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On forward invariance in Lyapunov stability theorem for local stability

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Document pages: 10 pages

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

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