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Symmetric Liapunov center theorem

2017; Springer Science+Business Media; Volume: 56; Issue: 2 Linguagem: Inglês

10.1007/s00526-017-1120-1

1432-0835

Ernesto Pérez-Chavela, Sławomir Rybicki, Daniel Strzelecki,

Nonlinear Waves and Solitons

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 $$(*)\; \ddot{q}= -\nabla U(q)$$ , where U(q) is a $$\Gamma $$ -invariant potential and $$\Gamma $$ is a compact Lie group acting linearly on $${\mathbb {R}}^n$$ . If 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 neighborhood of $$\Gamma (q_0)$$ there is a non-stationary periodic orbit of solutions of system $$(*)$$ .