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Stochastic Perturbations of Stable Dynamical Systems: Trajectory-Wise Approach

Abstract

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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Correspondence to O. A. Sultanov.

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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). https://doi.org/10.1007/s10958-019-04428-1

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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