Phase Reduction for Stochastic Oscillators

Peter Thomas
Department of Mathematics
Case Western Reserve University

Stochastic oscillations play a role in many fields of science, and reduction of many-dimensional oscillators to a low-dimension phase description is an important tool for their analysis.  I will discuss two approaches to extending the phase description to the stochastic setting.
1. Schwabedal and Pikovsky [Phys. Rev. Lett., 2013] proposed a phase reduction scheme for stochastic oscillators by seeking a family of Poincare sections with the “mean–return-time” (MRT) property: a trajectory started at one such “isochron” would return to the same isochron, having completed a single orbit, with an average transit time uniform across initial conditions. In [Cao, Lindner, Thomas (2020), SIAM Journal on Applied Mathematics] we established a mathematical framework for MRT isochrons based on a first passage time/PDE framework, which I will describe.
2. Thomas and Lindner [Phys. Rev. Lett., 2014] proposed an alternative phase reduction based on a spectral decomposition of the generator of the Markov process.  The complex argument of the eigenfunction for the slowest decaying eigenmode provides a natural generalization of the asymptotic phase of an oscillator.  We recently showed that the spectral asymptotic phase can be defined not only for stochastic systems with an underlying deterministic attracting limit cycle or a stable heteroclinic cycle, but also for a quasicycle system, i.e. a linear Ornstein-Uhlenbeck process with complex eigenvalues.  [Thomas and Lindner, Phys. Rev. E. 2019].
Joint work with Benjamin Lindner, Humboldt University (Berlin).