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On forward invariance in Lyapunov stability theorem for local stability

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  • Dai Zusai

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. "On forward invariance in Lyapunov stability theorem for local stability," Papers 2006.04280, arXiv.org.
    • Handle: RePEc:arx:papers:2006.04280
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    References listed on IDEAS

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    1. , & , H. & ,, 2015. "Sampling best response dynamics and deterministic equilibrium selection," Theoretical Economics, Econometric Society, vol. 10(1), January.
    2. Dai Zusai, 2018. "Tempered best response dynamics," International Journal of Game Theory, Springer;Game Theory Society, vol. 47(1), pages 1-34, March.
    3. Dai Zusai, 2013. "Tempered Best Response Dynamics," DETU Working Papers 1301, Department of Economics, Temple University.
    4. Hofbauer, Josef & Sandholm, William H., 2009. "Stable games and their dynamics," Journal of Economic Theory, Elsevier, vol. 144(4), pages 1665-1693.4, July.
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    Keywords

    lyapunov function; local stability; forward invariance; evolutionary dynamics.;
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