terça-feira, 22 de março de 2011

Palestra no ICMC: Morse Decomposition and Lyapunov Functions

O Instituto de Ciências Matemáticas e Computação (ICMC), da USP São Carlos convida a todos para a palestra Morse Decomposition and Lyapunov Functions, a ser ministrada por Éder Ritis Aragão Costa. O evento acontecerá nesta quarta-feira (23), às 10h30min, na sala 5-003.

Eder Ritis possui licenciatura em Matemática pela UNESP (2006) e atualmente é doutorando em Matemática pelo ICMC, onde desenvolve pesquisa na área de sistemas dinâmicos não-lineares sob a orientação do professor Alexandre Nolasco de Carvalho.

Confira abaixo o resumo da palestra, em inglês:

In this seminary, given a gradient-like nonlinear semigroup (in the sense of [CL]) in a general metric space, we construct a differentiable Lyapunov function for it proving that gradient-like nonlinear semigroups are in fact gradient semigroups. This is done without any compactness assumption on the associated group or semigroup and any additional assumption on the phase space in which it is defined. Our proofs, in comparison with the classical works as [C] and [R], are quit different and considerably extends the results there. We adopt an approach that that enable us to use the results on stability of gradient-like (see [CL]) semigroups to obtain the stability of gradient semigroups under perturbations.

For the construction of the Lyapunov function we will firstly prove that the disjoint family of isolated invariant sets of a gradient-like semigroup on a general metric space can be reordered in such a way that it becomes a Morse decomposition for the global attractor. Again, the proofs are intuitive, focused on the dynamics of the semigroup and, for instance, not based on chain recurrence and related more classical notions in this theory. A refinement of the results from [C] would lead us to define a generalized Lyapunov function, not only on the attractor but on the whole phase space. Indeed, we will say that a semigroup {T(t):t>=0} with a global attractor A and a disjoint family of isolated invariant sets E={E_1,... ,E_n} is a generalized gradient semigroup with respect to E if there exists a continuous function V:X--> IR such that the function that send each t>=0 in V(T(t)x) is decreasing for each x in X, the function V is constant in E_i for each i in {1,...n} and V(T(t)x)=V(x) for all t>=0 if and only if x belongs to the union of E_1,...E_n.

Our main result can be stated as follows Theorem. {T(t):t>=0} is a generalized gradient semigroup with respect to E if and only if it is a generalized gradient-like semigroup with respect to E. In addition the Lyapunov function V:X--> IR of a generalized gradient-like semigroup may be chosen in such a way that V(E_k)=k for each k=1,2,... ,n.