  The focus of this work is on local stability of a class of nonlinear ordinary
differential equations (ODE) that describe limits of empirical measures
associated with finite-state weakly interacting N-particle systems. Local
Lyapunov functions are identified for several classes of such ODE, including
those associated with systems with slow adaptation and Gibbs systems. Using
results from [5] and large deviations heuristics, a partial differential
equation (PDE) associated with the nonlinear ODE is introduced and it is shown
that positive definite subsolutions of this PDE serve as local Lyapunov
functions for the ODE. This PDE characterization is used to construct explicit
Lyapunov functions for a broad class of models called locally Gibbs systems.
This class of models is significantly larger than the family of Gibbs systems
and several examples of such systems are presented, including models with
nearest neighbor jumps and models with simultaneous jumps that arise in
applications.
