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Approximation of Random Slow Manifolds and Settling of Inertial Particles Under Uncertainty

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Abstract

A method is provided for approximating random slow manifolds of a class of slow–fast stochastic dynamical systems. Thus approximate, low dimensional, reduced slow systems are obtained analytically in the case of sufficiently large time scale separation. To illustrate this dimension reduction procedure, the impact of random environmental fluctuations on the settling motion of inertial particles in a cellular flow field is examined. It is found that noise delays settling for some particles but enhances settling for others. A deterministic stable manifold is an agent to facilitate this phenomenon. Overall, noise appears to delay the settling in an averaged sense.

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Acknowledgments

This work was partially supported by the NSF Grant 1025422, the NSFC Grants 11371367, 11271290 and 11271013, the Central University Fundamental Research Fund (Grant 2014QT005), and the Office of Naval Research under the Grant N00014-12-1-0257.

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Authors and Affiliations

  1. College of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

    Jian Ren

  2. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Jian Ren & Jinqiao Duan

  3. Institute for Pure and Applied Mathematics, UCLA, Los Angeles, CA, 90095, USA

    Jinqiao Duan

  4. Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, 60616, USA

    Jinqiao Duan

  5. Department of Mathematics, University of North Carolina-Chapel Hill, Chapel Hill, NC, 27599-3250, USA

    Christopher K. R. T. Jones

Authors
  1. Jian Ren
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  2. Jinqiao Duan
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  3. Christopher K. R. T. Jones
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Corresponding author

Correspondence to Christopher K. R. T. Jones.

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This paper is in memory of Klaus Kirchgässner, an exemplary scientist and a good friend and mentor.

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Ren, J., Duan, J. & Jones, C.K.T. Approximation of Random Slow Manifolds and Settling of Inertial Particles Under Uncertainty. J Dyn Diff Equat 27, 961–979 (2015). https://doi.org/10.1007/s10884-015-9452-z

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  • DOI: https://doi.org/10.1007/s10884-015-9452-z

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