Numerical study of stable and unstable canards in a slow–fast hard-type van der Pol oscillator

Abstract

This Letter investigates the canards generated by a slow–fast hard-type van der Pol oscillator numerically. “Hard-type” oscillators are those that can simultaneously exhibit a stable equilibrium, a stable relaxation oscillation, and an unstable limit cycle; the possibility of this coexistence is due to the structure of the oscillator that contains a nonlinear element characterized by a fifth-order polynomial. In this Letter, we create a two-parameter bifurcation diagram in which both supercritical and subcritical Andronov–Hopf bifurcations and canard explosions are confirmed. We observe that the stable canard in this oscillator exhibits two explosion events. Furthermore, the oscillator exhibits an unstable canard, which also exhibits an explosion. These stable and unstable canards transform into large oscillations and disappear via a saddle-node bifurcation. We note that both the two sequential canard explosions as well as the unstable canard observed here represent novel phenomena that have not been observed prior to this work.