I will consider deterministic and stochastic perturbations of various systems.

In a broad class of problems, the limiting long-time behavior of the perturbed system, in an appropriate

time scale, can be described as a motion on the simplex of invariant probability measures of the

non-perturbed system. The motion on the simplex is defined either by limit theorems for large deviations,

by a modified averaging principle, or by a diffusion approximation.

To be specific, I will, mostly, speak on perturbations of an oscillator with one degree of freedom.