The Liapunov Center Theorem for a Class of Equivariant Hamiltonian Systems

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 𝑆2=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.