Devaney Chaos Research Articles

Devaney chaos and stronger forms of sensitivity on the product of semiflows

In this paper, we discuss transitivity and sensitivity problems on product of two semiflows. Further, we study various chaos related properties and stronger forms of sensitivity on the countably infinite product of semiflows under the action of the most general possible monoids. Examples are provided wherever necessary to support our results.

Analyzing Devaney Chaos of a Sine–Cosine Compound Function System

The one-dimensional Sine map and Chebyshev map are classical chaotic maps, which have clear chaotic characteristics. In this paper, we establish a chaotic framework based on a Sine–Cosine compound function system by analyzing the existing one-dimensional Sine map and Chebyshev map. The sensitive dependence on initial conditions, topological transitivity and periodic-point density of this chaotic framework is proved, showing that the chaotic framework satisfies Devaney’s chaos definition. In order to illustrate the chaotic behavior of the chaotic framework, we propose three examples, called Cosine–Polynomial (C–P) map, Sine–Tangent (S–T) map and Sine–Exponent (S–E) map, respectively. Then, we evaluate the chaotic behavior with Sine map and Chebyshev map by analyzing bifurcation diagrams, Lyapunov exponents, correlation dimensions, Kolmogorov entropy and [Formula: see text] complexity. Experimental results show that the chaotic framework has better unpredictability and more complex chaotic behaviors than the classical Sine map and Chebyshev map. The results also verify the effectiveness of the theoretical analysis of the proposed chaotic framework.

A remark on banks theorem

ABSTRACTFor discrete autonomous dynamical systems, it was found that in the three conditions defining Devaney chaos, topological transitivity and dense periodic points together imply sensitive dependence on initial condition (Banks, Brooks, Cairns, Davis and Stacey, On Devaney's definition of chaos, The Amer. Math. Monthly 99 (1992), pp. 332–334). In this paper, we give the definition of finitely generated non-autonomous dynamical systems (NADS) and generalize the Banks et al. theorem to the finitely generated NADS.

Chaos theory from the mathematical point of view

This work is intended as an attempt to survey existingde finitions of chaos for discrete dynamical systems. Discussion is restricted to the settingof topological dynamics, while the measure-theoretic (ergodic theory) and smooth (differentiable dynamical systems) aspects are omitted as exceedingt he scope of this paper. Chaos theory is understood here as a part of topological dynamics, so aforementioned definitions of chaos are just examples of particular dynamical system properties, and are considered inside the framework of the mathematical theory of discrete dynamical systems. It is not the purpose of this article to study chaos theory understood as a new kind of interdisciplinary branch of science devoted to nonlinear phenomena. As for prerequisites, the reader is expected to possess some mathematical maturity, and to be familiar with basic topology of (compact) metric spaces. No preliminary knowledge of the dynamical systems theory is required, however some is recommended. The first two section are devoted to general discussion of the term „chaos” and contains authors opinion on this subject. To facilitate access to the rest of the article some relevant material from the dynamical system theory is briefly repeated in the third section. The next section (Section 4) introduces the notion of topological transitivity along with some stronger variants, namely topological mixing and weak mixing. Section 5 gives a detailed account of the famous Sharkovskii’s Theorem in its full generality. This is required for characterization of chaotic interval maps. Sections 6-13 are devoted to various notions of chaos or related to chaos in dynamical systems. Each section contains an attempt to motivate the notion, historical background and formal definition followed with a review of known properties, relations between various notions of chaos, and some relevant open problems. Section 6 is devoted to a sensitivity to initial conditions – a notion which is accepted as a basic indicator of chaotic behavior. Section 7 introduces a definition of chaos accordingt o Auslander and Yorke. Section 8 examines the notion of Li-Yorke pair and Li-Yorke chaos. Section 9 deals with the definition of chaos introduced in Devaney’s book (Devaney chaos). Section 10 recalls some facts connected with symbolic dynamics, which provides a rich source of examples for various interestingb ehavior, and it is an indispensable tool for exploration of many systems. Section 11 describes the so-called “topological horseshoes”, which are generalizations of the famous example due to Smale. The existence of a horseshoe in a given dynamical system proves the existence of a subsystem with a dynamics similar to some symbolic dynamical system, hence with a very complicated behavior. Section 12 gives a brief exposition of the topological entropy and its relation to chaos. The review of various notions of chaos ends with section 13, containingd escription of distributional chaos.

