Dimension reduction for slow-fast, piecewise-linear ODEs and obstacles to a general theory

Abstract

This paper concerns systems of ODEs that are continuous, piecewise-linear (with two pieces) and slow-fast. The limiting slow dynamics occurs on a piecewise-linear critical manifold that corresponds to equilibria of the fast dynamics (layer equations). When the critical manifold is attracting for each piece of the layer equations, one would like to know when the full dynamics of the system near this manifold is a regular perturbation of the limiting slow dynamics (reduced system). We identify three phenomena unique to the piecewise-linear setting that obstruct such an extension of Fenichel theory: at the boundary (switching manifold) the equilibrium of the layer equations can be unstable, coincide with another attractor, or attract orbits in a chaotic fashion. The first two phenomena (at least) can inhibit a regular perturbation and lead to the creation of canards. If a regular perturbation does occur for the piecewise-linear approximation to a boundary equilibrium bifurcation, dimension reduction can be achieved and we illustrate this with a three-dimensional model of ocean circulation. We also introduce a slow-fast version of the observer canonical form for piecewise-linear ODEs that simplifies the analysis.