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Stochastic Perturbation of Integrable Systems: A Window to Weakly Chaotic Systems

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Abstract

Integrable non-linear Hamiltonian systems perturbed by additive noise develop a Lyapunov instability, and are hence chaotic, for any amplitude of the perturbation. This phenomenon is related, but distinct, from Taylor’s diffusion in hydrodynamics. We develop expressions for the Lyapunov exponents for the cases of white and colored noise. The situation described here being ‘multi-resonance’—by nature well beyond the Kolmogorov–Arnold–Moser regime, it offers an analytic glimpse on the regime in which many near-integrable systems, such as some planetary systems, find themselves in practice. We show with the aid of a simple example, how one may model in some cases weakly chaotic deterministic systems by a stochastically perturbed one, with good qualitative results.

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Notes

  1. The exponent \(\frac{1}{3}\) is also familiar in the theory of products of random matrices, see: [1, 2]. This is not surprising considering that the noise has in this approximation no impact on the dynamical system but only on the tangent dynamics, and since (18) end up being random linear transformations.

  2. For the treatment of the diffusion away from a torus (in the case of the Standard Map), see: [15].

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Acknowledgments

We wish to thank G. Benettin, M. Chertkov, A. Politi, S. Ruffo, S. Tremaine and A. deWijn for clarifying remarks and suggestions.

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Authors and Affiliations

  1. PMMH-ESPCI, Paris, France

    Khanh-Dang Nguyen Thu Lam

  2. Laboratoire de Physique Statistique, École Normale Supérieure, CNRS UMR 8550, UPMC, UDD, 24 Rue Lhomond, 75231 , Paris, France

    Jorge Kurchan

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  1. Khanh-Dang Nguyen Thu Lam
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  2. Jorge Kurchan
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Correspondence to Jorge Kurchan.

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Lam, KD.N.T., Kurchan, J. Stochastic Perturbation of Integrable Systems: A Window to Weakly Chaotic Systems. J Stat Phys 156, 619–646 (2014). https://doi.org/10.1007/s10955-014-1030-y

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  • DOI: https://doi.org/10.1007/s10955-014-1030-y

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