FitzHugh-Nagumo revisited: Types of bifurcations, periodical forcing and stability regions by a Lyapunov functional

We study several aspects of FitzHugh-Nagumo's (FH-N) equations without diffusion. Some global stability results as well as the boundedness of solutions are derived by using a suitably defined Lyapunov functional. We show the existence of both supercritical and subcritical Hopf bifurcations. We demonstrate that the number of all bifurcation diagrams is 8 but that the possible sequential occurrences of bifurcation events is much richer. We present a numerical study of all example exhibiting a series of various bifurcations, including subcritical Hopf bifurcations, homoclinic bifurcations and saddle-node bifurcations of equilibria and of periodic solutions. Finally, we study periodically forced FH-N equations. We prove that phase-locking occurs independently of the magnitude of the periodic forcing.