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July  2015, 35(7): 2845-2861. doi: 10.3934/dcds.2015.35.2845

Morse decomposition of global attractors with infinite components

Tomás Caraballo 1, , Juan C. Jara 1, , José A. Langa 2, and  José Valero 3,

1. 

Dpto. Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain
2. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla
3. 

Centro de Investigación Operativa, Universidad Miguel Hernández de Elche, Avda. de la Universidad, s/n, 03202 Elche

Received  December 2013 Revised  December 2014 Published  January 2015

In this paper we describe some dynamical properties of a Morse decomposition with a countable number of sets. In particular, we are able to prove that the gradient dynamics on Morse sets together with a separation assumption is equivalent to the existence of an ordered Lyapunov function associated to the Morse sets and also to the existence of a Morse decomposition -that is, the global attractor can be described as an increasing family of local attractors and their associated repellers.
Keywords: Morse decomposition, infinite components, gradient dynamics, gradient-like semigroup., Lyapunov function.
Mathematics Subject Classification: 37B25, 37L99, 35B40, 35B4.
Citation: Tomás Caraballo, Juan C. Jara, José A. Langa, José Valero. Morse decomposition of global attractors with infinite components. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 2845-2861. doi: 10.3934/dcds.2015.35.2845
References:
[1]

E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Stability of gradient semigroups under perturbation, Nonlinearity, 24 (2011), 2099-2117. doi: 10.1088/0951-7715/24/7/010.  Google Scholar

[2]

E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Continuity of Lyapunov functions and of energy level for a generalized gradient system, Topological Methods Nonl. Anal., 39 (2012), 57-82.  Google Scholar

[3]

J. M. Arrieta, A. Rodríguez-Bernal and J. Valero, Dynamics of a reaction-diffusion equation with a discontinuous nonlinearity, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 2695-2984. doi: 10.1142/S0218127406016586.  Google Scholar

[4]

A. V. Babin and M. I. Vishik, Attractors in Evolutionary Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[5]

A. N. Carvalho and J. A. Langa, An extension of the concept of gradient semigroups which is stable under perturbation, J. Differential Equations, 246 (2009), 2646-2668. doi: 10.1016/j.jde.2009.01.007.  Google Scholar

[6]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Series, 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[7]

A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Suárez, Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J. Differential Equations, 236 (2007), 570-603. doi: 10.1016/j.jde.2007.01.017.  Google Scholar

[8]

C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38, American Mathematical Society, Providence, R.I., 1978.  Google Scholar

[9]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs Number, 25, American Mathematical Society, Providence, RI, 1988.  Google Scholar

[10]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin, 1981.  Google Scholar

[11]

M. Hurley, Chain recurrence, semiflows and gradients, J. Dyn. Diff. Equations, 7 (1995), 437-456. doi: 10.1007/BF02219371.  Google Scholar

[12]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[13]

D. E. Norton, The fundamental theorem of dynamical systems, Comment. Math., Univ. Carolinae, 36 (1995), 585-597.  Google Scholar

[14]

M. Patrao, Morse decomposition of semiflows on topological spaces, J. Dyn. Diff. Equations, 19 (2007), 181-198. doi: 10.1007/s10884-006-9033-2.  Google Scholar

[15]

M. Patrao and Luiz A.B. San Martin, Semiflows on topological spaces: Chain transitivity and semigroups, J. Dyn. Diff. Equations, 19 (2007), 155-180. doi: 10.1007/s10884-006-9032-3.  Google Scholar

[16]

J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, England, 2001. doi: 10.1007/978-94-010-0732-0.  Google Scholar

[17]

K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Universitext, Springer-Verlag, 1987. doi: 10.1007/978-3-642-72833-4.  Google Scholar

[18]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[19]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[20]

M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations, Cambridge University Press, Cambridge, England, 1992.  Google Scholar

show all references

References:
[1]

E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Stability of gradient semigroups under perturbation, Nonlinearity, 24 (2011), 2099-2117. doi: 10.1088/0951-7715/24/7/010.  Google Scholar

[2]

E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Continuity of Lyapunov functions and of energy level for a generalized gradient system, Topological Methods Nonl. Anal., 39 (2012), 57-82.  Google Scholar

[3]

J. M. Arrieta, A. Rodríguez-Bernal and J. Valero, Dynamics of a reaction-diffusion equation with a discontinuous nonlinearity, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 2695-2984. doi: 10.1142/S0218127406016586.  Google Scholar

[4]

A. V. Babin and M. I. Vishik, Attractors in Evolutionary Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[5]

A. N. Carvalho and J. A. Langa, An extension of the concept of gradient semigroups which is stable under perturbation, J. Differential Equations, 246 (2009), 2646-2668. doi: 10.1016/j.jde.2009.01.007.  Google Scholar

[6]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Series, 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[7]

A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Suárez, Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J. Differential Equations, 236 (2007), 570-603. doi: 10.1016/j.jde.2007.01.017.  Google Scholar

[8]

C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38, American Mathematical Society, Providence, R.I., 1978.  Google Scholar

[9]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs Number, 25, American Mathematical Society, Providence, RI, 1988.  Google Scholar

[10]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin, 1981.  Google Scholar

[11]

M. Hurley, Chain recurrence, semiflows and gradients, J. Dyn. Diff. Equations, 7 (1995), 437-456. doi: 10.1007/BF02219371.  Google Scholar

[12]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[13]

D. E. Norton, The fundamental theorem of dynamical systems, Comment. Math., Univ. Carolinae, 36 (1995), 585-597.  Google Scholar

[14]

M. Patrao, Morse decomposition of semiflows on topological spaces, J. Dyn. Diff. Equations, 19 (2007), 181-198. doi: 10.1007/s10884-006-9033-2.  Google Scholar

[15]

M. Patrao and Luiz A.B. San Martin, Semiflows on topological spaces: Chain transitivity and semigroups, J. Dyn. Diff. Equations, 19 (2007), 155-180. doi: 10.1007/s10884-006-9032-3.  Google Scholar

[16]

J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge, England, 2001. doi: 10.1007/978-94-010-0732-0.  Google Scholar

[17]

K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Universitext, Springer-Verlag, 1987. doi: 10.1007/978-3-642-72833-4.  Google Scholar

[18]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4757-5037-9.  Google Scholar

[19]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[20]

M. I. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations, Cambridge University Press, Cambridge, England, 1992.  Google Scholar

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