Singular Hopf Bifurcation to Relaxation Oscillations

Abstract

Relaxation oscillations characterized by two quite different time scales are described by mathematical models of the form $x_t = f(x,y,\lambda ,\varepsilon )$ and $y_t = \varepsilon g(x,y,\lambda ,\varepsilon )$ where $\varepsilon \ll 1$ and $\lambda $ is the control parameter. In this paper, we study the singular Hopf bifurcation from a basic steady state to these relaxation oscillations. Our bifurcation analysis shows how the harmonic oscillations near the bifurcation point progressively change to become pulsed, triangular oscillations. In the second part of the paper, we present a numerical study of the FitzHugh–Nagumo equations for nerve conduction. We first observe that the numerical results are in good agreement with the analytical predictions. We then consider the switching from a stable steady state to a stable periodic solution, or the reverse transition. Our purpose is to explain the annihilation experiments described in the nerve conduction literature.