Existence of Lipschitz continuous Lyapunov functions strict outside the strong chain recurrent set

TL;DR

This paper establishes the existence of Lipschitz continuous Lyapunov functions that are strictly decreasing outside the strong chain recurrent set for continuous flows on compact metric spaces, extending previous results and characterizing recurrence sets.

Contribution

It constructs explicit Lipschitz Lyapunov functions for flows, extending recent work, and provides new characterizations and conditions relating Lyapunov functions to recurrence sets.

Findings

Lipschitz Lyapunov functions are strict outside the strong chain recurrent set.

Characterization of the strong chain recurrent set via Lipschitz Lyapunov functions.

Conditions for the existence of $ ext{C}^{1,1}$ strict Lyapunov functions in vector field flows.

Abstract

The aim of this paper is to study in detail the relations between strong chain recurrence for flows and Lyapunov functions. For a continuous flow on a compact metric space, uniformly Lipschitz continuous on the compact subsets of the time, we first make explicit a Lipschitz continuous Lyapunov function strict -that is strictly decreasing- outside the strong chain recurrent set of the flow. This construction extends to flows some recent advances of Fathi and Pageault in the case of homeomorphisms; moreover, it improves Conley's result about the existence of a continuous Lyapunov function strictly decreasing outside the chain recurrent set of a continuous flow. We then present two consequences of this theorem. From one hand, we characterize the strong chain recurrent set in terms of Lipschitz continuous Lyapunov functions. From the other hand, in the case of a flow induced by a vector field, we establish a sufficient condition for the existence of a $\mathcal{C}^{1,1}$ strict Lyapunov function and we also discuss various examples. Moreover, for general continuous flows, we show that the strong chain recurrent set has only one strong chain transitive component if and only if the only Lipschitz continuous Lyapunov functions are the constants. Finally, we provide a necessary and sufficient condition to guarantee that the strong chain recurrent set and the chain recurrent one coincide.