Amplitude and phase dynamics of noisy oscillators

TL;DR

This paper derives rigorous stochastic differential equations for the phase and amplitude of noisy nonlinear oscillators, valid beyond weak noise limits, and provides formulas for key oscillation characteristics using Itô calculus.

Contribution

It introduces a comprehensive framework for describing amplitude and phase dynamics of noisy oscillators with rigorous equations valid for arbitrary noise intensities.

Findings

Derived stochastic equations for amplitude and phase that are valid beyond weak noise.

Provided formulas for expected frequency, amplitude, and variance.

Demonstrated efficient solutions via asymptotic expansions for small noise.

Abstract

A description in terms of phase and amplitude variables is given, for nonlinear oscillators subject to white Gaussian noise described by It\^o stochastic differential equations. The stochastic differential equations derived for the amplitude and the phase are rigorous, and their validity is not limited to the weak noise limit. If the noise intensity is small, the equations can be efficiently solved using asymptotic expansions. Formulas for the expected angular frequency, expected oscillation amplitude and amplitude variance are derived using It\^o calculus.