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