Notes
Abstract or Additional Information
Attracting periodic orbits of vector fields are important objects of study, both from a theory and an applications point of view. Especially when the underlying oscillation is far from sinusoidal— think pulsing lasers or spiking neurons — it is an interesting and challenging question to ask how the system relaxes back to the periodic orbit when it is subjected to a given perturbation. This question can be answered in a global and geometric way by considering the isochrons, each of which consists of all the points that converge to the periodic orbit in phase with a specific point on it. As such, the isochrons are stabe manifolds of the (fixed) points on the periodic orbit under the time-T map, where T is the period of the periodic orbit. In particular, they foliate the basin of attraction, and the geometric properties of this foliation encode the re-synchronisation behavior of the system after a perturbation.
We focus on the important case of planar systems, where the isochrons are smooth curves that can be computed efficiently with a boundary value problem set-up. We introduce backward-time isochrons of unstable periodic orbits and spiralling equilibria, which foliate the basin of repulsion. In the annulus where they both exist, the two foliations are typically either transverse to each other or have quadratic tangencies. The transition between these two cases is a cubic isochron foliation tangency, and we show that it can be interpreted as the onset of strong phase sensitivity. Moreover, we illustrate with several examples, including the planar FitzHugh-Nagumo model, that this bifurcation of isochron foliations emerges as a natural consequence of an increasing time-scale separation. We also briefly discuss a second source of phase sensitivity: the accumulation of isochrons on a nontrivial basin boundary.
This is joint work with James Hannam and Peter Langfield.
Host: Jon Rubin