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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Non-autonomous Morse-decomposition and Lyapunov functions for gradient-like processes


Authors: E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa
Journal: Trans. Amer. Math. Soc. 365 (2013), 5277-5312
MSC (2010): Primary 37B25, 37B35; Secondary 37B55, 35B40, 35B41
Published electronically: April 1, 2013
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Abstract: We define (time dependent) Morse-decompositions for non-autonomous evolution processes (non-autonomous dynamical systems) and prove that a non-autonomous gradient-like evolution process possesses a Morse-decomposition on the associated pullback attractor. We also prove the existence of an associated Lyapunov function which describes the gradient behavior of the system. Finally, we apply these abstract results to non-autonomous perturbations of autonomous gradient-like evolution processes (semigroups or autonomous dynamical systems).


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Additional Information

E. R. Aragão-Costa
Affiliation: Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil
Email: ritis@icmc.usp.br

T. Caraballo
Affiliation: Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla Spain
Email: caraball@us.es

A. N. Carvalho
Affiliation: Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil
Email: andcarva@icmc.usp.br

J. A. Langa
Affiliation: Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla Spain
Email: langa@us.es

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05810-2
PII: S 0002-9947(2013)05810-2
Keywords: Non-autonomous Morse decomposition, Lyapunov functions for non-autonomous dynamical systems, robustness of dynamical structures
Received by editor(s): April 8, 2011
Received by editor(s) in revised form: September 21, 2011, and February 3, 2012
Published electronically: April 1, 2013
Additional Notes: The first author was partially supported by CAPES/DGU 267/2008 and FAPESP 2008/50248-0, Brazil
The second author was partially supported by FEDER and Ministerio de Ciencia e Innovación grant # MTM2008-0088, # MTM2011-22411, PBH2006-0003-PC, and Junta de Andalucía grants # P07-FQM-02468, # FQM314 and HF2008-0039, Spain
The third author was partially supported by CNPq 302022/2008-2, CAPES/DGU 267/2008 and FAPESP 2008/55516-3, Brazil and Junta de Andalucía grant # P07-FQM-02468
The fourth author was partially supported by FEDER and Ministerio de Ciencia e Innovación grants # MTM2008-0088, # MTM2011-22411, # PBH2006-0003-PC, and Junta de Andalucía grants # P07-FQM-02468, # FQM314 and HF2008-0039, Spain
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.