Dynamics on Binary Relations over Topological Spaces

The existence of chaos and the quest of dense orbits have been recently considered for dynamical systems given by multivalued linear operators. We consider the notions of topological transitivity, topologically mixing property, hypercyclicity, periodic points, and Devaney chaos in the general case of binary relations on topological spaces, and we analyze how they can be particularized when they are represented with graphs and digraphs. The relations of these notions with different types of connectivity and with the existence of Hamiltonian paths are also exposed. Special attention is given to the study of dynamics over tournaments. Finally, we also show how disjointness can be introduced in this setting.

Disjoint Hypercyclicity and Topological Entropy of Composition Operators

In a previous paper, we characterized the Devaney chaos, frequent hypercyclicity and dense distributional chaos of composition operators induced by continuous self-maps on the real line. The present paper further investigates the disjoint hypercyclicity and topological entropy of these operators. It is shown that the composition operator is [Formula: see text]-transitive if and only if it is Cesàro-hypercyclic, if and only if it is supercyclic, if and only if it has the specification property on the whole space. Furthermore, sufficient and necessary conditions for a pair of composition operators to be disjoint hypercyclic (disjoint mixing, respectively) are obtained. Finally, sufficient conditions for the composition operator to admit infinite topological entropy are provided.

Sensitive dependence for nonautonomous discrete dynamical systems

Given a nonautonomous discrete dynamical system (NDS) (X,f1,∞) we show that transitivity and density of periodic points do not imply sensitivity in general, i.e., in the definition of Devaney chaos there are no redundant conditions for NDS. In addition, we show that if we also assume uniform convergence of the sequence (fn) that induces the NDS, then sensitivity follows. Furthermore, in contrast to the autonomous case, we show that there exist minimal NDS which are neither equicontinuous nor sensitive.

Chaotic behaviors of one dimensional wave equations with van der Pol nonlinear boundary conditions

For one-dimensional linear wave equations with van der Pol boundary conditions, there have existed several different ways in the literature to characterize the complexity of their solutions. In this paper, we give a new way to characterize the dynamical behaviors of the systems, including almost equicontinuity and Devaney chaos. In order to do this, we consider a one-dimensional system (I, f) and its functional envelope (L1(I,I),Hf) and show that Hf is weakly mixing if and only if f is; Hf is Devaney chaotic if f is weakly mixing; Hf is almost equicontinuous if f has an attracting 2-period orbit. Those abstract results have their own significance and can be applied to the linear wave equations with van der Pol boundary conditions.

Devaney's Chaos for Maps on $G$-spaces

We study the notion of sensitivity on $G$-spaces and through examples observe that $G$-sensitivity neither implies nor is implied by sensitivity. Further, we obtain necessary and sufficient conditions for a map to be $G$-sensitive. Next, we define the notion of Devaney's chaos on $G$-space and show that $G$-sensitivity is a redundant condition in the definition.

Set-Valued Chaos in Linear Dynamics

We study several notions of chaos for hyperspace dynamics associated to continuous linear operators. More precisely, we consider a continuous linear operator $$T : X \rightarrow X$$ on a topological vector space X, and the natural hyperspace extensions $$\overline{T}$$ and $$\widetilde{T}$$ of T to the spaces $$\mathcal {K}(X)$$ of compact subsets of X and $$\mathcal {C}(X)$$ of convex compact subsets of X, respectively, endowed with the Vietoris topology. We show that, when X is a complete locally convex space (respectively, a locally convex space), then Devaney chaos (respectively, topological ergodicity) is equivalent for the maps T, $$\overline{T}$$ and $$\widetilde{T}$$ . Also, under very general conditions, we obtain analogous equivalences for Li-Yorke chaos. Finally, some remarks concerning the topological transitivity and weak mixing properties are included, extending results in Banks (Chaos Solitons Fractals 25(3):681–685, 2005) and Peris (Chaos Solitons Fractals 26(1):19–23, 2005).

Chaotic semigroups from second order partial differential equations

We give general conditions on given parameters to ensure Devaney and distributional chaos for the solution C0-semigroup corresponding to a class of second-order partial differential equations. We also provide a critical parameter that led us to distinguish between stability and chaos for these semigroups. In the case of chaos, we prove that the C0-semigroup admits a strongly mixing measure with full support. We also give concrete examples of partial differential equations, such as the telegraph equation, whose solutions satisfy these properties.

Dynamics of multivalued linear operators

Abstract We introduce several notions of linear dynamics for multivalued linear operators (MLO’s) between separable Fréchet spaces, such as hypercyclicity, topological transitivity, topologically mixing property, and Devaney chaos. We also consider the case of disjointness, in which any of these properties are simultaneously satisfied by several operators. We revisit some sufficient well-known computable criteria for determining those properties. The analysis of the dynamics of extensions of linear operators to MLO’s is also considered.

