Title: Dimension reduction for stochastic oscillators: Investigating competing generalizations of phase and isochrons
Speaker: Alexander Cao (Case Western Reserve University)
Advisor: Peter Thomas (Associate Professor, Case Western Reserve University)
Abstract: A stable, finite-period limit cycle in a deterministic system has a set of Poincaré sections with two properties: all trajectories with initial conditions on a given leaf (i) converge to a common trajectory on the limit cycle and (ii) pass through the same leaf after an interval equalling one period. The leaves are the isochrons, and the timing of their movement defines the asymptotic phase of points converging to the limit cycle. In a Markovian stochastic setting, two generalizations of the asymptotic phase and the isochron foliation have been proposed. One is based on the spectral decomposition of the Kolmogorov Backward operator and the other on foliations defined by a uniform mean first-passage time property. We extend the latter generalization by reformulating the established numerical procedure calculating the leaves in terms of a partial differential equation. We also discuss the (non)equivalence of these two phase generalizations for noisy oscillators.