Seminario de Ecuaciones Diferenciales

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Título de la conferencia: Symmetric Lyapunov Center Theorem.
Conferenciante: Slawomir Rybicki (Universidad de Torun).
Let U:Ω⊂ RN R be a potential of the class C2 defined on an open subset Ω⊂RN and let q0Ω 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:γΓ}U1(0) is Γ -homeomorphic to Γ/Γq0( Γq0={γq0=q0:γΓ} is the stabilizer of q0), the critical points of the potential U usually are not isolated in U1(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 U1(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.