Abstract and Applied Analysis
Volume 2012 (2012), Article ID 530209, 12 pages
http://dx.doi.org/10.1155/2012/530209
Research Article

The Liapunov Center Theorem for a Class of Equivariant Hamiltonian Systems

Jia Li and Yanling Shi

Department of Mathematics, Southeast University, Nanjing 210096, China

Received 6 May 2011; Revised 14 November 2011; Accepted 17 November 2011

Academic Editor: Roman Simon Hilscher

Copyright © 2012 Jia Li and Yanling Shi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the existence of the periodic solutions in the neighbourhood of equilibria for 𝐶 ∞ equivariant Hamiltonian vector fields. If the equivariant symmetry 𝑆 acts antisymplectically and 𝑆 2 = 𝐼 , we prove that generically purely imaginary eigenvalues are doubly degenerate and the equilibrium is contained in a local two-dimensional flow-invariant manifold, consisting of a one-parameter family of symmetric periodic solutions and two two-dimensional flow-invariant manifolds each containing a one-parameter family of nonsymmetric periodic solutions. The result is a version of Liapunov Center theorem for a class of equivariant Hamiltonian systems.