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Necessary and sufficient conditions for existence of stationary solutions of some nonlinear stochastic systems

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Abstract

At the state of statistical stationarity, the response of a nonlinear system under multiplicative random excitations can be either trivial or non-trivial, depending on the spectral levels of the excitations and the values of certain system parameters. Assuming that the random excitations are Gaussian white noises, the two types of response may be investigated by way of their stationary densities, which are obtainable for first order dynamical systems and for higher order dynamical systems belonging to the class of generalized stationary potential. Alternatively, the Lyapunov exponents can be computed for perturbation from either the trivial or non-trivial solution, since a negative sign for the greatest Lyapunov exponent provides both the necessary and sufficient conditions for the stability of sample functions with probability one. It is shown in two specific examples, that the boundary at which the greatest Lyapunov exponent changes its sign coincides with the boundary for regularity (or being normalizable) for the probability density in both the trivial and non-trivial solutions. Thus, the stability conditions in the strong sense of probability one and the weak sense in distribution are identical in these cases.

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Authors and Affiliations

  1. Institute for Technical Mechanics, University of Karlsruhe, D-7500, Karlsruhe 1, F.R.G.

    W. V. Wedig

  2. Center for Applied Stochastics Research, Florida Atlantic University, 33431, Boca Raton, FL, U.S.A.

    Y. K. Lin & G. Q. Cai

Authors
  1. W. V. Wedig
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  2. Y. K. Lin
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  3. G. Q. Cai
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Wedig, W.V., Lin, Y.K. & Cai, G.Q. Necessary and sufficient conditions for existence of stationary solutions of some nonlinear stochastic systems. Nonlinear Dyn 1, 75–90 (1990). https://doi.org/10.1007/BF01857586

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