A criterion for timescale decomposition of external inputs for generalized phase reduction of limit-cycle oscillators

TL;DR

This paper introduces a simple criterion for decomposing external inputs into slow and weak components, enhancing the generalized phase reduction method for accurately predicting the dynamics of strongly driven limit-cycle oscillators.

Contribution

It proposes a new criterion for timescale decomposition that improves the applicability and accuracy of the generalized phase reduction method for various oscillators.

Findings

The criterion accurately predicts phase dynamics in numerical simulations.

It enables systematic application of the generalized phase reduction method.

The approach extends the method's robustness to strongly driven oscillators.

Abstract

The phase reduction method is a dimension reduction method for weakly driven limit-cycle oscillators, which has played an important role in the theoretical analysis of synchro- nization phenomena. Recently, we proposed a generalization of the phase reduction method [W. Kurebayashi et al., Phys. Rev. Lett. 111, 2013]. This generalized phase reduction method can robustly predict the dynamics of strongly driven oscillators, for which the conventional phase reduction method fails. In this generalized method, the external input to the oscillator should be properly decomposed into a slowly varying component and remaining weak fluctua- tions. In this paper, we propose a simple criterion for timescale decomposition of the external input, which gives accurate prediction of the phase dynamics and enables us to systematically apply the generalized phase reduction method to a general class of limit-cycle oscillators. The validity of the criterion is confirmed by numerical simulations.