Perturbation of Systems with a First Integral: Motion on the Reeb Graph

TL;DR

This paper investigates the long-term dynamics of systems near those with a smooth first integral, revealing a universal behavior influenced by the integral, perturbations, and initial conditions, with explicit limiting distributions.

Contribution

It introduces a universal framework for understanding long-time behavior of perturbed systems with first integrals, applicable across deterministic and stochastic cases.

Findings

Long-time behavior is determined by the first integral, perturbation, and initial point.

The limiting distribution can be explicitly calculated.

The behavior is universal across broad classes of noise and deterministic systems.

Abstract

We consider the long-time behavior of systems close to a system with a smooth first integral. Under certain assumptions, the limiting behavior, to some extent, turns out to be universal: it is determined by the first integral, the deterministic perturbation, and the initial point. Furthermore, it is the same for a broad class of noises. In particular, the long-time behavior of a deterministic system can, in a sense, be stochastic and stochastic systems can have a reduced stochasticity. The limiting distribution is calculated explicitly.