On the dynamical behaviour of FitzHugh–Nagumo systems: Revisited

The purpose of this paper is to analyse a general form of the FitzHugh–Nagumo model as completely as possible. The main result is that no more than two limit cycles can be bifurcated from the unique fixed point via Hopf bifurcation, and there exist parameters such that this upper bound is attained. For these parameters, the stability of the inner and outer cycle, together with the unique fixed point is also established. The results are approached through Lyapunov coefficients and rely on a theorem by Andronov and Aleksandrovic [A.A. Andronov, A.A. Aleksandrovic, Theory of Bifurcations of Dynamical System on a Plane, Wiley, 1971]. Based on singular perturbation theory a sufficient condition for existence of a unique stable limit cycle is given under certain assumptions.