Smooth Dynamics Becomes Hybrid in the Limit

Abstract

We show the appearance of an essentially nonlocal dynamics describing the limit behavior of trajectories of a class of dynamical systems defined by classical autonomous ODEs with smooth right-hand sides containing a small parameter and becoming discontinuous in the formal limit. The limit dynamics is shown to be described by an explicitly constructed Nerode--Kohn hybrid dynamical system consisting of a continuous plant (an ODE) and a finite state machine which are interacting and producing hybrid dynamics with possible memory effects. We remark, however, that, from the “statistical” point of view, the limit behavior of an ensemble of trajectories can still be described by an ODE, with possibly time-dependent and discontinuous right-hand side depending on the chosen ensemble.