American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdss.2020092

Morse decomposition for gradient-like multi-valued autonomous and nonautonomous dynamical systems

Yejuan Wang 1, and  Tomás Caraballo 2,

1. 

School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China
2. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012-Sevilla, Spain

Received  May 2018 Revised  August 2018

Fund Project: This work was supported by NSF of China (Grants No. 41875084, 11571153), and the Fundamental Research Funds for the Central Universities under Grant Nos. lzujbky-2018-it58 and lzujbky-2018-ot03, and partially supported by Ministerio de Economía y Competitividad (Spain), FEDER (European Community) under grant MTM2015-63723-P, and Consejería de Innovación Ciencia y Empresa de la Junta de Andalucía (Spain) under grant P12-FQM-1492

In this paper, we first prove that the property of being a gradient-like general dynamical system and the existence of a Morse decomposition are equivalent. Next, the stability of gradient-like general dynamical systems is analyzed. In particular, we show that a gradient-like general dynamical system is stable under perturbations, and that Morse sets are upper semi-continuous with respect to perturbations. Moreover, we prove that any solution of perturbed general dynamical systems should be close to some Morse set of the unperturbed gradient-like general dynamical system. We do not assume local compactness for the metric phase space $ X $, unlike previous results in the literature. Finally, we extend the Morse decomposition theory of single-valued nonautonomous dynamical systems to the multi-valued case, without imposing any compactness of the parameter spaces.

Keywords: Gradient-like general dynamical systems, Morse decomposition, nonautonomous multi-valued dynamical systems.
Mathematics Subject Classification: Primary: 35B40, 35B41, 37L30; Secondary: 58C06, 47H20.
Citation: Yejuan Wang, Tomás Caraballo. Morse decomposition for gradient-like multi-valued autonomous and nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020092
References:
[1]

E. R. Aragão-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Stability of gradient semigroups under perturbations, 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, Non-autonomous Morse-decomposition and lyapunov functions for gradient-like processes, Trans. Amer. Math. Soc., 365 (2013), 5277-5312.  doi: 10.1090/S0002-9947-2013-05810-2.  Google Scholar

[3]

T. Caraballo, P. E. Kloeden and P. Marín-Rubio, Weak pullback attractors of setvalued processes, J. Math. Anal. Appl., 288 (2003), 692-707.  doi: 10.1016/j.jmaa.2003.09.039.  Google Scholar

[4]

T. Caraballo, J. C. Jara, J. A. Langa and J. Valero, Morse decomposition of global attractors with infinite components, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 2845-2861.  doi: 10.3934/dcds.2015.35.2845.  Google Scholar

[5]

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

[6]

C. Conley, Isolated Invariant Sets and the Morse Index, In: Regional Conference Series in Mathematics 38, American Matematical Society, Providence, 1978.  Google Scholar

[7]

H. B. da Costa and J. Valero, Morse decompositions and Lyapunov functions for dynamically gradient multivalued semiflows, Nonlinear Dyn., 84 (2016), 19-34.  doi: 10.1007/s11071-015-2193-z.  Google Scholar

[8]

H. B. da Costa and J. Valero, Morse decompositions with infinite components for multivalued semiflows, Set-Valued Var. Anal., 25 (2017), 25-41.  doi: 10.1007/s11228-016-0363-x.  Google Scholar

[9]

F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.  Google Scholar

[10]

P. E. Kloeden, Asymptotic invariance and limit sets of general control systems, J. Diff. Eqns., 19 (1975), 91-105.  doi: 10.1016/0022-0396(75)90021-2.  Google Scholar

[11]

D. S. Li, On dynamical stability in general dynamical systems, J. Math. Anal. Appl., 263 (2001), 455-478.  doi: 10.1006/jmaa.2001.7620.  Google Scholar

[12]

D. S. Li, Morse decompositions for general dynamical systems and differential inclusions with applications to control systems, SIAM J. Control Optim., 46 (2007), 35-60.  doi: 10.1137/060662101.  Google Scholar

[13]

D. S. Li, Y. J. Wang and S. Y. Wang, On the dynamics of nonautonomous general dynamical systems and differential inclusions, Set-Valued Anal., 16 (2008), 651-671.  doi: 10.1007/s11228-007-0054-8.  Google Scholar

[14]

R. McGehee, Attractors for closed relations on compact Hausdorff spaces, Indiana Univ. Math. J., 41 (1992), 1165-1209.  doi: 10.1512/iumj.1992.41.41058.  Google Scholar

[15]

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

[16]

Y. J. Wang and D. S. Li, Morse decompositions for periodic general dynamical systems and differential inclusions, Set-Valued Var. Anal., 20 (2012), 519-549.  doi: 10.1007/s11228-012-0212-5.  Google Scholar

[17]

Y. J. Wang and D. S. Li, Morse decompositions for nonautonomous general dynamical systems, Set-Valued Var. Anal., 22 (2014), 117-154.  doi: 10.1007/s11228-013-0264-1.  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 perturbations, 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, Non-autonomous Morse-decomposition and lyapunov functions for gradient-like processes, Trans. Amer. Math. Soc., 365 (2013), 5277-5312.  doi: 10.1090/S0002-9947-2013-05810-2.  Google Scholar

[3]

T. Caraballo, P. E. Kloeden and P. Marín-Rubio, Weak pullback attractors of setvalued processes, J. Math. Anal. Appl., 288 (2003), 692-707.  doi: 10.1016/j.jmaa.2003.09.039.  Google Scholar

[4]

T. Caraballo, J. C. Jara, J. A. Langa and J. Valero, Morse decomposition of global attractors with infinite components, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 2845-2861.  doi: 10.3934/dcds.2015.35.2845.  Google Scholar

