Symmetric Liapunov center theorem

Abstract

In this article, using an infinite-dimensional equivariant Conley index, we prove a generalization of the profitable Liapunov center theorem for symmetric potentials. Consider the system $\ddot{q}=-\nabla U(q),$ where $U(q)$ is a $\Gamma$-symmetric potential, where $\Gamma$ is a compact Lie group acting linearly on ${\mathbb{R}}^n$. If the system possess a non-degenerate orbit of stationary solutions $\Gamma (q_0)$ with trivial isotropy group, such that there exists at least one positive eigenvalue of the Hessian ${\nabla }^{2}U(q_0)$, then in any neighbourhood of orbit $\Gamma (q_0)$ there is a periodic orbit of solutions of the system.

22 pages

Cited by in corpus (4)

• Periodic solutions of symmetric Hamiltonian systems
• Parameterized splitting theorems and bifurcations for potential operators, Part I: Abstract theory
• Symmetric Liapunov center theorem for orbit with nontrivial isotropy group
• Global bifurcation index of critical orbit of strongly indefinite functional