The Liapunov Center Theorem for a Class of Equivariant Hamiltonian Systems

Abstract EN

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.