove that the whole space is a transitive extremal distributionally scrambled set in this system.