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We investigate three types of averaging principles and the normal deviationfor multi-scale stochastic differential equations (in short, SDEs) withpolynomial nonlinearity. More specifically, we first demonstrate the strongconvergence of the solution of SDEs, which involves highly oscillatingcomponents and fast processes, to that of the averaged equation. Then weinvestigate the small fluctuations of the system around its average, and showthat the normalized difference weakly converges to an Ornstein-Uhlenbeck typeprocess, which can be viewed as a functional central limit theorem.Additionally, we show that the attractor of the original system tends to thatof the averaged equation in probability measure space as the time scale$\varepsilon$ goes to zero. Finally, we establish the second Bogolyubovtheorem; that is to say, we prove that there exists a quasi-periodic solutionin a neighborhood of the stationary solution of the averaged equation when the$\varepsilon$ is small.

Averaging principle and normal deviation for multi-scale SDEs with polynomial nonlinearity