Smooth Morse-Lyapunov Functions and Morse Theory of Strong Attractors for Nonsmooth Dynamical Systems

TL;DR

This paper develops smooth Morse-Lyapunov functions for nonsmooth dynamical systems, explores the homotopy type of attractor neighborhoods, and introduces critical groups and Morse inequalities for Morse sets.

Contribution

It introduces a method to construct smooth Morse-Lyapunov functions for nonsmooth systems and establishes Morse theory results for strong attractors.

Findings

All open attractor neighborhoods share the same homotopy type.

Defined critical groups for Morse sets of attractors.

Established Morse inequalities and equations for nonsmooth dynamical systems.

Abstract

In this paper we first construct smooth Morse-Lyapunov functions of attractors for nonsmooth dynamical systems. Then we prove that all open attractor neighborhoods of an attractor have the same homotopy type. Based on this basic fact we finally introduce the concept of critical group for Morse sets of an attractor and establish Morse inequalities and equations.