Dimension reduction for slow-fast, piecewise-smooth, continuous systems of ODEs

TL;DR

This paper develops a method for dimension reduction in slow-fast, piecewise-smooth ODE systems by analyzing stability and invariant neighborhoods of critical manifolds, with applications to ocean circulation models.

Contribution

It introduces a new approach to analyze non-differentiable critical manifolds in piecewise-smooth systems, extending geometric singular perturbation theory.

Findings

Global stability implies forward invariance near the critical manifold

Dimension reduction is possible through regular perturbation in stable regions

Methodology applied to analyze boundary bifurcations in ocean circulation model

Abstract

The limiting slow dynamics of slow-fast, piecewise-linear, continuous systems of ODEs occurs on critical manifolds that are piecewise-linear. At points of non-differentiability, such manifolds are not normally hyperbolic and so the fundamental results of geometric singular perturbation theory do not apply. In this paper it is shown that if the critical manifold is globally stable then the system is forward invariant in a neighbourhood of the critical manifold. It follows that in this neighbourhood the dynamics is given by a regular perturbation of the dynamics on the critical manifold and so dimension reduction can be achieved. If the attraction is instead non-global, additional dynamics involving canards may be generated. For boundary equilibrium bifurcations of piecewise-smooth, continuous systems, the results are used to establish a general methodology by which such bifurcations can be analysed. This approach is illustrated with a three-dimensional model of ocean circulation.