American Institute of Mathematical Sciences

Advanced
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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[20]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[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.

[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.

[16]

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

[17]

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

[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.

[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.

[20]

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

[1]

Yejuan Wang, Tomás Caraballo. Morse decomposition for gradient-like multi-valued autonomous and nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2303-2326. doi: 10.3934/dcdss.2020092
[2]

Ming-Chia Li. Stability of parameterized Morse-Smale gradient-like flows. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 1073-1077. doi: 10.3934/dcds.2003.9.1073
[3]

Peter Takáč. Stabilization of positive solutions for analytic gradient-like systems. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 947-973. doi: 10.3934/dcds.2000.6.947
[4]

Elena Nozdrinova, Olga Pochinka. Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1101-1131. doi: 10.3934/dcds.2020311
[5]

Lev M. Lerman, Elena V. Gubina. Nonautonomous gradient-like vector fields on the circle: Classification, structural stability and autonomization. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1341-1367. doi: 10.3934/dcdss.2020076
[6]

Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2393-2416. doi: 10.3934/cpaa.2012.11.2393
[7]

Xuewei Ju, Desheng Li, Jinqiao Duan. Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1175-1197. doi: 10.3934/dcdsb.2019011
[8]

Matteo Tanzi, Lai-Sang Young. Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 6015-6041. doi: 10.3934/dcds.2020257
[9]

Rafael Ayala, Jose Antonio Vilches, Gregor Jerše, Neža Mramor Kosta. Discrete gradient fields on infinite complexes. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 623-639. doi: 10.3934/dcds.2011.30.623
[10]

J. K. Krottje. On the dynamics of a mixed parabolic-gradient system. Communications on Pure and Applied Analysis, 2003, 2 (4) : 521-537. doi: 10.3934/cpaa.2003.2.521
[11]

Qinghua Ma, Zuoliang Xu, Liping Wang. Recovery of the local volatility function using regularization and a gradient projection method. Journal of Industrial and Management Optimization, 2015, 11 (2) : 421-437. doi: 10.3934/jimo.2015.11.421
[12]

Yanfei Wang, Qinghua Ma. A gradient method for regularizing retrieval of aerosol particle size distribution function. Journal of Industrial and Management Optimization, 2009, 5 (1) : 115-126. doi: 10.3934/jimo.2009.5.115
[13]

Sergey A. Denisov. Infinite superlinear growth of the gradient for the two-dimensional Euler equation. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 755-764. doi: 10.3934/dcds.2009.23.755
[14]

Mauro Patrão, Luiz A. B. San Martin. Morse decomposition of semiflows on fiber bundles. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 561-587. doi: 10.3934/dcds.2007.17.561
[15]

Yigui Ou, Haichan Lin. A class of accelerated conjugate-gradient-like methods based on a modified secant equation. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1503-1518. doi: 10.3934/jimo.2019013
[16]

Carlos Cabrera, Peter Makienko, Peter Plaumann. Semigroup representations in holomorphic dynamics. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1333-1349. doi: 10.3934/dcds.2013.33.1333
[17]

Esmail Abdul Fattah, Janet Van Niekerk, Håvard Rue. Smart Gradient - An adaptive technique for improving gradient estimation. Foundations of Data Science, 2022, 4 (1) : 123-136. doi: 10.3934/fods.2021037
[18]

Arnaud Goullet, Shaun Harker, Konstantin Mischaikow, William D. Kalies, Dinesh Kasti. Efficient computation of Lyapunov functions for Morse decompositions. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2419-2451. doi: 10.3934/dcdsb.2015.20.2419
[19]

Delio Mugnolo, René Pröpper. Gradient systems on networks. Conference Publications, 2011, 2011 (Special) : 1078-1090. doi: 10.3934/proc.2011.2011.1078
[20]

Yuhong Dai, Ya-xiang Yuan. Analysis of monotone gradient methods. Journal of Industrial and Management Optimization, 2005, 1 (2) : 181-192. doi: 10.3934/jimo.2005.1.181

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (99)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy