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A topological characterization of the $\omega$-limit sets of analytic vector fields on open subsets of the sphere
Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations
1. | Department of Mathematics, School of Science, Civil Aviation University of China, Tianjin 300300, China |
2. | School of Mathematics, Tianjin University, Tianjin 300072, China |
3. | Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA |
We consider the nonautonomous perturbation $ x_t+Ax = f(x)+\varepsilon h(t) $ of a gradient-like system $ x_t+Ax = f(x) $ in a Banach space $ X $, where $ A $ is a sectorial operator with compact resolvent. Assume the non-perturbed system $ x_t+Ax = f(x) $ has an attractor $ {\mathscr A} $. Then it can be shown that the perturbed one has a pullback attractor $ {\mathscr A} _\varepsilon $ near $ {\mathscr A} $. If all the equilibria of the non-perturbed system in $ {\mathscr A} $ are hyperbolic, we also infer from [
References:
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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.
doi: 10.1090/S0002-9947-2013-05810-2. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam, 1992. |
[3] |
P. Brunovský and P. Poláčik,
The Morse–Smale structure of a generic reaction–diffusion equation in higher space dimension, J. Differential Equations, 135 (1997), 129-181.
doi: 10.1006/jdeq.1996.3234. |
[4] |
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. |
[5] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional NonAutonomous Dynamical Systems, Appl. Math. Sci. 182, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[6] |
A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Su$\acute{a}$rez,
Characterization of nonautonomous attractors of a perturbed gradient system, J. Differential Equations, 236 (2007), 570-603.
doi: 10.1016/j.jde.2007.01.017. |
[7] |
T. Caraballo, J. C. Jara, J. A. Langa and Z. X. Liu,
Morse decomposition of attractors for
non-autonomous dynamical systems, Adv. Nonlinear Stud., 13 (2013), 309-329.
doi: 10.1515/ans-2013-0204. |
[8] |
T. Caraballo, J. A. Langa and Z. X. Liu,
Gradient infinite-dimensional random dynamical
system, SIAM J. Appl. Dyn. Syst., 11 (2012), 1817-1847.
doi: 10.1137/120862752. |
[9] |
D. Cheban, P. Kloeden and B. Schmalfuss,
The relation between pullback and global attractors for nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory, 2 (2002), 125-144.
|
[10] |
D. Cheban, C. Mammana and E. Michetti,
The structure of global attractors for nonautonomous perturbations of discrete gradient-like dynamical systems, J. Difference Equ. Appl., 22 (2016), 1673-1697.
doi: 10.1080/10236198.2016.1234616. |
[11] |
X. Chen and J. Q. Duan,
State space decomposition for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 957-974.
doi: 10.1017/S0308210510000661. |
[12] |
V. V. Chepyzhov and M. I. Vishik,
Attractors of nonautonmous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.
|
[13] |
V. V. Chepyzhov and M. I. Vishik, Attractors of Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002. |
[14] |
H. Crauel, A. Debussche and F. Flandoli,
Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[15] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr. 25, Amer. Math. Soc., R.I., 1988. |
[16] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, 1981. |
[17] |
L. Kapitanski and I. Rodnianski,
Shape and morse theory of attractors, Comm. Pure Appl. Math., 53 (2000), 218-242.
doi: 10.1002/(SICI)1097-0312(200002)53:2<218::AID-CPA2>3.0.CO;2-W. |
[18] |
P. E. Kloeden and T. Lorenz,
Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268.
doi: 10.1090/proc/12735. |
[19] |
P. E. Kloeden and H. M. Rodrigues,
Dynamics of a class of ODEs more general than almost
periodic, Nonlinear Anal., 74 (2011), 2695-2719.
doi: 10.1016/j.na.2010.12.025. |
[20] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge Univ. Press, Cambridge, New-York, 1991.
doi: 10.1017/CBO9780511569418. |
[21] |
D. S. Li and X. X. Zhang,
On dynamical properties of general dynamical systems and differential inclusions, J. Appl. Math. Anal. Appl., 274 (2002), 705-724.
doi: 10.1016/S0022-247X(02)00352-9. |
[22] |
D. S. Li and J. Q. Duan,
Structure of the set of bounded solutions for a class of nonautonomous
second-order differential equations, J. Differential Equations, 246 (2009), 1754-1773.
doi: 10.1016/j.jde.2008.10.031. |
[23] |
D. S. Li, Morse theory of attractors via Lyapunov functions, preprint, arXiv: 1003.0305v1. |
[24] |
M. Rasmussen,
Morse decompositions of nonautonomous dynamical systems, Trans. Amer. Math. Soc., 359 (2007), 5091-5115.
doi: 10.1090/S0002-9947-07-04318-8. |
[25] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, 2001.
doi: 10.1007/978-94-010-0732-0. |
[26] |
G. R. Sell and Y. C. You, Dynamics of Evolution Equations, Springer-Verlag, New York, 2002.
doi: 10.1007/978-1-4757-5037-9. |
[27] |
W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc., 136 (1998), x+93 pp.
doi: 10.1090/memo/0647. |
[28] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition,
Springer Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[29] |
M. I. Vishik, Asymptotic Behavior of Solutions of Evlutionary Equations, Cambridge Univ. Press, Cambriage, England, 1992. |
[30] |
M. I. Vishik, S. V. Zelik and V. V. Chepyzhov,
Regular attractors and nonautonomous perturbations of them, Mat. Sb., 204 (2013), 1-42.
