Devaney Chaos Research Articles

On n-scrambled sets

This paper investigates n-scrambled sets in the sense of distribution and Li-Yorke, focusing on the equivalence of n-scrambled sets and the existence of uncountable n-scrambled sets. For the equivalence, n-scrambled sets can imply m-scrambled sets in both senses for any 2≤m<n, but their inverses require certain conditions to hold. A method is provided for constructing an uncountable n-scrambled set in the distribution and Li-Yorke senses, which also verifies the equivalence between n-scrambled sets. For the existence, it is shown that distributionally n-scrambled tuples can be derived from the weak specification property (WSP), a 1-periodic and a non-1-periodic point. And the Li-Yorke (n+1)-scrambled tuple can be characterized by an n-scrambled tuple utilizing the SP. Finally, the uncountable Li-Yorke n-scrambled set is established based on the weak mixing, and the uncountable mean Li-Yorke n-scrambled set is induced by the shadowing and Devaney chaos.

Weaker forms of specification for maps on uniform spaces

We introduce and study here some weaker forms of specification property for uniformly continuous surjective self-maps on uniform spaces, namely topological quasi-weak specification property, topological semi-weak specification property and topological periodic quasi-weak specification property. We also introduce and study the notion of topological specification point for uniformly continuous surjective self-maps on uniform spaces. It is proved that for uniformly continuous surjective self-maps on uniform spaces the pointwise topological periodic specification property implies Devaney chaos. Moreover, the relation between pointwise notions of mixing, topological shadowing and topological specification is explored. It is shown that for uniformly continuous self-maps on uniform spaces the existence of two distinct topological specification points implies that the map has positive uniform entropy. Also, the limiting behaviour of a topological specification point under orbital convergence of maps is studied.

Chaotic Behaviors of a Delay Partial Difference Equation with a Delay Controller

A delay partial difference equation with a delay controller is studied in this paper. We show that it can lead to both Devaney chaos and Li–Yorke chaos by applying the modified Marotto’s theorem, with three criteria established for generating chaos. Computer simulations of chaotic behaviors and spatiotemporal graphs are performed with three examples.

Devaney chaos of a set-valued map and its inverse limit

We study relationships between a set-valued map and its inverse limit about the notion of periodic point set, transitivity, sensitivity and Devaney chaos. We show that periodic point set of a set-valued map is dense if and only if periodic point set of the inverse limit with the set-valued map is dense. Sensitivity of a set-valued map and its inverse limit are independent of each other. If the inverse limit with the set-valued map is chaotic in the sense of Devaney (respectively, transitive), then the set-valued map is chaotic in the sense of Devaney (respectively, transitive).

Lyapunov Stability in the Cournot Duopoly Game

This paper studies Lyapunov stability at a point and its application in the Cournot duopoly game. We first explore the relationship between Lyapunov stability at a point, nonsensitivity and non-Devaney chaos and find that a dynamical system is nonsensitive and non-Devaney chaotic if there is a point in this system such that it is Lyapunov stable at that point. We next prove a group of equivalent characterizations of Lyapunov stability at a point to deduce the composite theorems and product theorems of Lyapunov stability at a point, and then we prove three equivalent characterizations of the Cournot duopoly system to demonstrate that this system is Lyapunov stable at its unique nonzero fixed point (Cournot equilibrium point) when the unit costs of the Cournot double oligarchies satisfy certain conditions. Therefore, we conclude that the Cournot duopoly system is safe relative to both sensitivity and Devaney chaos. The robustness of our results are also verified by conducting numerical simulations in the Cournot duopoly game.

Dynamical analysis in controlled globally coupled map lattices

This paper investigates a globally coupled map lattice. Rigorous proofs to the existence of chaos in the sense of both Li–Yorke and Devaney in two controlled globally coupled map lattices are presented. In addition, the existence of Li–Yorke chaos and Devaney chaos for a general discrete dynamical system in {mathbf {R}}^{N} and l^{infty} is considered. For illustration, two examples are provided.

