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TUPD-2021-003

TITLE Lyapunov’s direct method for stability of a set and its transitivity under a differential inclusion
AUTHOR Dai ZUSAI

Associate Professor, Graduate School of Economics and Management, Tohoku University

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ABSTRUCT

We present a version of Lyapunov’s direct method for stability of a set under a differential inclusion. We pay careful attention to the assumption of forward invariance of a basin of attraction, which is often overlooked when applying the method to local stability. Even if the value of a local Lyapunoov function monotonically changes in some neighborhood of the limit set, this alone does not prevent a trajectory from escaping from this particular neighborhood. In this note, we verify that we can construct a smaller but forward invariant neighborhood. As a corollary, we obtain a transitivity theorem on basins of attractions without requiring forward invariance.

KEYWORDS Lyapunov function, stability of a set, forward invariance, evolutionary dynamics.
ISSUED May 2021

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