We consider isochrons of a periodic orbit in the planar FitzHugh-Nagumo system; these are curves of points that converge to the periodic orbit in phase with each other. We extend this notion to isochrons of a focus equilibrium and also define what we call backward-time isochrons, that is, isochrons of the reversed-time system. We show that a cubic tangency occurs between a set of forward-time and backward-time isochrons, which we call a cubic isochron foliation tangency (CIFT). This phenomenon is not a local feature but happens globally throughout the annulus where both sets of isochrons exist. We construct and discuss examples of three mechanisms for a CIFT: a global time-scale separation; a perturbation that increases the velocity along trajectories in a local region of phase space; and a canard explosion.

This is joint work with Peter Langfield.