A nonlinear oscillator with parametric coloured noise: some analytical results

The asymptotic behaviour of a nonlinear oscillator subject to a multiplicative Ornstein–Uhlenbeck noise is investigated. When the dynamics is expressed in terms of energy–angle coordinates, it is observed that the angle is a fast variable as compared to the energy. Thus, an effective stochastic dynamics for the energy can be derived if the angular variable is averaged out. However, the standard elimination procedure, performed earlier for a Gaussian white noise, fails when the noise is coloured because of correlations between the noise and the fast angular variable. We develop here a specific averaging scheme that retains these correlations. This allows us to calculate the probability distribution function (PDF) of the system and to derive the behaviour of physical observables in the long time limit.