A Phase Description of Mutually Coupled Chaotic Oscillators

TL;DR

This paper extends phase reduction theory to strongly chaotic oscillators, enabling the inference of meaningful coupling functions from data even without limit cycles, thus broadening the applicability of phase-based analysis.

Contribution

It derives a new phase description for coupled chaotic oscillators using Poincaré sections and invariant measures, justifying phase analysis in chaotic regimes.

Findings

Theoretical coupling functions closely match those inferred from data.

Phase description remains meaningful for chaotic oscillators without limit cycles.

Results validate the use of phase-based methods in chaotic systems.

Abstract

The synchronization of rhythms is ubiquitous in both natural and engineered systems, and the demand for data-driven analysis is growing. When rhythms arise from limit cycles, phase reduction theory shows that their dynamics are universally modeled as coupled phase oscillators under weak coupling. This simple representation enables direct inference of inter-rhythm coupling functions from measured time-series data. However, strongly rhythmic chaos can masquerade as noisy limit cycles. In such cases, standard estimators still return plausible coupling functions even though a phase-oscillator model lacks a priori justification. We therefore extend the phase description to the chaotic oscillators. Specifically, we derive a closed equation for the phase difference by defining the phase on a Poincar\'e section and averaging the phase dynamics over invariant measures of the induced return maps. Numerically, the derived theoretical functions are in close agreement with those inferred from time-series data. Consequently, our results justify the applicability of phase description to coupled chaotic oscillators and show that data-driven coupling functions retain clear dynamical meaning in the absence of limit cycles.