Phase reduction theory for hybrid nonlinear oscillators

TL;DR

This paper extends phase reduction theory to hybrid nonlinear oscillators with discrete switching, enabling analysis and control of complex rhythmic systems that are otherwise difficult to study.

Contribution

It develops a general phase reduction framework for hybrid systems, broadening the scope of analysis for nonlinear oscillators beyond smooth models.

Findings

Hybrid oscillators exhibit ultrafast and robust synchronization.

Logarithmic scaling observed at synchronization transition.

Application demonstrated on a biped locomotion model.

Abstract

Hybrid dynamical systems characterized by discrete switching of smooth dynamics have been used to model various rhythmic phenomena. However, the phase reduction theory, a fundamental framework for analyzing the synchronization of limit-cycle oscillations in rhythmic systems, has mostly been restricted to smooth dynamical systems. Here we develop a general phase reduction theory for weakly perturbed limit cycles in hybrid dynamical systems that facilitates analysis, control, and optimization of nonlinear oscillators whose smooth models are unavailable or intractable. On the basis of the generalized theory, we analyze injection locking of hybrid limit-cycle oscillators by periodic forcing and reveal their characteristic synchronization properties, such as ultrafast and robust entrainment to the periodic forcing and logarithmic scaling at the synchronization transition. We also illustrate the theory by analyzing the synchronization dynamics of a simple physical model of biped locomotion.