Stability under persistent perturbation by white noise

Physics and Mathematics of Nonlinear Phenomena 2013

Kalyakin, L.^1

Institute of Mathematics RAS, Chernyshevskii Str., 112, Ufa, 450008, Russia^1

关键词: Deterministic dynamical systems;    Equilibrium positions;    Length of intervals;    Lyapunov's stability;    Parabolic Equations;    Perturbation parameters;    Stable equilibrium;    Unperturbed systems;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/482/1/012019/pdf
DOI  :  10.1088/1742-6596/482/1/012019

来源: IOP

【 摘 要 】

Deterministic dynamical system which has an asymptotical stable equilibrium is considered under persistent perturbation by white noise. It is well known that if the perturbation does not vanish in the equilibrium position then there is not Lyapunov's stability. The trajectories of the perturbed system diverge from the equilibrium to arbitrarily large distances with probability 1 in finite time. New concept of stability on a large time interval is discussed. The length of interval agrees the reciprocal quantity of the perturbation parameter. The measure of stability is the expectation of the square distance from the trajectory till the equilibrium position. The method of parabolic equation is applied to both estimate the expectation and prove such stability. The main breakthrough is the barrier function derived for the parabolic equation. The barrier is constructed by using the Lyapunov function of the unperturbed system.