Lower dimensional invariant tori for multi-scale Hamiltonian systems

TL;DR

This paper investigates the persistence of lower dimensional invariant tori in multi-scale Hamiltonian systems, extending classical results and establishing a quasi-periodic Poincaré theorem for resonant systems.

Contribution

It extends Arnold's fundamental theorem to multi-scale systems by proving the persistence of lower dimensional invariant tori and establishing a quasi-periodic Poincaré theorem.

Findings

Persistence of lower dimensional invariant tori in multi-scale systems

At least 2^{m_0} resonant tori survive small perturbations

Extension of classical KAM results to resonant multi-scale Hamiltonian systems

Abstract

The ``Fundamental Theorem" given by Arnold in [2] asserts the persistence of full dimensional invariant tori for 2-scale Hamiltonian systems. However, persistence in multi-scale systems is much more complicated and difficult. In this paper, we explore the persistence of lower dimensional invariant tori for multi-scale Hamiltonian systems, which play an important role in dynamics of resonant Hamiltonian systems. Moreover, using the corresponding results we give a quasi-periodic Poincar\'{e} Theorem for multi-scale Hamiltonian systems, i.e., at least $2^{m_0}$ families resonant tori survive small perturbations, where the integer $m_0$ is the multiplicity of resonance.