Canard-like phenomena in piecewise-smooth Van der Pol systems

Abstract

We show that a nonlinear, piecewise-smooth, planar dynamical system can exhibit canard phenomena. Canard solutions and explosion in nonlinear, piecewise-smooth systems can be qualitatively more similar to the phenomena in smooth systems than piecewise-linear systems, since the nonlinearity allows for canards to transition from small cycles to canards ``with heads." The canards are born of a bifurcation that occurs as the slow-nullcline coincides with the splitting manifold. However, there are conditions under which this bifurcation leads to a phenomenon called super-explosion, the instantaneous transition from a globally attracting periodic orbit to relaxations oscillations. Also, we demonstrate that the bifurcation---whether leading to canards or super-explosion---can be subcritical.

17 pages, 11 figures

References in corpus (3)

• Andronov-Hopf Bifurcations in Planar, Piecewise-Smooth, Continuous Flows
• Relaxation oscillations in an idealized ocean circulation model
• Mixed mode oscillations in a conceptual climate model

Cited by in corpus (4)

• Relaxation oscillations in an idealized ocean circulation model
• The instantaneous local transition of a stable equilibrium to a chaotic attractor in piecewise-smooth systems of differential equations
• Dimension reduction for slow-fast, piecewise-smooth, continuous systems of ODEs
• Canards, folded nodes and mixed-mode oscillations in piecewise-linear slow-fast systems