A Conley-type Lyapunov function for the strong chain recurrent set

TL;DR

This paper constructs a continuous Lyapunov function for continuous flows on compact metric spaces, which decreases outside the strong chain recurrent set, generalizing Conley's theorem.

Contribution

It provides a constructive proof of a Lyapunov function that characterizes the strong chain recurrent set, extending Conley's fundamental theorem.

Findings

Existence of a continuous Lyapunov function for the strong chain recurrent set.

The Lyapunov function is strictly decreasing outside the set.

Generalization of Conley's theorem to the strong chain recurrent set.

Abstract

Let $\phi:X\times\mathbb{R} \rightarrow X$ be a continuous flow on a compact metric space $(X,d)$. In this article we constructively prove the existence of a continuous Lyapunov function for $\phi$ which is strictly decreasing outside $\mathcal{SCR}_d(\phi)$. Such a result generalizes Conley's Fundamental Theorem of Dynamical Systems for the strong chain recurrent set.