The Hénon and Heiles Poblem in Three Dimesions I: Periodic Orbits near the Origin

  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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Revista:
International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

ISSN: 0218-1274

Ano de publicación: 1998

Volume: 8

Número: 6

Páxinas: 1199-1213

Tipo: Artigo

DOI: 10.1142/S0218127498000942 SCOPUS: 2-s2.0-0032093326 WoS: WOS:000077123200008 GOOGLE SCHOLAR

Outras publicacións en: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

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Resumo

This paper is the first part of a study of the Hénon and Heiles problem in three dimensions. Due to the axial symmetry of the Hamiltonian, the third component of the angular momentum is an integral and the system is considered as a Hamiltonian with two degrees of freedom. As functions of that integral, we show the existence of three circular trajectories around the axis Oz and a domain for which we have bounded motions. In the part of that domain near the origin, the corresponding dynamical system is treated as a perturbed harmonic oscillator in 1-1-1 resonance. We present some numerical studies searching for periodic orbits, showing the corresponding Poincaré surfaces of section. In addition, we obtain some natural families of periodic orbits associated with the relative equilibria of the fourth order normalized system.