2020/06/07 by Dai Zusai, Zusai, Dai
Biochemistry, Genetics and Molecular Biology · Decision Sciences · Social Sciences · #Gene Regulatory Network Analysis #Game Theory and Applications #Evolutionary Game Theory and Cooperation
paper · pdf · doi:10.48550/arxiv.2006.04280
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.