Detalles de Evento
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.
