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Exact stationary solutions independent of energy for strongly nonlinear stochastic systems of multiple degrees of freedom

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Abstract

A new procedure is proposed to construct strongly nonlinear systems of multiple degrees of freedom subjected to parametric and/or external Gaussian white noises, whose exact stationary solutions are independent of energy. Firstly, the equivalent Fokker-Planck-Kolmogorov (FPK) equations are derived by using exterior differentiation. The main difference between the equivalent FPK equation and the original FPK equation lies in the additional arbitrary antisymmetric diffusion matrix. Then the exact stationary solutions and the structures of the original systems can be obtained by using the coefficients of antisymmetric diffusion matrix. The obtained exact stationary solutions, which are generally independent of energy, are for the most general class of strongly nonlinear stochastic systems multiple degrees of freedom (MDOF) so far, and some classes of the known ones dependent on energy belong to the special cases of them.

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Authors and Affiliations

  1. Department of Mechanics, Zhejiang University, Hangzhou, 310027, China

    ZhiLong Huang & XiaoLing Jin

  2. The State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou, 310027, China

    ZhiLong Huang

Authors
  1. ZhiLong Huang
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  2. XiaoLing Jin
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Correspondence to ZhiLong Huang.

Additional information

Supported by the National Natural Science Foundation of China (Grant No. 10672142) and the Program for New Century Excellent Talents in University

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Huang, Z., Jin, X. Exact stationary solutions independent of energy for strongly nonlinear stochastic systems of multiple degrees of freedom. Sci. China Ser. E-Technol. Sci. 52, 2424–2431 (2009). https://doi.org/10.1007/s11431-008-0186-6

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  • DOI: https://doi.org/10.1007/s11431-008-0186-6

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