Skip to main content

Stochastic stability of a dynamical system perturbed by white noise

Abstract

The effect of small constantly acting random perturbations of white noise type on a dynamical system with locally stable fixed point is studied. The perturbed system is considered in the form of Itô stochastic differential equations, and it is assumed that the perturbation does not vanish at a fixed point. In this case, the trajectories of the stochastic system issuing from points near the stable fixed point exit from the neighborhood of equilibrium with probability 1. Classes of perturbations such that the equilibrium of a deterministic system is stable in probability on an asymptotically large time interval are described.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    N. N. Krasovskii, Stability of Motion. Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay (Stanford Univ. Press, Stanford, 1963).

    MATH  Google Scholar 

  2. 2.

    L. A. Kalyakin and O. A. Sultanov, “Stability of autoresonance models,” Differ. Uravn. 49 (3), 279–293 (2013) [Differ. Equ. 49 (3), 267–281 (2013)].

    MathSciNet  MATH  Google Scholar 

  3. 3.

    O. A. Sultanov, “Lyapunov functions for nonautonomous systems close to Hamiltonian,” Ufim. Mat. Zh. 2 (4), 88–98 (2010).

    MATH  Google Scholar 

  4. 4.

    B. Oksendal, Stochastic Differential Equations. An Introduction with Applications, in Universitext (Springer-Verlag, Berlin, 1998).

    Google Scholar 

  5. 5.

    R. Khasminskii, Stochastic Stability of Differential Equations, in Stoch. Model. Appl. Probab. (Springer-Verlag, Berlin, 2012), Vol. 66.

    Google Scholar 

  6. 6.

    I. Ya. Kats and A. A. Martynyuk, Stability and Stabilization Nonlinear Systems with Random Structures, in Stability and Control: Theory, Methods Appl. (Taylor and Francis, London, 2002), Vol. 18.

    Google Scholar 

  7. 7.

    H. J. Kushner, Stochastic Stability and Control (Academic Press, New York, 1967), Vol. 33.

    MATH  Google Scholar 

  8. 8.

    X. Mao, Exponential Stability of Stochastic Differential Equations, inMonogr. Textbooks in Pure Appl. Math. (Marcel Dekker, New York, 1994), Vol. 182.

    Google Scholar 

  9. 9.

    M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, in Grundlehren Math. Wiss. (Springer-Verlag, Berlin, 1998), Vol. 260.

    Google Scholar 

  10. 10.

    R. Z. Khas’minskii, “On the stability under constantly acting random perturbations,” in Information Transmission Theory. Pattern Recognition (Nauka, Moscow, 1965) [in Russian].

    Google Scholar 

  11. 11.

    M. M. Khapaev, Asymptotic Methods and Stability in the Theory of Nonlinear Vibrations (Vyssh. ShkolaMoscow, 1988) [in Russian].

    Google Scholar 

  12. 12.

    A. D. Venttsel’, A Course in the Theory of Random Processes (Nauka, Moscow, 1996) [in Russian].

    MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to O. A. Sultanov.

Additional information

Original Russian Text © O. A. Sultanov, 2017, published in Matematicheskie Zametki, 2017, Vol. 101, No. 1, pp. 130–139.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sultanov, O.A. Stochastic stability of a dynamical system perturbed by white noise. Math Notes 101, 149–156 (2017). https://doi.org/10.1134/S0001434617010151

Download citation

Keywords

  • dynamical system
  • perturbation
  • white noise
  • stability