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The focus of this work is on local stability of a class of nonlinear ordinarydifferential equations (ODE) that describe limits of empirical measuresassociated with finite-state weakly interacting N-particle systems. LocalLyapunov functions are identified for several classes of such ODE, includingthose associated with systems with slow adaptation and Gibbs systems. Usingresults from [5] and large deviations heuristics, a partial differentialequation (PDE) associated with the nonlinear ODE is introduced and it is shownthat positive definite subsolutions of this PDE serve as local Lyapunovfunctions for the ODE. This PDE characterization is used to construct explicitLyapunov functions for a broad class of models called locally Gibbs systems.This class of models is significantly larger than the family of Gibbs systemsand several examples of such systems are presented, including models withnearest neighbor jumps and models with simultaneous jumps that arise inapplications.

Local stability of Kolmogorov forward equations for finite state nonlinear Markov processes