Singular Hopf Bifurcation to Relaxation Oscillations. II

Abstract

This paper is a continuation of our study of singular Hopf bifurcations to relaxation oscillations [SIAM Journal on Applied Mathematics, 46 (1986), pp. 721–739]. In our previous analysis, a system of two first-order nonlinear equatins is investigated, and it is shown how the bifurcation problem can be reduced to a weakly perturbed conservative system of two first-order equations. In this paper, a method, based on analyzing nearly conservative quantities, was developed to find the bifurcation diagram of the periodic solutions emerging from singular Hopf bifurcation points. Specifically, the subcritical Hopf bifurcation of the FitzHugh–Nagumo equations is analyzed. The extension from the earlier bifurcation analysis is achieved by performing a perturbation analysis from the first integral of the leading-order nonlinear oscillator. At a critical value of a parameter, the first integral exists up to an additional order in the perturbed equation. This condition permits the resolution of features determined by higher-order terms and reveals the “kneebend” in the bifurcation curve.