Stochastic parametric modulation of linear and non-linear oscillators: Perturbation theory of the response function

TL;DR

This paper develops a perturbation theory to analyze how stochastic parametric modulation affects linear and nonlinear oscillators, revealing conditions for stabilization and large responses due to noise characteristics.

Contribution

It introduces a diagrammatic perturbation approach to study stochastic parametric modulation effects on oscillators, including stabilization and resonance phenomena.

Findings

Coloured noise can stabilize an unstable oscillator configuration.

White noise modulation can induce large responses at specific parameter values.

Theoretical predictions are experimentally testable.

Abstract

We study a stochastically driven, damped nonlinear oscillator whose frequency is modulated by a white or coloured noise. Using diagrammatic perturbation theory, we find that in the absence of nonlinearity, parametric modulation by a coloured noise can lead to a Kapitza pendulum-like stabilization of an unstable configuration provided the noise is anti-correlated. Further, we show that for modulation by a white noise of amplitude $\lambda$ and correlation strength $F$, the system will have an extremely large response if the product of $\lambda^{2}F$ equals a specific combination of the frequency and the damping coefficient. This prediction can be experimentally tested.