Quasi-periodic stability of normally resonant tori

TL;DR

This paper investigates the stability of quasi-periodic tori in systems with normal-internal resonance, establishing conditions for their persistence across reversible, Hamiltonian, and dissipative systems.

Contribution

It introduces generalized non-degeneracy conditions ensuring the stability of resonant tori, extending classical results to more complex resonant scenarios.

Findings

Conditions for quasi-periodic torus stability established

Results applicable to reversible, Hamiltonian, and dissipative systems

Extension of classical KAM theory to resonant cases

Abstract

We study quasi-periodic tori under a normal-internal resonance, possibly with multiple eigenvalues. Two non-degeneracy conditions play a role. The first of these generalizes invertibility of the Floquet matrix and prevents drift of the lower dimensional torus. The second condition involves a Kolmogorov-like variation of the internal frequencies and simultaneously versality of the Floquet matrix unfolding. We focus on the reversible setting, but our results carry over to the Hamiltonian and dissipative contexts.