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A nonlinear oscillator with parametric coloured noise: some analytical results

Kirone Mallick1 and Philippe Marcq2

1 Service de Physique Th?orique, Centre d'?tudes de Saclay, 91191 Gif-sur-Yvette Cedex, France
2 Institut de Recherche sur les Ph?nom?nes Hors ?quilibre, Universit? de Provence, 49 rue Joliot-Curie, BP 146, 13384 Marseille Cedex 13, France

E-mail: mallick@spht.saclay.cea.fr and marcq@irphe.univ-mrs.fr
Received 24 March 2005
Published 15 June 2005
Print publication: Issue 26 (1 July 2005)

Abstract. 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.

PACS numbers: 05.10.Gg, 05.40.-a, 05.45.-a

URL: http://stacks.iop.org/0305-4470/38/5913
PII: S0305-4470(05)96976-X

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