The Hénon and Heiles Poblem in Three Dimesions II: Relative Equilibria and Bifucartions in the Reduced System

  1. Ferrer, S. 3
  2. Lara, M. 4
  3. Palacián, J. 1
  4. San Juan, J.F. 2
  5. Viartola, A. 3
  6. Yanguas, P. 1
  1. 1 Universidad Pública de Navarra
    info

    Universidad Pública de Navarra

    Pamplona, España

    ROR https://ror.org/02z0cah89

  2. 2 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

  3. 3 Universidad de Zaragoza
    info

    Universidad de Zaragoza

    Zaragoza, España

    ROR https://ror.org/012a91z28

  4. 4 Real Inst. y Observ. de la A., 11110 San Fernando (Cádiz), Spain
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Journal:
International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

ISSN: 0218-1274

Year of publication: 1998

Volume: 8

Issue: 6

Pages: 1215-1229

Type: Article

DOI: 10.1142/S0218127498000954 SCOPUS: 2-s2.0-0032094211 WoS: WOS:000077123200009 GOOGLE SCHOLAR

More publications in: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

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Abstract

The second part of a research on the Hénon and Heiles system in three dimensions is presented. We focus on motions around the origin, where the system may be treated as a case of three perturbed isotropic harmonic oscillators. It is the only Hamiltonian of the family of axially symmetric cubic potentials in 1-1-1 resonance, which needs at least order four in the normalization to reach the features of the system. The fourth order normalized system is analyzed, considering orbits at any inclination. We study and classify the relative equilibria of the system finding pitchfork, centre-saddle and cusp bifurcations in the reduced orbit space.