Abstract
We study the existence of periodic solutions in the neighbourhood ofsymmetric (partially) elliptic equilibria in purely reversible Hamiltonian vectorfields. These are Hamiltonian vector fields with an involutory reversing symmetry$R$. We contrast the cases where $R$ acts symplectically and anti-symplectically.In case $R$ acts anti-symplectically, generically purely imaginary eigenvaluesare isolated, and the equilibrium is contained in a local two-dimensional invariantmanifold containing symmetric periodic solutions encircling the equilibriumpoint.In case $R$ acts symplectically, generically purely imaginary eigenvaluesare doubly degenerate, and the equilibrium is contained in two two-dimensional invariantmanifolds containing nonsymmetric periodic solutions encirclingthe equilibrium point. In addition, there exists a three-dimensional invariantsurface containing a two-parameter family of symmetric periodic solutions.
Highlights
It is well known that reversible and Hamiltonian dynamical systems have many striking features in common, see for instance [18, 17, 13]
It is well known [1] that generic elliptic equilibria in Hamiltonian vector fields are contained in two-dimensional manifolds containing one-parameter families of periodic solutions
In this paper we present a Liapunov Center theorem for purely reversible Hamiltonian vector fields, that are vector fields which are Hamiltonian and reversible at the same time, describing all periodic solutions in the neighbourhood of typical elliptic equilibria
Summary
To cite this version: Claudio Buzzi, Jeroen Lamb. Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2004, 10.3934/dcdsb.2005.5.51. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés
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