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

Jan Andres,Jerzy Jezierski

doi:10.3390/math8091602

Abstract

The main aim of this article is two-fold: (i) to generalize into a multivalued setting the classical Ivanov theorem about the lower estimate of a topological entropy in terms of the asymptotic Nielsen numbers, and (ii) to apply the related inequality for admissible pairs to impulsive differential equations and inclusions on tori. In case of a positive topological entropy, the obtained result can be regarded as a nontrivial contribution to deterministic chaos for multivalued impulsive dynamics.

Highlights

In 1982, Ivanov formulated in [1] his remarkable inequality in the form of the following theorem.Theorem 1. Let f be a continuous self-map of a compact polyhedron
The notion of a topological entropy can be understood in the sense of both [2,3]
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]

Summary

Introduction

In 1982, Ivanov formulated in [1] his remarkable inequality in the form of the following (slightly more precise) theorem. (cf [1] Theorem) Let f be a (single-valued) continuous self-map of a compact polyhedron. We need a new definition of a topological entropy for the class of (multivalued) admissible maps. On this basis, we will be able to generalize Theorem 1 in a desired way, i.e., in order to be applied to impulsive differential inclusions on tori. ANR-spaces and admissible maps in the sense of Górniewicz, for which we will newly define the topological entropy. We are in position to define the topological entropy for admissible pairs ( p, q), when following the definition of Bowen [3] for single-valued maps.

Elements of the Nielsen Theory for Multivalued Maps

Ivanov’s Theorem for Admissible Pairs

Application to Impulsive Differential Inclusions on Tori

Concluding Remarks