Bifurcation analysis of a periodically forced relaxation oscillator: differential model versus phase-resetting map

We compare the dynamics of the periodically forced FitzHugh-Nagumo oscillator in its relaxation regime to that of a one-dimensional discrete map of the circle derived from the phase-resetting response of this oscillator (the "phase-resetting map"). The forcing is a periodic train of Gaussian-shaped pulses, with the width of the pulses much shorter than the intrinsic period of the oscillator. Using numerical continuation techniques, we compute bifurcation diagrams for the periodic solutions of the full differential equations, with the stimulation period being the bifurcation parameter. The period-1 solutions, which belong either to isolated loops or to an everywhere-unstable branch in the bifurcation diagram at sufficiently small stimulation amplitudes, merge together to form a single branch at larger stimulation amplitudes. As a consequence of the fast-slow nature of the oscillator, this merging occurs at virtually the same stimulation amplitude for all the period-1 loops. Again using continuation, we show that this stimulation amplitude corresponds, in the circle map, to a change of topological degree from one to zero. We explain the origin of this coincidence, and also discuss the translational symmetry properties of the bifurcation diagram.