Numerical computation of critical surfaces for the breakup of invariant tori in Hamiltonian systems

TL;DR

This paper numerically computes the critical surfaces for invariant tori breakup in Hamiltonian systems, comparing renormalization and conjugation methods, revealing cusps in three-dimensional cases.

Contribution

It introduces and compares two computational methods for critical surface determination in Hamiltonian systems with multiple degrees of freedom.

Findings

Cusps found in the critical surface for 3D invariant tori.

Critical surface for 2D tori is smooth.

Comparison of renormalization-group and conjugation methods.

Abstract

We compute the critical surface for the existence of invariant tori of a family of Hamiltonian systems with two and three degrees of freedom. We use and compare two methods to compute the critical surfaces: renormalization-group transformations and conjugation in configuration space. We unveil the presence of cusps in the critical surface for the breakup of three-dimensional invariant tori, whereas the critical surface of two-dimensional invariant tori is expected to be smooth.