On forward invariance in Lyapunov stability theorem for local stability

Dai Zusai

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arXiv · 2006.04280

On forward invariance in Lyapunov stability theorem for local stability

Dai Zusai

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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.

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Dai Zusai. 2020-06-07. On forward invariance in Lyapunov stability theorem for local stability. https://arxiv.org/abs/2006.04280

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