Phase and amplitude dynamics of noisy oscillators described by Itô stochastic differential equations

Abstract

We present a novel phase-amplitude model for noisy oscillators described by Itô stochastic differential equations. The model is completely rigorous and it holds for any value of the noise intensity. The phase and amplitude equations depend on the choice of an appropriate set of basis vectors. We show that using Floquet's basis, a phase-amplitude description is obtained analogous to others, previously proposed. We also show how, using moment closure techniques, information on the expected angular frequency, oscillation amplitude and amplitude variance can be obtained from the phase-amplitude model without solving the equations explicitly.

Amplitude and phase dynamics of noisy oscillators

“…The amplitude function is interpreted as an orbital deviation from the limit cycle. The following theorem represents the generalization of a classical result for ODE to the case of Itô SDE.Theorem Consider the Itô SDE such that the ODE obtained setting ϵ = 0 admits a nontrivial T –periodic limit cycle x s ( t ). Let { u 1 ( t ),…, u n ( t )} and { v 1 ( t ),…, v n ( t )} be two reciprocal bases such that u 1 ( t ) satisfies and such that the biorthogonality condition holds.…”

“…The amplitude ‡ function is interpreted as an orbital deviation from the limit cycle. The following theorem [29,30] represents the generalization of a classical result for ODE [24] to the case of Itô SDE.…”