On the continuation of degenerate periodic orbits via normal form: full dimensional resonant tori

TL;DR

This paper introduces a normal form method to continue degenerate periodic orbits in Hamiltonian systems, especially those emerging from resonant torus breakings, extending classical approaches to degenerate cases.

Contribution

A novel normal form construction enabling the continuation and approximation of degenerate periodic orbits beyond classical averaging methods.

Findings

Algorithm effectively continues degenerate orbits at leading order.

Method applies to resonant tori in Hamiltonian systems.

Potential for analyzing localized periodic orbits in oscillator chains.

Abstract

We reconsider the classical problem of the continuation of degenerate periodic orbits in Hamiltonian systems. In particular we focus on periodic orbits that arise from the breaking of a completely resonant maximal torus. We here propose a suitable normal form construction that allows to identify and approximate the periodic orbits which survive to the breaking of the resonant torus. Our algorithm allows to treat the continuation of approximate orbits which are at leading order degenerate, hence not covered by classical averaging methods. We discuss possible future extensions and applications to localized periodic orbits in chains of weakly coupled oscillators.