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A Wong–Zakai approximation for random slow manifolds with application to parameter estimation

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Abstract

We study a Wong–Zakai approximation for the random slow manifold of a slow–fast stochastic dynamical system. We first deduce the existence of the random slow manifold about an approximation system driven by an integrated Ornstein–Uhlenbeck process. Then, we compute the slow manifold of the approximation system, in order to gain insights of the long time dynamics of the original stochastic system. By restricting this approximation system to its slow manifold, we thus get a reduced slow random system. This reduced slow random system is used to accurately estimate a system parameter of the original system. An example is presented to illustrate this approximation.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 11531006, 11771449 and 11971186). The authors thank Xiujun Cheng, Jian Ren and Xiaoli Chen for useful discussions about the program for parameter estimation, and Hua Zhang for pointing out the theoretical basis about the computation of stochastic integration.

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Authors and Affiliations

  1. Center for Mathematical Sciences, School of Mathematics and Statistics, Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, China

    Ziying He

  2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Xinyong Zhang

  3. Collaborative Innovation Center of China Pilot Reform Exploration and Assessment, Hubei Sub-Center, Hubei University of Economics, Wuhan, 430205, China

    Tao Jiang

  4. School of Mathematics and Statistics, Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, China

    Xianming Liu

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  1. Ziying He
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Correspondence to Xianming Liu.

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He, Z., Zhang, X., Jiang, T. et al. A Wong–Zakai approximation for random slow manifolds with application to parameter estimation. Nonlinear Dyn 98, 403–426 (2019). https://doi.org/10.1007/s11071-019-05201-4

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