Non linear stability in resonant cases: A geometrical approach.

  1. Elipe, A. 2
  2. Lanchares, V. 1
  3. López-Moratalla, T. 3
  4. Riaguas, A. 22
  1. 1 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

  2. 2 Universidad de Zaragoza
    info

    Universidad de Zaragoza

    Zaragoza, España

    ROR https://ror.org/012a91z28

  3. 3 Real Inst. y Observ. de la Armada, 11110 San Fernando, Spain
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Journal:
Journal of Nonlinear Science

ISSN: 0938-8974

Year of publication: 2001

Volume: 11

Issue: 3

Pages: 211-222

Type: Article

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DOI: 10.1007/S00332-001-0001-Z SCOPUS: 2-s2.0-0004226772 WoS: WOS:000171107100003 GOOGLE SCHOLAR

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Abstract

In systems with two degrees of freedom, Arnold's theorem is used for studying nonlinear stability of the origin when the quadratic part of the Hamiltonian is a nondefinite form. In that case, a previous normalization of the higher orders is needed, which reduces the Hamiltonian to homogeneous polynomials in the actions. However, in the case of resonances, it could not be possible to bring the Hamiltonian to the normal form required by Arnold's theorem. In these cases, we determine the stability from analysis of the normalized phase flow. Normalization up to an arbitrary order by Lie-Deprit transformation is carried out using a generalization of the Lissajous variables.