doi: 10.1070/SM2013v204n01ABEH004290. |
[31] |
Y. J. Wang, D. S. Li and P. E. Kloeden,
On the asymptotical behavior of nonautonomous
dynamical systems, Nonlinear Anal., 59 (2004), 35-53.
doi: 10.1016/j.na.2004.03.035. |
show all references
References:
[1] |
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. |
[2] |
A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam, 1992. |
[3] |
P. Brunovský and P. Poláčik,
The Morse–Smale structure of a generic reaction–diffusion equation in higher space dimension, J. Differential Equations, 135 (1997), 129-181.
doi: 10.1006/jdeq.1996.3234. |
[4] |
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. |
[5] |
A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional NonAutonomous Dynamical Systems, Appl. Math. Sci. 182, 2013.
doi: 10.1007/978-1-4614-4581-4. |
[6] |
A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Su$\acute{a}$rez,
Characterization of nonautonomous attractors of a perturbed gradient system, J. Differential Equations, 236 (2007), 570-603.
doi: 10.1016/j.jde.2007.01.017. |
[7] |
T. Caraballo, J. C. Jara, J. A. Langa and Z. X. Liu,
Morse decomposition of attractors for
non-autonomous dynamical systems, Adv. Nonlinear Stud., 13 (2013), 309-329.
doi: 10.1515/ans-2013-0204. |
[8] |
T. Caraballo, J. A. Langa and Z. X. Liu,
Gradient infinite-dimensional random dynamical
system, SIAM J. Appl. Dyn. Syst., 11 (2012), 1817-1847.
doi: 10.1137/120862752. |
[9] |
D. Cheban, P. Kloeden and B. Schmalfuss,
The relation between pullback and global attractors for nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory, 2 (2002), 125-144.
|
[10] |
D. Cheban, C. Mammana and E. Michetti,
The structure of global attractors for nonautonomous perturbations of discrete gradient-like dynamical systems, J. Difference Equ. Appl., 22 (2016), 1673-1697.
doi: 10.1080/10236198.2016.1234616. |
[11] |
X. Chen and J. Q. Duan,
State space decomposition for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 957-974.
doi: 10.1017/S0308210510000661. |
[12] |
V. V. Chepyzhov and M. I. Vishik,
Attractors of nonautonmous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.
|
[13] |
V. V. Chepyzhov and M. I. Vishik, Attractors of Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002. |
[14] |
H. Crauel, A. Debussche and F. Flandoli,
Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.
doi: 10.1007/BF02219225. |
[15] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr. 25, Amer. Math. Soc., R.I., 1988. |
[16] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, 1981. |
[17] |
L. Kapitanski and I. Rodnianski,
Shape and morse theory of attractors, Comm. Pure Appl. Math., 53 (2000), 218-242.
doi: 10.1002/(SICI)1097-0312(200002)53:2<218::AID-CPA2>3.0.CO;2-W. |
[18] |
P. E. Kloeden and T. Lorenz,
Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268.
doi: 10.1090/proc/12735. |
[19] |
P. E. Kloeden and H. M. Rodrigues,
Dynamics of a class of ODEs more general than almost
periodic, Nonlinear Anal., 74 (2011), 2695-2719.
doi: 10.1016/j.na.2010.12.025. |
[20] |
O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge Univ. Press, Cambridge, New-York, 1991.
doi: 10.1017/CBO9780511569418. |
[21] |
D. S. Li and X. X. Zhang,
On dynamical properties of general dynamical systems and differential inclusions, J. Appl. Math. Anal. Appl., 274 (2002), 705-724.
doi: 10.1016/S0022-247X(02)00352-9. |
[22] |
D. S. Li and J. Q. Duan,
Structure of the set of bounded solutions for a class of nonautonomous
second-order differential equations, J. Differential Equations, 246 (2009), 1754-1773.
doi: 10.1016/j.jde.2008.10.031. |
[23] |
D. S. Li, Morse theory of attractors via Lyapunov functions, preprint, arXiv: 1003.0305v1. |
[24] |
M. Rasmussen,
Morse decompositions of nonautonomous dynamical systems, Trans. Amer. Math. Soc., 359 (2007), 5091-5115.
doi: 10.1090/S0002-9947-07-04318-8. |
[25] |
J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, 2001.
doi: 10.1007/978-94-010-0732-0. |
[26] |
G. R. Sell and Y. C. You, Dynamics of Evolution Equations, Springer-Verlag, New York, 2002.
doi: 10.1007/978-1-4757-5037-9. |
[27] |
W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc., 136 (1998), x+93 pp.
doi: 10.1090/memo/0647. |
[28] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition,
Springer Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[29] |
M. I. Vishik, Asymptotic Behavior of Solutions of Evlutionary Equations, Cambridge Univ. Press, Cambriage, England, 1992. |
[30] |
M. I. Vishik, S. V. Zelik and V. V. Chepyzhov,
Regular attractors and nonautonomous perturbations of them, Mat. Sb., 204 (2013), 1-42.
doi: 10.1070/SM2013v204n01ABEH004290. |
[31] |
Y. J. Wang, D. S. Li and P. E. Kloeden,
On the asymptotical behavior of nonautonomous
dynamical systems, Nonlinear Anal., 59 (2004), 35-53.
doi: 10.1016/j.na.2004.03.035. |
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