On forward invariance in Lyapunov stability theorem for local stability

Dai Zusai

Papers from arXiv.org

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.

Date: 2020-06
New Economics Papers: this item is included in nep-gth
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