Estimating Phase from Observed Trajectories Using the Temporal 1-Form

TL;DR

This paper introduces a novel data-driven method to estimate the asymptotic phase of oscillators from short observed trajectories, enabling detailed phase and isochron geometry analysis.

Contribution

The method uses a series expansion to directly compute phase response curves and can estimate phase from shorter data segments than previous approaches.

Findings

Accurately estimates phase from short trajectories

Recovers phase response curves and isochron curvature

Applicable to models from measured or simulated data

Abstract

Oscillators are ubiquitous in nature, and usually associated with the existence of an asymptotic phase that governs the long-term dynamics of the oscillator. % We show that asymptotic phase can be estimated using a carefully chosen series expansion which directly computes the phase response curve and provide an algorithm for estimating the co-efficients of this series. Unlike all previously available data driven phase estimation methods, our algorithm can: (i) use observations that are much shorter than a cycle; (ii) recover phase within any forward invariant region for which sufficient data are available; (iii) recover the phase response curves (PRC-s) that govern weak oscillator coupling; (iv) show isochron curvature, and recover nonlinear features of isochron geometry. Our method may find application wherever models of oscillator dynamics need to be constructed from measured or simulated time-series.