Forward-Time and Backward-Time Isochrons and Their Interactions

We consider isochrons of a periodic orbit in the planar FitzHugh-Nagumo system, that is, the curves of points that converge to the periodic orbit in phase with each other, and we extend this notion to isochrons of a focus equilibrium. We introduce the notion of backward-time isochrons, which are isochrons of the reversed-time system, and show that a cubic tangency occurs between a set of forward-time and backward-time isochrons. We call this moment of tangency a cubic isochron foliation tangency (CIFT). This phenomenon is not a local feature but happens globally throughout the annulus where both sets of isochrons coexist. We construct and discuss examples for 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 (in the Van der Pol system).