On Intervals, Transitivity = Chaos

Vellekoop, Michel; Berglund, Raoul

doi:10.1080/00029890.1994.11996955

Cited by 66 publications

(41 citation statements)

References 2 publications

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“…The notion of chaos is very appealing, and it has intrigued many scientists (see [2,3,14,17,20] for some works on the properties that characterize a chaotic process). In the case of discrete time dynamical systems (DTDS) defined on a metric space, many definitions of chaos are based on the notion of sensitivity (see for example [8,13,17]).…”

Section: Introductionmentioning

confidence: 99%

Lyapunov Exponents versus Expansivity and Sensitivity in Cellular Automata

Finelli¹,

Manzini

Margara³

1998

Journal of Complexity

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We establish a connection between the theory of Lyapunov exponents and the properties of expansivity and sensitivity to initial conditions for a particular class of discrete time dynamical systems; cellular automata (CA). The main contribution of this paper is the proof that all expansive cellular automata have positive Lyapunov exponents for almost all the phase space configurations. In addition, we provide an elementary proof of the non-existence of expansive CA in any dimension greater than 1. In the second part of this paper we prove that expansivity in dimension greater than 1 can be recovered by restricting the phase space to a suitable subset of the whole space. To this extent we describe a 2-dimensional CA which is expansive over a dense uncountable subset of the whole phase space. Finally, we highlight the different behavior of expansive and sensitive CA for what concerns the speed at which perturbations propagate.

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Section: Introductionmentioning

confidence: 99%

Lyapunov Exponents versus Expansivity and Sensitivity in Cellular Automata

Finelli¹,

Manzini

Margara³

1998

Journal of Complexity

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“…However, in a surprising way, Banks et al has proved that transitivity and periodically density imply sensitivity dependence (for details see [3]). Furthermore, for continuous functions on real intervals, Vellekoop and Berglund in [4] show that transitivity by itself is sufficient to get chaos. This last result is not necessarily true in other type of metric spaces (see Example 4.1 in [5]).…”

Section: Is Mixing Iff Given Two Non-empty Open Subsets U and V Of X mentioning

confidence: 99%

Chaos on Set-Valued Dynamics and Control Sets

Román-Flores¹,

Ayala²

2018

Chaos Theory

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The aim of this chapter is threefold. First, we show some advances in complexity dynamics of set-valued discrete systems in connection with the Devaney'snotionofchaos.Secondly , we start to explore some relationships between control sets for the class of linear control systems on Lie groups with chaotic sets. Finally, through several open problems, we invite the readers to give a contribution to this beauty theory.

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“… Finally, we move to the main result of the paper, showing that f(x) = x 3 + x,  < -3, is chaotic on F. By the popular definition of a chaotic function by R. L. Devaney [2], a real function f on set F is chaotic if a) f has sensitive dependence on initial conditions on F. b) f is topologically transitive on F and c) periodic points of f are dense in F. However the following result by Banks, Brooks, Cairns, David and Stacey [4] suggests that under some circumstances, if f satisfies the conditions of topological transitivity and density of periodic points, then f is chaotic, i.e. the sensitivity dependence follows automatically if the other two conditions are satisfied.…”

Section: The Construction Of Domain Of Chaosmentioning

confidence: 99%

“…A real quadratic family [2] f(x) = k x ( 1 -x ), k  R has indicated that even the simplest looking functions may have the complex dynamics. This paper has aimed to study the behavior of cubic family functions and its chaotic behavior.…”

Section:  1 Nmentioning

confidence: 99%

“…In fact, some such maps illustrate virtually every important phenomenon that occur in the Dynamical Systems. [2] A real quadratic family [3] f(x) = k x ( 1 -x ), k  R (set of real numbers), has resolutely indicated that even the simplest looking functions may have the complex dynamics. This indication has put every real function under suspicion and the simple looking cubic family functions of the form f : R → R, f(x) = x 3 + x,   R, are no exceptions.…”

Section: Introductionmentioning

confidence: 99%

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