Topological entropy for local processes

Abstract

The aim of this article is to formalize definition of chaos (in terms of topological entropy) for dynamics of processes described by nonautonomous differential equations. We state a formal definition of topological entropy in this setting and provide tools for estimation of its value (its upper or lower bounds) in terms of Poincaré sections.

Ivanov’s Theorem for Admissible Pairs Applicable to Impulsive Differential Equations and Inclusions on Tori

“…For a positive topological entropy, Theorem 1 was effectively applied to discrete chaotic dynamics in [5][6][7][8] and to chaotic impulsive differential equations on tori in [9]. Let us note that the results dealing with the topological entropy for dynamic processes and differential equations from another perspective are rather rare (see e.g., [10][11][12][13][14]), and those for multivalued dynamics are even more delicate (see e.g., [15]).…”

Entropy of Nonautonomous Dynamical Systems

“…Together with Misiurewicz, they generalized this concept to sequences of maps between possibly different metric spaces in [15] and proved analogues of the Misiurewicz-Szlenk formula for the entropy of piecewise monotone interval maps. Further work on topological entropy of nonautonomous systems has been done in [18,20,21,22,23,24,25] by several researchers with different motivations and partially independently of [14,15]. An essential difference to the classical theory that should be mentioned is that the nonautonomous version of topological entropy is not a purely topological quantity.…”

Metric Entropy of Nonautonomous Dynamical Systems

“…The second exception is the quantity introduced in Kolyada and Snoha [13], the topological entropy of a nonautonomous system given as a discrete-time deterministic process on a compact topological space. The theory founded in [13] has been further developed in [9,10,14,20,22,26,28,29] by several authors. In some of these articles, the definition of entropy has been generalized, in particular to continuous-time systems, to systems with noncompact state space, systems with time-dependent state space, and to local processes.…”