Symmetric Liapunov center theorem

In this article, using an infinite-dimensional equivariant Conley index, we prove a generalization of the profitable Liapunov center theorem for symmetric potentials. Consider a system ( ∗ ) q ¨ = - ∇ U ( q ) , where U ( q ) is a Γ -invariant potential and Γ is a compact Lie group acting linearly on R n . If system ( ∗ ) possess a non-degenerate orbit of stationary solutions Γ ( q 0 ) with trivial isotropy group, such that there exists at least one positive eigenvalue of the Hessian ∇ 2 U ( q 0 ) , then in any neighborhood of Γ ( q 0 ) there is a non-stationary periodic orbit of solutions of system ( ∗ ) .