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

TL;DR

This paper develops a normal form method to continue degenerate periodic orbits in nearly integrable Hamiltonian systems, especially after the breaking of lower dimensional resonant tori, and analyzes their stability.

Contribution

It extends previous results to lower dimensional resonant tori and introduces a constructive scheme for identifying and approximating continued periodic orbits, including degenerate cases.

Findings

Normal form scheme for lower dimensional resonant tori

Algorithm for degenerate periodic orbits continuation

Stability analysis based on spectrum of approximate orbits

Abstract

We consider the classical problem of the continuation of periodic orbits surviving to the breaking of invariant lower dimensional resonant tori in nearly integrable Hamiltonian systems. In particular we extend our previous results (presented in CNSNS, 61:198-224, 2018) for full dimensional resonant tori to lower dimensional ones. We develop a constructive normal form scheme that allows to identify and approximate the periodic orbits which continue to exist after the breaking of the resonant torus. A specific feature of our algorithm consists in the possibility of dealing with degenerate periodic orbits. Besides, under suitable hypothesis on the spectrum of the approximate periodic orbit, we obtain information on the linear stability of the periodic orbits feasible of continuation. A pedagogical example involving few degrees of freedom, but connected to the classical topic of discrete solitons in dNLS-lattices, is also provided.