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The Poincaré Map of Randomly Perturbed Periodic Motion

Journal of Nonlinear Science volume 23, pages 835–861 (2013)Cite this article

Abstract

A system of autonomous differential equations with a stable limit cycle and perturbed by small white noise is analyzed in this work. In the vicinity of the limit cycle of the unperturbed deterministic system, we define, construct, and analyze the Poincaré map of the randomly perturbed periodic motion. We show that the time of the first exit from a small neighborhood of the fixed point of the map, which corresponds to the unperturbed periodic orbit, is well approximated by the geometric distribution. The parameter of the geometric distribution tends to zero together with the noise intensity. Therefore, our result can be interpreted as an estimate of the stability of periodic motion to random perturbations.

In addition, we show that the geometric distribution of the first exit times translates into statistical properties of solutions of important differential equation models in applications. To this end, we demonstrate three distinct examples from mathematical neuroscience featuring complex oscillatory patterns characterized by the geometric distribution. We show that in each of these models the statistical properties of emerging oscillations are fully explained by the general properties of randomly perturbed periodic motions identified in this paper.

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Notes

  1. Lemma 2.1 in Freidlin and Wentzell (1998) is stated for a system of the form (2.5) with P(x)=id. However, the argument used in the proof of this lemma applies to systems with P(x) satisfying the assumptions in Sect. 2.1 after a suitable modification of the action functional.

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Acknowledgements

This work was partially supported by a grant from the Simons Foundation (grant No. 208766 to PH) and an NSF grant (DMS 1109367 to GM).

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Authors and Affiliations

  1. Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, PA, 19104, USA

    Pawel Hitczenko & Georgi S. Medvedev

Authors
  1. Pawel Hitczenko
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  2. Georgi S. Medvedev
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Correspondence to Georgi S. Medvedev.

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Communicated by P. Newton.

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Hitczenko, P., Medvedev, G.S. The Poincaré Map of Randomly Perturbed Periodic Motion. J Nonlinear Sci 23, 835–861 (2013). https://doi.org/10.1007/s00332-013-9170-9

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  • DOI: https://doi.org/10.1007/s00332-013-9170-9

Keywords

  • Random perturbations
  • Limit cycle
  • First return map
  • Mixed-mode oscillations
  • Bursting

Mathematics Subject Classification

  • 34C15
  • 60H10
  • 92B25