# A mathematical framework for amplitude and phase noise analysis of coupled oscillators

## TL;DR

This paper develops a rigorous mathematical framework to analyze how noise affects the amplitude and phase synchronization of coupled oscillators, enabling phase-only modeling even under strong noise conditions.

## Contribution

It introduces a novel set of equations for amplitude and phase deviations in noisy coupled oscillators, valid beyond the weak noise approximation, using Floquet theory for partial decoupling.

## Key findings

- Decoupling of amplitude and phase dynamics via Floquet theory.
- Reduced phase model accurately describes oscillator behavior under noise.
- Insights into noise influence on oscillator frequency.

## Abstract

Synchronization of coupled oscillators is a paradigm for complexity in many areas of science and engineering. Any realistic network model should include noise effects. We present a description in terms of phase and amplitude deviation for nonlinear oscillators coupled together through noisy interactions. In particular, the coupling is assumed to be modulated by white Gaussian noise. The equations derived for the amplitude deviation and the phase are rigorous, and their validity is not limited to the weak noise limit. We show that using Floquet theory, a partial decoupling between the amplitude and the phase is obtained. The decoupling can be exploited to describe the oscillator's dynamics solely by the phase variable. We discuss to what extent the reduced model is appropriate and some implications on the role of noise on the frequency of the oscillators.