Devaney chaos, Li–Yorke chaos, and multi-dimensional Li–Yorke chaos for topological dynamics

Let π:T×X→X, written T↷πX, be a topological semiflow/flow on a uniform space X with T a multiplicative topological semigroup/group not necessarily discrete. We then prove:•If T↷πX is non-minimal topologically transitive with dense almost periodic points, then it is sensitive to initial conditions. As a result of this, Devaney chaos ⇒ Sensitivity to initial conditions, for this very general setting. Let R+↷πX be a C0-semiflow on a Polish space; then we show:•If R+↷πX is topologically transitive with at least one periodic point p and there is a dense orbit with no nonempty interior, then it is multi-dimensional Li–Yorke chaotic; that is, there is a uncountable set Θ⊆X such that for any k≥2 and any distinct points x1,…,xk∈Θ, one can find two time sequences sn→∞,tn→∞ withsn(x1,…,xk)→(x1,…,xk)∈Xkandtn(x1,…,xk)→(p,…,p)∈ΔXk. Consequently, Devaney chaos ⇒ Multi-dimensional Li–Yorke chaos. Moreover, let X be a non-singleton Polish space; then we prove:•Any weakly-mixing C0-semiflow R+↷πX is densely multi-dimensional Li–Yorke chaotic.•Any minimal weakly-mixing topological flow T↷πX with T abelian is densely multi-dimensional Li–Yorke chaotic.•Any weakly-mixing topological flow T↷πX is densely Li–Yorke chaotic. We in addition construct a completely Li–Yorke chaotic minimal SL(2,R)-acting flow on the compact metric space R∪{∞}. Our various chaotic dynamics are sensitive to the choices of the topology of the phase semigroup/group T.

Chaotic Dynamics of Composition Operators on the Space of Continuous Functions

In this paper, the chaotic dynamics of composition operators on the space of real-valued continuous functions is investigated. It is proved that the hypercyclicity, topologically mixing property, Devaney chaos, frequent hypercyclicity and the specification property of the composition operator are equivalent to each other and are stronger than dense distributional chaos. Moreover, the composition operator [Formula: see text] exhibits dense Li–Yorke chaos if and only if it is densely distributionally chaotic, if and only if the symbol [Formula: see text] admits no fixed points. Finally, the long-time behaviors of the composition operator with affine symbol are classified in detail.

The specification property and infinite entropy for certain classes of linear operators

We study the specification property and infinite topological entropy for two specific types of linear operators: translation operators on weighted Lebesgue function spaces and weighted backward shift operators on sequence F-spaces. It is known, from the work of Bartoll, Martínez-Giménez, Murillo-Arcila, and Peris, that for weighted backward shift operators, the existence of a single non-trivial periodic point is sufficient for specification. We show this also holds for translation operators on weighted Lebesgue function spaces. This implies, in particular, that for these operators, the specification property is equivalent to Devaney chaos. We show that these forms of chaos (unsurprisingly) imply infinite topological entropy, but that (surprisingly) the converse does not hold.

Devaney Chaos Induced by Turbulent and Erratic Functions

Devaney Chaos Induced by Turbulent and Erratic Functions

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.

Operators with the specification property

We study a version of the specification property for linear dynamics. Operators having the specification property are investigated, and relationships with other well known dynamical notions such as mixing, Devaney chaos, and frequent hypercyclicity are obtained.

On Devaney Chaos and Dense Periodic Points: Period 3 and Higher Implies Chaos

We look at density of periodic points and Devaney Chaos. We prove that if f is Devaney Chaotic on a compact metric space with no isolated points, then the set of points with prime period at least n is dense for each n. Conversely, we show that if f is a continuous function from a closed interval to itself, for which the set of points with prime period at least n is dense for each n, then there is a decomposition of the interval into closed subintervals on which either f or f2 is Devaney Chaotic. (In fact, this result holds if the set of points with prime period at least 3 is dense.)

On Different Types of Chaos for $\mathbb{Z}^d$-Actions

In this paper we obtain a characterization of $k$-typetransitivity for a $\mathbb{Z}^d$-action on certain spaces andthen prove that $k$-type SDIC is redundant in the definition of$k$-type Devaney chaos for $\mathbb{Z}^d$-actions on infinitemetric spaces. We define different types of chaos for$\mathbb{Z}^d$-actions and prove results related to theirpreservations under conjugacy and uniform conjugacy. Finally wediscuss $k$-type properties on product spaces.