Seminario de Ecuaciones Diferenciales

Título de la conferencia: Symmetric Lyapunov Center Theorem.
Conferenciante: Slawomir Rybicki (Universidad de Torun).
Abstract:
Let U:Ω⊂ RN → R be a potential of the class C2 defined on an open subset Ω⊂RN and let q0∈Ω be an isolated critical point of U i.e. U′(q0)=0.\\ The Lyapunov center theorem gives sufficient conditions for the existence of non-stationary periodic solutions of the system (∗)q¨(t)=−U′(q(t))in any neighborhood of q0. The aim of may talk is to present the Lyapunov center theorem for symmetric potentials. More precisely speaking, assume additionally that Ω⊂RN is an open and invariant subset of an orthogonal representation RN of a compact Lie group Γ and that the potential U:Ω→ RΓ-invariant i.e. it is constant on the orbits of the group Γ. Since the orbit of critical points Γ(q0)={γq0:γ∈Γ}⊂U′−1(0) is Γ -homeomorphic to Γ/Γq0( Γq0={γq0=q0:γ∈Γ} is the stabilizer of q0), the critical points of the potential U usually are not isolated in U′−1(0) and therefore we can not apply the Lyapunov center theorem to the study of non-stationary periodic solutions of the system (∗).Assume that the orbit Γ(q0)is isolated in U′−1(0). We will formulate sufficient conditions for the existence of periodic orbits of solutions of the equation (∗)in any neighborhood of the orbitΓ(q0)..Moreover, we will estimate the minimal period of these solutions. The basic idea of the proof is to apply the infinite-dimensional generalization of the(Γ×S1)-equivariant Conley index theory.