A Partial Differential Equation for the Mean--Return-Time Phase of Planar Stochastic Oscillators

Stochastic oscillations are ubiquitous in many systems. For deterministic systems, the oscillator's phase has been widely used as an effective one-dimensional description of a higher dimensional dynamics, particularly for driven or coupled systems. Hence, efforts have been made to generalize the phase concept to the stochastic framework. One notion of phase due to Schwabedal and Pikovsky is based on the mean-return time (MRT) of the oscillator but has so far been described only in terms of a numerical algorithm. Here we develop the boundary condition by which the partial differential equation for the MRT has to be complemented in order to obtain the isochrons (lines of equal phase) of a two-dimensional stochastic oscillator, and rigorously establish the existence and uniqueness of the MRT isochron function (up to an additive constant). We illustrate the method with a number of examples: the stochastic heteroclinic oscillator (which would not oscillate in the absence of noise); the isotropic Stuart-Landau oscillator, the Newby-Schwemmer oscillator, and the Stuart-Landau oscillator with polarized noise. For selected isochrons we confirm by extensive stochastic simulations that the return time from an isochron to the same isochron (after one complete rotation) is always the mean first-passage time (irrespective of the initial position on the isochron). Put differently, we verify that Schwabedal and Pikovsky's criterion for an isochron is satisfied. In addition, we discuss how to extend the construction to arbitrary finite dimensions. Our results will enable development of analytical tools to study and compare different notions of phase for stochastic oscillators.