Multiple Periodic Solutions and Fractal Attractors of Differential Equations with n-Valued Impulses

Jan Andres

doi:10.3390/math8101701

Abstract

Ordinary differential equations with n-valued impulses are examined via the associated Poincaré translation operators from three perspectives: (i) the lower estimate of the number of periodic solutions on the compact subsets of Euclidean spaces and, in particular, on tori; (ii) weakly locally stable (i.e., non-ejective in the sense of Browder) invariant sets; (iii) fractal attractors determined implicitly by the generating vector fields, jointly with Devaney’s chaos on these attractors of the related shift dynamical systems. For (i), the multiplicity criteria can be effectively expressed in terms of the Nielsen numbers of the impulsive maps. For (ii) and (iii), the invariant sets and attractors can be obtained as the fixed points of topologically conjugated operators to induced impulsive maps in the hyperspaces of the compact subsets of the original basic spaces, endowed with the Hausdorff metric. Five illustrative examples of the main theorems are supplied about multiple periodic solutions (Examples 1–3) and fractal attractors (Examples 4 and 5).

Highlights

The theory of impulsive differential equations and inclusions has been systematically developed, among other things, especially because of many practical applications
The topological fixed point theory for multivalued maps has been developed in two main directions: (i) for admissible maps and their particular cases like acyclic maps, Rδ -maps, and so forth, and (ii) for n-valued maps and their generalizations like n-acyclic maps and weighted maps
If φ : S1 ( S1 is an n-valued map of degree Deg( φ), N ( φ) := |n − Deg( φ)| holds for the Nielsen number N ( φ) of φ, and there is an n-valued map, say ψ, homotopic to φ (i.e., ψ ∼ φ), that has exactly |n − Deg( φ)| = |n − Deg(ψ)| fixed points

Summary

Introduction

The theory of impulsive differential equations and inclusions has been systematically developed (see e.g., the monographs [1,2,3,4], and the references therein), among other things, especially because of many practical applications (see e.g., References [1,4,5,6,7,8,9,10,11]). Relaxing the strict requirement of exactly n-values in Definition 1 below, we can consider the union operators of n single-valued maps, called the Hutchinson-Barnsley operators These operators play a crucial role in constructing the fractals as attractors of iterated function systems (see References [23,24]). The application of the deep results for the iterated function systems, including the chaotic dynamics on the Hutchinson-Barnsley (fractal) attractors, to impulsive differential equations via the Poincaré translation operators along the trajectories is quite original Besides these two novelty applications, our research in this field can be justified by a simple argument that the n-valued impulses extend with no doubts the variability in practical applications.

Preliminaries

N-Valued Maps

Hutchinson-Barnsley’s Operators

Application to Impulsive Differential Equations in Rn

Fractals and Chaos Determined by Impulsive Differential Equations

Concluding Remarks