Monday 21 Jul 2014: Interactions of forward- and backward-time isochrons
Peter Langfield -
Harrison 101 15:00-16:00
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 the high curvature features. We present two normal-form-type models that feature a cubic tangency bifurcation between forward- and backward-time isochrons, generated via two different mechanisms: the introduction of a global time-scale separation, and a local perturbation to the velocity along trajectories. This study makes use of a boundary value problem formulation to compute isochrons accurately as parametrised curves.