Continuation-based Computation of Global Isochrons

Isochrons are foliations of phase space that extend the notion of phase of a stable periodic orbit to the basin of attraction of this periodic orbit. Each point in the basin of attraction lies on only one isochron, and two points on the same isochron converge to the periodic orbit with the same phase. Global isochrons, that is, isochrons extended into the full basin of attraction rather than just a neighborhood of the periodic orbit, can form remarkable foliations. For example, accumulations of all isochrons can occur in arbitrarily small regions of phase space; the limit of such an accumulation is called the phaseless set, which lies on the boundary of the basin of attraction of the periodic orbit. Since global isochrons must typically be approximated numerically, such complicated geometries are often difficult to realize for actual examples. Indeed, the computation of global isochrons can be challenging, particularly for systems with multiple time scales. We present a novel method for computing isochrons via the continuation of a two-point boundary value problem, which is particularly effective for systems with multiple time scales. We use this method to compute global isochrons for a two-dimensional reduced Hodgkin–Huxley model and illustrate that the one-dimensional isochrons for a planar multiple-time-scale system can accumulate in the interior of the basin of attraction of the periodic orbit in a way similar to two-dimensional isochrons accumulating on the boundary of a three-dimensional basin of attraction.