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Polarization dynamics in a resonant optical medium with initial coherence between degenerate states
Morse decomposition for gradient-like multi-valued autonomous and nonautonomous dynamical systems
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 |
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.
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Stability of gradient semigroups under perturbations, Nonlinearity, 24 (2011), 2099-2117.
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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.
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T. Caraballo, P. E. Kloeden and P. Marín-Rubio,
Weak pullback attractors of setvalued processes, J. Math. Anal. Appl., 288 (2003), 692-707.
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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. |
[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. |
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C. Conley, Isolated Invariant Sets and the Morse Index, In: Regional Conference Series in Mathematics 38, American Matematical Society, Providence, 1978. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer, Berlin, 1987.
doi: 10.1007/978-3-642-72833-4. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
C. Conley, Isolated Invariant Sets and the Morse Index, In: Regional Conference Series in Mathematics 38, American Matematical Society, Providence, 1978. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer, Berlin, 1987.
doi: 10.1007/978-3-642-72833-4. |
[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. |
[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. |
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