Rigorous KAM results around arbitrary periodic orbits for Hamiltonian Systems

TL;DR

This paper develops a computer-assisted methodology to rigorously prove the existence and KAM stability of arbitrary periodic orbits in Hamiltonian systems, demonstrated through applications to systems with 2 and 3 degrees of freedom.

Contribution

It introduces a novel computational approach for establishing the existence and stability of periodic orbits in Hamiltonian systems, including complex cases like the figure eight orbit.

Findings

Verified tiny elliptic islands inside chaotic domains for a quartic potential.

Proved KAM stability of the figure eight orbit in the three-body problem.

Provided additional theoretical and numerical insights into the dynamics of the studied systems.

Abstract

We set up a methodology for computer assisted proofs of the existence and the KAM stability of an arbitrary periodic orbit for Hamiltonian systems. We give two examples of application for systems with 2 and 3 degrees of freedom. The first example verifies the existence of tiny elliptic islands inside large chaotic domains for a quartic potential. In the 3-body problem we prove the KAM stability of the well-known figure eight orbit and two selected orbits of the so called family of rotating Eights. Some additional theoretical and numerical information is also given for the dynamics of both examples.