[5]

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

[6]

C. Conley, Isolated Invariant Sets and the Morse Index, In: Regional Conference Series in Mathematics 38, American Matematical Society, Providence, 1978.  Google Scholar

[7]

H. B. da Costa and J. Valero, Morse decompositions and Lyapunov functions for dynamically gradient multivalued semiflows, Nonlinear Dyn., 84 (2016), 19-34.  doi: 10.1007/s11071-015-2193-z.  Google Scholar

[8]

H. B. da Costa and J. Valero, Morse decompositions with infinite components for multivalued semiflows, Set-Valued Var. Anal., 25 (2017), 25-41.  doi: 10.1007/s11228-016-0363-x.  Google Scholar

[9]

F. Flandoli and B. Schmalfuß, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise, Stoch. Stoch. Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.  Google Scholar

[10]

P. E. Kloeden, Asymptotic invariance and limit sets of general control systems, J. Diff. Eqns., 19 (1975), 91-105.  doi: 10.1016/0022-0396(75)90021-2.  Google Scholar

[11]

D. S. Li, On dynamical stability in general dynamical systems, J. Math. Anal. Appl., 263 (2001), 455-478.  doi: 10.1006/jmaa.2001.7620.  Google Scholar

[12]

D. S. Li, Morse decompositions for general dynamical systems and differential inclusions with applications to control systems, SIAM J. Control Optim., 46 (2007), 35-60.  doi: 10.1137/060662101.  Google Scholar

[13]

D. S. Li, Y. J. Wang and S. Y. Wang, On the dynamics of nonautonomous general dynamical systems and differential inclusions, Set-Valued Anal., 16 (2008), 651-671.  doi: 10.1007/s11228-007-0054-8.  Google Scholar

[14]

R. McGehee, Attractors for closed relations on compact Hausdorff spaces, Indiana Univ. Math. J., 41 (1992), 1165-1209.  doi: 10.1512/iumj.1992.41.41058.  Google Scholar

[15]

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

[16]

Y. J. Wang and D. S. Li, Morse decompositions for periodic general dynamical systems and differential inclusions, Set-Valued Var. Anal., 20 (2012), 519-549.  doi: 10.1007/s11228-012-0212-5.  Google Scholar

[17]

Y. J. Wang and D. S. Li, Morse decompositions for nonautonomous general dynamical systems, Set-Valued Var. Anal., 22 (2014), 117-154.  doi: 10.1007/s11228-013-0264-1.  Google Scholar

[1]

Yejuan Wang. On the upper semicontinuity of pullback attractors for multi-valued noncompact random dynamical systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3669-3708. doi: 10.3934/dcdsb.2016116
[2]

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

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

Ming-Chia Li. Stability of parameterized Morse-Smale gradient-like flows. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 1073-1077. doi: 10.3934/dcds.2003.9.1073
[5]

Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587
[6]

Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Approximation of attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 215-238. doi: 10.3934/dcdsb.2005.5.215
[7]

Chunqiu Li, Desheng Li, Xuewei Ju. On the forward dynamical behavior of nonautonomous systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 473-487. doi: 10.3934/dcdsb.2019190
[8]

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 & Applied Analysis, 2012, 11 (6) : 2393-2416. doi: 10.3934/cpaa.2012.11.2393
[9]

Björn Schmalfuss. Attractors for nonautonomous and random dynamical systems perturbed by impulses. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 727-744. doi: 10.3934/dcds.2003.9.727
[10]

David Cheban. Global attractors of nonautonomous quasihomogeneous dynamical systems. Conference Publications, 2001, 2001 (Special) : 96-101. doi: 10.3934/proc.2001.2001.96
[11]

Hongyong Cui, Peter E. Kloeden, Meihua Yang. Forward omega limit sets of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-12. doi: 10.3934/dcdss.2020065
[12]

Rubén Caballero, Alexandre N. Carvalho, Pedro Marín-Rubio, José Valero. Robustness of dynamically gradient multivalued dynamical systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1049-1077. doi: 10.3934/dcdsb.2019006
[13]

Yejuan Wang, Lin Yang. Global exponential attraction for multi-valued semidynamical systems with application to delay differential equations without uniqueness. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1961-1987. doi: 10.3934/dcdsb.2018257
[14]

S.Durga Bhavani, K. Viswanath. A general approach to stability and sensitivity in dynamical systems. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 131-140. doi: 10.3934/dcds.1998.4.131
[15]

Yancong Xu, Deming Zhu, Xingbo Liu. Bifurcations of multiple homoclinics in general dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 945-963. doi: 10.3934/dcds.2011.30.945
[16]

Yun Zhao, Wen-Chiao Cheng, Chih-Chang Ho. Q-entropy for general topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2059-2075. doi: 10.3934/dcds.2019086
[17]

Lev M. Lerman, Elena V. Gubina. Nonautonomous gradient-like vector fields on the circle: Classification, structural stability and autonomization. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-27. doi: 10.3934/dcdss.2020076
[18]

Marta Štefánková. Inheriting of chaos in uniformly convergent nonautonomous dynamical systems on the interval. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3435-3443. doi: 10.3934/dcds.2016.36.3435
[19]

João Ferreira Alves, Michal Málek. Zeta functions and topological entropy of periodic nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 465-482. doi: 10.3934/dcds.2013.33.465
[20]

Xin Li, Wenxian Shen, Chunyou Sun. Invariant measures for complex-valued dissipative dynamical systems and applications. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2427-2446. doi: 10.3934/dcdsb.2017124

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (29)
  • HTML views (238)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences