On forward invariance in Lyapunov stability theorem for local stability

TL;DR

This paper addresses a gap in Lyapunov stability proofs by ensuring forward invariance of neighborhoods, enabling more natural proofs of local stability and basins of attraction without tracking individual solutions.

Contribution

It introduces a method to find smaller forward invariant neighborhoods, strengthening Lyapunov stability arguments and proving a transitivity theorem for basins of attraction.

Findings

Established a procedure to identify forward invariant neighborhoods

Enhanced Lyapunov stability proofs by avoiding solution path tracking

Proved a transitivity theorem for basins of attraction

Abstract

Forward invariance of a basin of attraction is often overlooked when using a Lyapunov stability theorem to prove local stability; even if the Lyapunov function decreases monotonically in a neighborhood of an equilibrium, the dynamic may escape from this neighborhood. In this note, we fix this gap by finding a smaller neighborhood that is forward invariant. This helps us to prove local stability more naturally without tracking each solution path. Similarly, we prove a transitivity theorem about basins of attractions without requiring forward invariance. Keywords: Lyapunov function, local stability, forward invariance, evolutionary dynamics.