** 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 orbit

Γ(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.