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Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. (English) Zbl 1335.34090
Summary: This paper investigates the stochastic averaging of slow-fast dynamical systems driven by fractional Brownian motion with the Hurst parameter $$H$$ in the interval $$(\frac{1}{2},1)$$. We establish an averaging principle by which the obtained simplified systems (the so-called averaged systems) will be applied to replace the original systems approximately through their solutions. Here, the solutions to averaged equations of slow variables which are unrelated to fast variables can converge to the solutions of slow variables to the original slow-fast dynamical systems in the sense of mean square. Therefore, the dimension reduction is realized since the solutions of uncoupled averaged equations can substitute that of coupled equations of the original slow-fast dynamical systems, namely, the asymptotic solutions dynamics will be obtained by the proposed stochastic averaging approach.

##### MSC:
 34F05 Ordinary differential equations and systems with randomness 37H10 Generation, random and stochastic difference and differential equations 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 34E15 Singular perturbations, general theory for ordinary differential equations 34C29 Averaging method for ordinary differential equations
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