Symmetric Lyapunov center theorem for orbit with nontrivial isotropy group

Abstract

In this article, we prove two versions of the Lyapunov center theorem for symmetric potentials. We consider a second order autonomous system $$ \ddot q(t)=-\nabla U(q(t)) $$ in the presence of symmetries of a compact Lie group $\Gamma.$ We look for non-stationary periodic solutions of this system in a neighborhood of a $\Gamma$-orbit of critical points of the $\Gamma$-invariant potential $U.$ Our results generalize that of [13, 14]. As a topological tool, we use an infinite-dimensional generalization of the equivariant Conley index due to Izydorek, see [9].