# Phase oscillator model for noisy oscillators

## TL;DR

This paper derives a phase oscillator model for noisy oscillators, showing that phase noise behaves as a drift-diffusion process and that amplitude fluctuations influence the expected frequency, extending the Kuramoto model.

## Contribution

It introduces a phase reduction for noisy oscillators that accounts for amplitude fluctuations and their impact on frequency shifts, providing a more accurate model than traditional approaches.

## Key findings

- Phase noise is modeled as a drift-diffusion process.
- Amplitude fluctuations affect the expected oscillator frequency.
- Different approximation levels improve phase reduction accuracy.

## Abstract

The Kuramoto model has become a paradigm to describe the dynamics of nonlinear oscillator under the influence of external perturbations, both deterministic and stochastic. It is based on the idea to describe the oscillator dynamics by a scalar differential equation, that defines the time evolution for the phase of the oscillator. Starting from a phase and amplitude description of noisy oscillators, we discuss the reduction to a phase oscillator model, analogous to the Kuramoto model. The model derived shows that the phase noise problem is a drift-diffusion process. Even in the case where the expected amplitude remains unchanged, the unavoidable amplitude fluctuations do change the expected frequency, and the frequency shift depends on the amplitude variance. We discuss different degrees of approximation, yielding increasingly accurate phase reduced descriptions of noisy oscillators.