KAM Theorems for Multi-scale Torus

TL;DR

This paper investigates the persistence of invariant tori in multiscale Hamiltonian systems, demonstrating KAM theorem extensions and implications for ergodicity in such complex dynamical systems.

Contribution

It introduces a Hamiltonian system with multiscale rotation and establishes new KAM type theorems, including isoenergetic results, for these systems.

Findings

Invariant tori persist under certain conditions in multiscale Hamiltonian systems

KAM theorems are extended to systems with multiple scales

Boltzmann's ergodicity hypothesis does not hold for these systems

Abstract

In present paper, from the viewpoint of physical intuition we introduce a Hamiltonian system with multiscale rotation, which describes many systems, for example, the forced pendulum with fast rotation, weakly coupled $N$-oscillators with quasiperiodic force and so on. We study the persistence of invariant tori for this Hamiltonian system, and establish some KAM type results including the isoenergetic type. As consequences, we can show that Boltzmann's ergodicity hypothesis is also not true for this Hamiltonian system.