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Bifurcation Analysis of a Stochastically Driven Limit Cycle

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Abstract

We establish the existence of a bifurcation from an attractive random equilibrium to shear-induced chaos for a stochastically driven limit cycle, indicated by a change of sign of the first Lyapunov exponent. This relates to an open problem posed by Lin and Young (Nonlinearity 21:899–922, 2008) and Young (Nonlinearity 21:245–252, 2008), extending results by Wang and Young (Commun Math Phys 240(3):509–529, 2003) on periodically kicked limit cycles to the stochastic context.

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Acknowledgements

The authors would like to thank the referee for constructive comments which substantially helped improving the quality of the paper, in particular strengthening the main result. We also express our gratitude to Martin Hairer and Nils Berglund for suggesting the suitable generalization of the original model. In addition, the authors thank Alexis Arnaudon, Darryl Holm, Aleksandar Mijatovic, Nikolas Nüsken, Grigorios Pavliotis and Sebastian Wieczorek for useful discussions. Maximilian Engel was supported by a Roth Scholarship from the Department of Mathematics at Imperial College London and the SFB Transregio 109 “Discretization in Geometry and Dynamcis” sponsored by the German Research Foundation (DFG). Jeroen S.W. Lamb acknowledges the support by Nizhny Novgorod University through the Grant RNF 14-41-00044, and Martin Rasmussen was supported by an EPSRC Career Acceleration Fellowship EP/I004165/1. This research has also been supported by EU Marie-Curie IRSES Brazilian-European Partnership in Dynamical Systems (FP7-PEOPLE-2012-IRSES 318999 BREUDS) and EU Marie-Skłodowska-Curie ITN Critical Transitions in Complex Systems (H2020-MSCA-2014-ITN 643073 CRITICS).

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Authors and Affiliations

  1. Zentrum Mathematik der TU München, Boltzmannstr. 3, 85748, Garching bei München, Germany

    Maximilian Engel

  2. Department of Mathematics, Imperial College London, 180 Queens Gate, London, SW7 2AZ, UK

    Jeroen S. W. Lamb & Martin Rasmussen

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  1. Maximilian Engel
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  2. Jeroen S. W. Lamb
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  3. Martin Rasmussen
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Correspondence to Maximilian Engel.

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Communicated by M. Hairer

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Engel, M., Lamb, J.S.W. & Rasmussen, M. Bifurcation Analysis of a Stochastically Driven Limit Cycle. Commun. Math. Phys. 365, 935–942 (2019). https://doi.org/10.1007/s00220-019-03298-7

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  • DOI: https://doi.org/10.1007/s00220-019-03298-7

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