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A four-thirds law for phase randomization of stochastically perturbed oscillators and related phenomena

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Abstract

LetI be a set of invariants and θ be a set of angle variables for a system of differential equations with anO(ε) vector field. When time dependent stochastic perturbations, also ofO(ε), are added to the system, we have shown that under suitable conditionsI becomes a stochastic adiabatic invariant satisfying a diffusion equation on time scales of order 1/ɛ2, in the limit as ε»0. Here we show that the angle variables converge weakly to a Gaussian Markov process on an O(ɛ-4/3) time scale, and thus the phase becomes randomized at these times. Application to nearly integrable Hamiltonian systems is considered.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, The University of New Mexico, 87131, Albuquerque, New Mexico, USA

    Robert Cogburn & James A. Ellison

Authors
  1. Robert Cogburn
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  2. James A. Ellison
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Additional information

Communicated by J.L. Lebowitz

Supported by NSF grant DMR-8704348

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Cogburn, R., Ellison, J.A. A four-thirds law for phase randomization of stochastically perturbed oscillators and related phenomena. Commun.Math. Phys. 166, 317–336 (1994). https://doi.org/10.1007/BF02112318

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  • DOI: https://doi.org/10.1007/BF02112318

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