Continuity of Lyapunov functions and of energy level for generalized gradient semigroup
Contribuinte(s) |
UNIVERSIDADE DE SÃO PAULO |
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Data(s) |
23/10/2013
23/10/2013
2012
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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. CAPES/DGU [267/2008] CAPES/DGU FAPESP (Brazil) FAPESP, Brazil [2008/50248-0, 2008/55516-3] Ministerio de Ciencia e Innovacion [MTM2008-00088, PBH2006-0003-PC] Ministerio de Ciencia e Innovacion Junta de Andalucia, Spain Junta de Andalucia, Spain [P07-FQM-02468, FQM314, HF2008-0039] CNPq [302022/2008-2] CNPq Junta de Andalucia [P07-FQM-02468] Junta de Andalucia |
Identificador |
TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS, TORUN, v. 39, n. 1, pp. 57-82, MAR, 2012 1230-3429 |
Idioma(s) |
eng |
Publicador |
JULIUSZ SCHAUDER CTR NONLINEAR STUDIES TORUN |
Relação |
TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS |
Direitos |
closedAccess Copyright JULIUSZ SCHAUDER CTR NONLINEAR STUDIES |
Palavras-Chave | #MORSE DECOMPOSITION #GLOBAL ATTRACTOR #DYNAMICAL SYSTEMS #LYAPUNOV FUNCTION #TOPOLOGICAL-SPACES #SEMIFLOWS #EQUAÇÕES DIFERENCIAIS ORDINÁRIAS #EQUAÇÕES DIFERENCIAIS PARCIAIS #MATHEMATICS |
Tipo |
article original article publishedVersion |