Diffusion approximation for noise-induced evolution of first integrals in multifrequency systems

TL;DR

This paper develops a diffusion approximation model for the evolution of first integrals in multifrequency dynamical systems with fast oscillations, especially near resonance regions, using open book space structures.

Contribution

It introduces a novel diffusion approximation framework for noise-induced evolution of first integrals in systems with multifrequency oscillations, including cases with piece-wise action-angle coordinates.

Findings

Diffusion approximation valid when resonance tori are sparse.

Open book space model describes the process when coordinates are piece-wise defined.

Differential operators govern the diffusion process on each page.

Abstract

We consider fast oscillating perturbations of dynamical systems in regions where one can introduce action-angle type coordinates. In an appropriate time scale, a diffusion approximation of the first-integrals evolution is described under the assumption that the set of resonance tori is small enough. If the action-angle coordinates can be introduced just piece-wise , the limiting diffusion process should be considered on an open book space. Such a process can be described by differential operators, one on each page, supplemented by some gluing conditions on the binding of the open book.