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Low dimensional completely resonant tori in Hamiltonian Lattices and a Theorem of Poincaré

1 Department of Mathematics “F. Enriques”, Milano University, via Saldini 50, 20133 - Milano, Italy
2 GNFM (Gruppo Nazionale di Fisica Matematica) – Indam (Istituto Nazionale di Alta Matematica “F. Severi”), Roma, Italy

This contribution is part of the Special Issue: Modern methods in Hamiltonian perturbation theory
Guest Editors: Marco Sansottera; Ugo Locatelli
Link: www.aimspress.com/mine/article/5514/special-articles

Special Issues: Modern methods in Hamiltonian perturbation theory

We present an extension of a classical result of Poincaré (1892) about continuation of periodic orbits and breaking of completely resonant tori in a class of nearly integrable Hamiltonian systems, which covers most Hamiltonian Lattice models. The result is based on the fixed point method of the period map and exploits a standard perturbation expansion of the solution with respect to a small parameter. Two different statements are given, about existence and linear stability: a first one, in the so called non-degenerate case, and a second one, in the completely degenerate case. A pair of examples inspired to the existence of localized solutions in the discrete NLS lattice is provided.
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