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Mathematics in Engineering

2021, Volume 3, Issue 4: 1-20. doi: 10.3934/mine.2021029
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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
  • Received: 20 February 2020 Accepted: 31 May 2020 Published: 03 August 2020

  • 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.

    Citation: Tiziano Penati, Veronica Danesi, Simone Paleari. Low dimensional completely resonant tori in Hamiltonian Lattices and a Theorem of Poincaré[J]. Mathematics in Engineering, 2021, 3(4): 1-20. doi: 10.3934/mine.2021029

    Related Papers:

  • 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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