Morse decomposition of scalar differential equations with state-dependent delay

TL;DR

This paper develops a Morse decomposition framework for scalar differential equations with state-dependent delays, revealing the structure of global attractors and dynamics through Lyapunov functions related to sign changes.

Contribution

It generalizes previous results for constant delays by constructing Morse decompositions for state-dependent delays under new conditions.

Findings

Morse sets relate to level sets of a sign-change counting Lyapunov function

Results apply to two major types of state-dependent delays

Provides insight into the global dynamics of such delay differential equations

Abstract

We consider state-dependent delay differential equations of the form $$\dot{x}(t) = f(x(t), x(t - r(x_t))),$$ where $f$ is continuously differentiable and fulfills a negative feedback condition in the delayed term. Under suitable conditions on $r$ and $f$, we construct a Morse decomposition of the global attractor, giving some insight into the global dynamics. The Morse sets in the decomposition are closely related to the level sets of an integer valued Lyapunov function that counts the number of sign changes along solutions on intervals of length of the delay. This generalizes former results for constant delay. We also give two major types of state-dependent delays for which our results apply.