GENERIC PLANAR PHASE RESETTING NEAR A PHASELESS POINT

Abstract We study the planar FitzHugh–Nagumo system with an attracting periodic orbit that surrounds a repelling focus equilibrium. When the associated oscillation of the system is perturbed, in a given direction and with a given amplitude, there will generally be a change in phase of the perturbed oscillation with respect to the unperturbed one. This is recorded by the phase transition curve (PTC), which relates the old phase (along the periodic orbit) to the new phase (after perturbation). We take a geometric point of view and consider the phase-resetting surface comprising all PTCs as a function of the perturbation amplitude. This surface has a singularity when the perturbation maps a point on the periodic orbit exactly onto the repelling focus, which is the only point that does not return to stable oscillation. We also consider the PTC as a function of the direction of the perturbation and present how the corresponding phase-resetting surface changes with increasing perturbation amplitude. In this way, we provide a complete geometric interpretation of how the PTC changes for any perturbation direction. Unlike other examples discussed in the literature so far, the FitzHugh–Nagumo system is a generic example and, hence, representative for planar vector fields.