A continuation method for computing 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. These properties allow one to define so-called phase models that reduce the dimension of an oscillating system. Global isochrons, that is, isochrons extended into the full basin of attraction rather than just a neighborhood of the periodic orbit, can typically only be approximated numerically. Unfortunately, their computation is rather difficult, 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. We view the isochron associated with a particular phase point γ on the periodic orbit as the Poincaré section of the time-T return map that has γ as its fixed point; here, the time T is the period of the periodic orbit. The boundary value problem set-up uses the orbit segment that defines the periodic orbit starting at γ as the first known solution on the isochron; other solutions are found by continuation along the linear approximation of the isochron in a prespecified small neighborhood of the periodic orbit. This means that the computational error is fully controlled by the accuracy settings of the boundary value solver and the maximal distance along the linear approximation. This method can easily be implemented for two-dimensional systems, and we are able to compute global isochrons of planar slow-fast systems in unprecedented detail. We show how complicated the structure of isochrons can be already for such low-dimensional systems and discuss how the isochrons determine the phase sensitivity of the system.