Kolmogorov's Theorem for Low-Dimensional Invariant Tori of Hamiltonian Systems

TL;DR

This paper proves that in Hamiltonian systems, hyperbolic invariant tori persist under small perturbations if the unperturbed system has a saddle point, establishing a correspondence with critical points of a specific function.

Contribution

It introduces a new correspondence between hyperbolic invariant tori and critical points of a function, extending Kolmogorov's theorem to low-dimensional cases.

Findings

Persistence of hyperbolic tori under perturbations

Existence of a one-to-one correspondence with critical points of a function

Persistence guaranteed if the unperturbed system has a saddle point

Abstract

In this paper the problem of persistence of invariant tori under small perturbations of integrable Hamiltonian systems is considered. The existence of one-to-one correspondence between hyperbolic invariant tori and critical points of the function $\Psi$ of two variables defined on semi-cylinder is established. It is proved that if unperturbed Hamiltonian has a saddle point, then under arbitrary perturbations there persists at least one hyperbolic torus.