Morse decomposition for semi-dynamical systems with an application to systems of state-dependent delay differential equations

TL;DR

This paper develops a unified framework for Morse decompositions in semi-dynamical systems with state-dependent delays, generalizing previous results and applying to threshold-type delay systems.

Contribution

It introduces a general approach to construct Morse decompositions using discrete Lyapunov functions for semi-dynamical systems on metric spaces.

Findings

Unified framework for Morse decompositions in semi-dynamical systems

Generalizes previous delay differential equation results

Application to systems with state-dependent delays

Abstract

Understanding the structure of the global attractor is crucial in the field of dynamical systems, where Morse decompositions provide a powerful tool by partitioning the attractor into finitely many invariant Morse sets and gradient-like connecting orbits. Building on Mallet-Paret's pioneering use of discrete Lyapunov functions for constructing Morse decompositions in delay differential equations, similar approaches have been extended to various delay systems, also including state-dependent delays. In this paper, we develop a unified framework assuming the existence and some properties of a discrete Lyapunov function for a semi-dynamical system on an arbitrary metric space, and construct a Morse decomposition of the global attractor in this general setting. We demonstrate that our findings generalize previous results; moreover, we apply our theorem to a cyclic system of differential equations with threshold-type state-dependent delay.