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.

Low dimensional completely resonant tori in Hamiltonian Lattices and a Theorem of Poincaré / T. Penati, V. Danesi, S. Paleari. - In: MATHEMATICS IN ENGINEERING. - ISSN 2640-3501. - 3:4(2021), pp. 1-20. [10.3934/mine.2021029]

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

T. Penati
Primo
;
V. Danesi
Secondo
;
S. Paleari
Ultimo
2021

Abstract

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.
Hamiltonian lattices; perturbation theory; average methods; resonant tori; periodic orbits; linear stability;
Settore MAT/07 - Fisica Matematica
3-ago-2020
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/757193
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