Journal of Nonlinear Science

, Volume 21, Issue 6, pp 835–874 | Cite as

Bifurcations of the Hamiltonian Fourfold 1:1 Resonance with Toroidal Symmetry

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Abstract

This paper deals with the analysis of Hamiltonian Hopf as well as saddle-center bifurcations in 4-DOF systems defined by perturbed isotropic oscillators (1:1:1:1 resonance), in the presence of two quadratic symmetries Ξ and L 1. When we normalize the system with respect to the quadratic part of the energy and carry out a reduction with respect to a three-torus group we end up with a 1-DOF system with several parameters on the thrice reduced phase space. Then, we focus our analysis on the evolution of relative equilibria around singular points of this reduced phase space. In particular, dealing with the Hamiltonian Hopf bifurcation the ‘geometric approach’ is used, following the steps set up by one of the authors in the context of 3-DOF systems. In order to see the interplay between integrals and physical parameters in the analysis of bifurcations, we consider as a perturbation a one-parameter family, which in particular includes one of the classical Stark–Zeeman models (parallel case) in three dimensions.

Keywords

Hamiltonian system Fourfold 1:1 resonance Bifurcation Normal form Reduction Hamiltonian Hopf bifurcation 

Mathematics Subject Classification (2000)

37S20 37S15 53D20 53D05 53D17 70H33 70H12 34C14 
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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Departamento de Matemática AplicadaUniversidad de MurciaEspinardoSpain
  2. 2.Faculteit Wiskunde en InformaticaTechnische Universiteit EindhovenEindhovenThe Netherlands

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