Continuation of relative periodic orbits in a class of triatomic Hamiltonian systems

  • Guillaume James

    Institut National Polytechnique de Grenoble and CNRS, Laboratoire Jean Kuntzmann (UMR 5224), tour IRMA, BP 53, 38041 Grenoble Cedex 9, France
  • Pascal Noble

    Université de Lyon, Université Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43, boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France
  • Yannick Sire

    Laboratoire d'Analyse Topologie Probabilités, UMR 6632, Université Aix-Marseille III, Faculté des Sciences et Techniques Saint-Jérôme, Avenue Escadrille Normandie-Niemen, 13397 Marseille Cedex 20, France

Abstract

We study relative periodic orbits (i.e. time-periodic orbits in a frame rotating at constant velocity) in a class of triatomic Euclidean-invariant (planar) Hamiltonian systems. The system consists of two identical heavy atoms and a light one, and the atomic mass ratio is treated as a continuation parameter. Under some nondegeneracy conditions, we show that a given family of relative periodic orbits existing at infinite mass ratio (and parametrized by phase, rotational degree of freedom and period) persists for sufficiently large mass ratio and for nearby angular velocities (this result is valid for small angular velocities). The proof is based on a method initially introduced by Sepulchre and MacKay [J.-A. Sepulchre, R.S. MacKay, Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators, Nonlinearity 10 (1997) 679–713] and further developed by Muñoz-Almaraz et al. [F.J. Muñoz-Almaraz, et al., Continuation of periodic orbits in conservative and Hamiltonian systems, Physica D 181 (2003) 1–38] for the continuation of normal periodic orbits in Hamiltonian systems. Our results provide several types of relative periodic orbits, which extend from small amplitude relative normal modes [J.-P. Ortega, Relative normal modes for nonlinear Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 665–704] up to large amplitude solutions which are not restrained to a small neighborhood of a stable relative equilibrium. In particular, we show the existence of large amplitude motions of inversion, where the light atom periodically crosses the segment between heavy atoms. This analysis is completed by numerical results on the stability and bifurcations of some inversion orbits as their angular velocity is varied.

Cite this article

Guillaume James, Pascal Noble, Yannick Sire, Continuation of relative periodic orbits in a class of triatomic Hamiltonian systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 4, pp. 1237–1264

DOI 10.1016/J.ANIHPC.2008.10.002