Canard explosion and relaxation oscillation in planar, piecewise-smooth, continuous systems

TL;DR

This paper extends the understanding of canard explosions and relaxation oscillations to general planar, piecewise-smooth, continuous systems, identifying conditions for bifurcation types and the occurrence of super-explosions versus canards.

Contribution

It generalizes previous results by establishing conditions for bifurcation types in non-smooth systems beyond piecewise-linear cases.

Findings

Conditions for smooth and nonsmooth bifurcations identified

Criteria for super-explosion bifurcations established

Extension of canard explosion theory to broader classes of systems

Abstract

Classical canard explosion results in smooth systems require the vector field to be at least $C^3$, since canard cycles are created as the result of a Hopf bifurcation. The work on canards in nonsmooth, planar systems is recent and has thus far been restricted to piecewise-linear or piecewise-smooth Van der Pol systems, where an extremum of the critical manifold arises from the nonsmoothness. In both of these cases, a canard (or canard-like) explosion is created through a nonsmooth bifurcation as the slow nullcline passes through a corner of the critical manifold. Additionally, it is possible for these systems to exhibit a superexplosion bifurcation where the canard explosion is skipped. This paper extends the results to more general piecewise-smooth systems, finding conditions for when a periodic orbit is created through either a smooth or nonsmooth bifurcation. In the case the bifurcation is nonsmooth, conditions determining whether the bifurcation is a super-explosion or canards are created.