CONTINUITY OF LYAPUNOV FUNCTIONS AND OF ENERGY LEVEL FOR A GENERALIZED GRADIENT SEMIGROUP

Autor(es)

Aragao-Costa, Eder R. ^[1] ; Caraballo, Tomas ^[2] ; Carvalho, Alexandre N. ^[1] ; Langa, Jose A. ^[2]

Afiliação do(s) autor(es)

^[1] Univ Sao Paulo, Inst Ciencias Matemat & Comp, BR-13560970 Sao Carlos, SP - Brazil

^[2] Univ Seville, Dept Ecuac Diferenciales & Anal Numer, E-41080 Seville - Spain

• Tipo de documento: Artigo Científico
• Fonte: TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS; v. 39, n. 1, p. 57-82, MAR 2012.
• Citações Web of Science: 10

Resumo

The global attractor of a gradient-like semigroup has a Morse decomposition. Associated to this Morse decomposition there is a Lyapunov function (differentiable along solutions)-defined on the whole phase space- which proves relevant information on the structure of the attractor. In this paper we prove the continuity of these Lyapunov functions under perturbation. On the other hand, the attractor of a gradient-like semigroup also has an energy level decomposition which is again a Morse decomposition but with a total order between any two components. We claim that, from a dynamical point of view, this is the optimal decomposition of a global attractor; that is, if we start from the finest Morse decomposition, the energy level decomposition is the coarsest Morse decomposition that still produces a Lyapunov function which gives the same information about the structure of the attractor. We also establish sufficient conditions which ensure the stability of this kind of decomposition under perturbation. In particular, if connections between different isolated invariant sets inside the attractor remain under perturbation, we show the continuity of the energy level Morse decomposition. The class of Morse-Smale systems illustrates our results. (AU)

• Processo FAPESP: 08/50248-0 - Atratores para semigrupos alpha-vezes integrados semilineares parcialmente acoplados e sua caracterização
• Beneficiário: Éder Ritis Aragão Costa
• Modalidade de apoio: Bolsas no Brasil - Doutorado Direto
• Processo FAPESP: 08/55516-3 - Sistemas dinâmicos não lineares em espaços de dimensão infinita
• Beneficiário: Alexandre Nolasco de Carvalho
• Modalidade de apoio: Auxílio à Pesquisa - Temático