Interactions of forward- and backward-time isochrons

Peter LangfieldMcGill University

In the 1970s Winfree introduced the concept of an isochron as the set of all points in the basin of an attracting periodic orbit that converge to the periodic orbit in forward time with the same asymptotic phase. It has been observed that in slow-fast systems, such as the FitzHugh-Nagumo model, the isochrons of such systems can have complicated geometric features; in particular, regions with high curvature that are related to sensitivity in the system. In order to understand where these features come from, we introduce backward-time isochrons that exist in the basin of a repelling periodic orbit, and we consider their interactions with the forward-time isochrons. We show that a cubic tangency between the two sets of isochrons is responsible for creating high curvature features. This study makes use of a boundary value problem formulation to compute isochrons accurately as parametrised curves.