A remark on the onset of resonance overlap

TL;DR

This paper introduces two simple Hamiltonian systems to rigorously illustrate Chirikov's resonance overlap criterion, demonstrating how it predicts global instability and symbolic dynamics within certain parameter regions.

Contribution

The paper provides a rigorous demonstration of Chirikov's criterion using new Hamiltonian models, clarifying its applicability in detecting global instability.

Findings

Global instability observed in the models

Existence of symbolic dynamics in certain parameters

Resonance overlap correlates with instability regions

Abstract

Chirikov's celebrated criterion of resonance overlap has been widely used in celestial mechanics and Hamiltonian dynamics to detect global instability, but is rarely rigourous. We introduce two simple Hamiltonian systems, each depending on two parameters measuring respectively the distance to resonance overlap and non-integrability. Within some thin region of the parameter plane, classical perturbation theory shows the existence of global instability and symbolic dynamics, thus illustrating Chirikov's criterion.