New definitions of Devaney's chaos

We give equivalent definitions of a Devaney chaotic semiflow on a metric space in terms of eventual sensitivity and topological sensitivity. As a consequence, we prove two versions of Auslander-Yorke dichotomy for the case of not necessarily compact phase spaces. We also raise some questions.

Chaos and Shadowing in General Systems

In this paper we describe some basic notions of topological dynamical systems for maps of type f : X × X → X named general systems. This is proved that every uniformly expansive general system has the shadowing property and every uniformly contractive general system has the (asymptotic) average shadowing and shadowing properties. In the rest, Devaney chaos for general systems is considered. Also, we show that topological transitivity and density of periodic points of a general systems imply topological ergodicity. We also obtain some results on the topological mixing and sensitivity for general systems.

Dynamics of interval maps generated by erasing substitutions

We study the discrete dynamics of interval maps generated by the action of erasing block substitutions on the binary expansion. We propose a classification of erasing block substitutions in a hierarchy of classes displaying progressively stronger erasing character. We investigate how this affects the dynamics of the corresponding interval maps, showing that the richest dynamical behavior (Devaney and Li-Yorke chaos, infinite topological entropy) is achieved at a precise step in this hierarchy, which we name completely erasing substitutions.

Li-Yorke chaos and topological distributional chaos in a sequence

We study here the topological notion of Li-Yorke chaos defined for uniformly continuous self-maps defined on uniform Hausdorff spaces, which are not necessarily compact metrizable. We prove that a weakly mixing uniformly continuous self-map defined on a second countable Baire uniform Hausdorff space without isolated points is Li-Yorke chaotic. Further, we define and study the notion of topological distributional chaos in a sequence for uniformly continuous self-maps defined on uniform Hausdorff spaces. We prove that Li-Yorke chaos is equivalent to topological distributional chaos in a sequence for uniformly continuous self-maps defined on second countable Baire uniform Hausdorff space without isolated points. As a consequence, we obtain that Devaney chaos implies topological distributional chaos in a sequence.

Chaos on Fuzzy Dynamical Systems

Given a continuous map f:X→X on a metric space, it induces the maps f¯:K(X)→K(X), on the hyperspace of nonempty compact subspaces of X, and f^:F(X)→F(X), on the space of normal fuzzy sets, consisting of the upper semicontinuous functions u:X→[0,1] with compact support. Each of these spaces can be endowed with a respective metric. In this work, we studied the relationships among the dynamical systems (X,f), (K(X),f¯), and (F(X),f^). In particular, we considered several dynamical properties related to chaos: Devaney chaos, A-transitivity, Li–Yorke chaos, and distributional chaos, extending some results in work by Jardón, Sánchez and Sanchis (Mathematics 2020, 8, 1862) and work by Bernardes, Peris and Rodenas (Integr. Equ. Oper. Theory 2017, 88, 451–463). Especial attention is given to the dynamics of (continuous and linear) operators on metrizable topological vector spaces (linear dynamics).

A type of shadowing and distributional chaos

For any continuous self-map of a compact metric space, we prove a saturation of distributionally scrambled Mycielski sets under a type of shadowing and the chain transitivity. This extends a result of J. Li, J. Li, and S. Tu, Devaney chaos plus shadowing implies distributional chaos, Chaos 26 (2016), 093103, 6 pp.

Transitive abelian nonautonomous dynamical systems

We consider abelian nonautonomous dynamical systems F=(fn) on compact spaces X which are moreover transitive; that is, the limit set of some point x∈X coincides with the whole space X, and we extend some results of [1]. More precisely, we show that such F must be either pointwise sensitive or uniformly rigid and almost equicontinuous. Moreover, we prove that if the set of equicontinuity points of F has nonempty interior, then it coincides with the set of transitive points and F will be equicontinuous if and only if it is minimal. Next, we assume that the sequence (fn) converges uniformly to a transitive map f, and we prove that if f is minimal or equicontinuous then F is either sensitive or equicontinuous, minimal and uniformly rigid. We also study chaos for F, and we prove that Devaney chaos implies Li-Yorke chaos.

