# Smooth dynamics becomes hybrid in the limit

created by stepanov on 25 Feb 2016
modified on 13 Jan 2017

[BibTeX]

Published Paper

Inserted: 25 feb 2016
Last Updated: 13 jan 2017

Journal: SIAM J. Control and Optimization
Volume: 55
Number: 1
Pages: 29-50
Year: 2017
Doi: 10.1137/16M1064611

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 \emph{hybrid} dynamical system consisting of a continuous plant (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.