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Approximations in Mean Square Analysis of Stochastically Forced Equilibria for Nonlinear Dynamical Systems

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  • Irina Bashkirtseva

    (Institute of Natural Sciences and Mathematics, Ural Federal University, Lenina 51, 620000 Ekaterinburg, Russia)

Abstract

Motivated by important applications to the analysis of complex noise-induced phenomena, we consider a problem of the constructive description of randomly forced equilibria for nonlinear systems with multiplicative noise. Using the apparatus of the first approximation systems, we construct an approximation of mean square deviations that explicitly takes into account the presence of multiplicative noises, depending on the current system state. A spectral criterion of existence and exponential stability of the stationary second moments for the solution of the first approximation system is presented. For mean square deviation, we derive an expansion in powers of the small parameter of noise intensity. Based on this theory, we derive a new, more accurate approximation of mean square deviations in a general nonlinear system with multiplicative noises. This approximation is compared with the widely used approximation based on the stochastic sensitivity technique. The general mathematical results are illustrated with examples of the model of climate dynamics and the van der Pol oscillator with hard excitement.

Suggested Citation

  • Irina Bashkirtseva, 2024. "Approximations in Mean Square Analysis of Stochastically Forced Equilibria for Nonlinear Dynamical Systems," Mathematics, MDPI, vol. 12(14), pages 1-12, July.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:14:p:2199-:d:1434401
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    References listed on IDEAS

    as
    1. Sun, Yahui & Hong, Ling & Jiang, Jun, 2017. "Stochastic sensitivity analysis of nonautonomous nonlinear systems subjected to Poisson white noise," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 508-515.
    2. Bashkirtseva, I. & Ryashko, L., 2019. "Stochastic sensitivity analysis of chaotic attractors in 2D non-invertible maps," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 78-84.
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