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Stochastic Perturbations of Stable Dynamical Systems: Trajectory-Wise Approach


We study stochastic perturbations of a dynamical system with a locally stable fixed point. The perturbed system has the form of Ito stochastic differential equations. We assume that perturbations do not vanish at the equilibrium of the deterministic system. Using the approach based on consideration of trajectories to the analysis of stochastic differential equations, we find restrictions for perturbations under which the stability of the equilibrium is preserved with probability 1.

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  1. 1.

    A. N. Borodin and P. Salminen, Handbook of Brownian Motion. Facts and Formulae, Birkhäuser-Verlag, Basel–Boston–Berlin (2002).

    Book  MATH  Google Scholar 

  2. 2.

    M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, Springer-Verlag, New York–Heidelberg–Berlin (1998).

    Book  MATH  Google Scholar 

  3. 3.

    M. M. Khapaev, Asymptotic Mathods and Stability in the Theory of Nonlinear Oscillations [in Russian], Vysshaya Shkola, Moscow (1988).

    Google Scholar 

  4. 4.

    R. Z. Khasminskii, “On the stability under permanent perturbations,” in: Theory of Information Transfer. Identification of Images [in Russian], Nauka, Moscow (1965), pp. 74–87.

  5. 5.

    R. Khasminskii, Stochastic Stability of Differential Equations, Springer-Verlag, Berlin–Heidelberg (2012).

    Book  MATH  Google Scholar 

  6. 6.

    I. G. Malkin, Theory of Stability of Motion [in Russian], GITTL, Moscow–Leningrad (1952).

    Google Scholar 

  7. 7.

    F. S. Nasyrov, Local Times, Symmetric Integrals, and Stochastic Analysis [in Russian], Fizmatlit, Moscow (2011).

    Google Scholar 

  8. 8.

    B. ∅ksendal, Stochastic Differential Equations. An Introduction with Applications, Springer-Verlag, New York–Heidelberg–Berlin (1998).

  9. 9.

    O. A. Sultanov, “Stability of autoresonance models subject to random perturbations for systems of nonlinear oscillation equations,” Zh. Vychisl. Mat. Mat. Fiz., 54, No. 1, 65–79 (2014).

    MathSciNet  MATH  Google Scholar 

  10. 10.

    O. Sultanov, “White noise perturbation of locally stable dynamical systems,” Stochast. Dynam., 17, No. 1, C. 1750002 (2017); arXiv:1509.07323

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Corresponding author

Correspondence to O. A. Sultanov.

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 139, Differential Equations. Mathematical Physics, 2017.

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Sultanov, O.A. Stochastic Perturbations of Stable Dynamical Systems: Trajectory-Wise Approach. J Math Sci 241, 340–353 (2019).

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Keywords and phrases

  • dynamical system
  • perturbation
  • white noise
  • stochastic differential equation
  • stability with probability 1

AMS Subject Classification

  • 93E15
  • 34D10
  • 60H10