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August  2020, 13(8): 2303-2326. 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, 2020, 13 (8) : 2303-2326. 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

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