Sensitivity and chaos on product and on hyperspatial semiflows

ABSTRACT We study stronger forms of sensitivity on the arbitrary product of semiflows on Hausdorff uniform spaces. Moreover, Devaney chaos, Poincaré chaos, and their related properties are studied on product semiflows. Furthermore, we study relations between the chaos-related properties of a semiflow and that of its induced hyperspatial semiflow. Some results regarding stronger forms of sensitivity are also obtained on hyperspatial semiflows.

On transitive and chaotic dynamics of linear semiflows

In this paper, we study the underlying relationship between transitivity and chaos for linear semiflows, which provides a better understanding of chaotic dynamics in topological spaces. We show that syndetically transitive linear semiflows with abelian acting semigroups are thickly syndetically-transitive and weakly mixing. As a result, Devaney chaos implies the weak mixing for linear semiflows with abelian acting semigroups. Besides, we give the characterizations for multi-dimensional Li-Yorke chaotic linear semiflows and show that if a linear semiflow (S,X) on a Polish space satisfies one of the following conditions: (i) S is discrete; (ii) dimX=∞; (iii) S is abelian, then Devaney chaos implies multi-dimensional Li-Yorke chaos. This answers a latest question posed in [12] (Question 2.19) to some extent. We also show that any topologically transitive linear semiflow (S,X) on a Polish space X with a translation-invariant metric appears to be densely Li-Yorke chaotic if IntX(Bx)=∅ for any compact subset B of S and any transitive point x∈X. Moreover, if the acting semigroup S is abelian, we present a weaker condition for the densely multi-dimensional Li-Yorke chaos.

Abstract similarity, fractals and chaos

<p style='text-indent:20px;'>A new mathematical concept of abstract similarity is introduced and is illustrated in the space of infinite strings on a finite number of symbols. The problem of chaos presence for the Sierpinski fractals, Koch curve, as well as Cantor set is solved by considering a natural similarity map. This is accomplished for Poincaré, Li-Yorke and Devaney chaos, including multi-dimensional cases. Original numerical simulations illustrating the results are presented.

The C1 persistence of heteroclinic repellers in [formula omitted

In this paper, we show that if f is a C1-map from Rn into itself and has heteroclinic repellers, then g also has heteroclinic repellers with ‖f−g‖C1 being small enough and exhibits Devaney's chaos. The results demonstrate C1 structural stability of heteroclinic repellers in Euclidean spaces. In the end, we give some examples to illustrate our theoretical results.

Ultragraph shift spaces and chaos

Motivated by C*-algebra theory, ultragraph edge shift spaces generalize shifts of finite type to the infinite alphabet case. In this paper we study several notions of chaos for ultragraph shift spaces. More specifically, we show that Li-Yorke, Devaney and distributional chaos are equivalent conditions for ultragraph shift spaces, and characterize this condition in terms of a combinatorial property of the underlying ultragraph. Furthermore, we prove that such properties imply the existence of a compact, perfect set which is distributionally scrambled of type 1 in the ultragraph shift space (a result that is not known for a labelled edge shift (with the product topology) of an infinite graph).

Some Chaos Notions on Dendrites

Transitivity is a key element in a chaotic dynamical system. In this paper, we present some relations between transitivity, stronger and alternative notions of it on compact and dendrite spaces. The relation between Auslander and Yorke chaos and Devaney chaos on dendrites is also discussed. Moreover, we prove that Devaney chaos implies strong dense periodicity on dendrites while the converse is not true.

Dynamics of Weakly Mixing Nonautonomous Systems

For a commutative nonautonomous dynamical system we show that topological transitivity of the nonautonomous system induced on probability measures (hyperspaces) is equivalent to the weak mixing of the induced systems. Several counter examples are given for the results which are true in autonomous but need not be true in nonautonomous systems. Wherever possible sufficient conditions are obtained for the results to hold true. For a commutative periodic nonautonomous system on intervals, it is proved that weak mixing implies Devaney chaos. Given a periodic nonautonomous system, it is shown that sensitivity is equivalent to some stronger forms of sensitivity on a closed unit interval.