# Linear response theory for coupled phase oscillators with general coupling functions

## TL;DR

This paper introduces a comprehensive linear response theory for coupled phase oscillators that accommodates diverse coupling functions, natural frequency distributions, phase-lags, and time-delays, surpassing previous limited models.

## Contribution

The authors develop a general linear response framework applicable to a broad class of coupled oscillators, extending beyond the constraints of prior methods like Ott--Antonsen ansatz.

## Key findings

- Theory accurately predicts oscillator behavior in simulations.
- Applicable to systems with arbitrary coupling functions and parameters.
- Extends the analytical tools available for studying synchronization.

## Abstract

We develop a linear response theory by computing the asymptotic value of the order parameter from the linearized equation of continuity around the nonsynchronized reference state using the Laplace transform in time. The proposed theory is applicable to a wide class of coupled phase oscillator systems and allows for any coupling functions, any natural frequency distributions, any phase-lag parameters, and any values for the time-delay parameter. This generality is in contrast to the limitation of the previous methods of the Ott--Antonsen ansatz and the self-consistent equation for an order parameter, which are restricted to a model family whose coupling function consists of only a single sinusoidal function. The theory is verified by numerical simulations.