Switching and metastability in singularly perturbed, nonlinear differential equations

We examine deterministic, nonlinear, singularly perturbed differential equations whose fast time scale behavior exhibits metastability: long periods of stability separated by rapid transitions between equilibria. The slow time scale behavior exhibits switching: discontinuous (and in some cases, nondeterministic) jumps. Metastability and switching can occur in many systems which are large collections of coupled, nonlinear subsystems such as one finds in models of electric generators coupled together by a transmission network. The approach taken here, based on bifurcation analysis, extends conventional asymptotic approximations (e.g., singular perturbation or multiple time scale analysis) to a large class of essentially non-linear systems including some multimachine models used in power system analysis. This approach also identifies some sources of nondeterminisim in otherwise deterministic-seeming systems, and it suggests new random and nondeterministic reduced order models of such systems.