Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations

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March 2019, 24(3): 1175-1197. doi: 10.3934/dcdsb.2019011

Forward attraction of pullback attractors and synchronizing behavior of gradient-like systems with nonautonomous perturbations

Xuewei Ju ^1, , Desheng Li ^2,, and Jinqiao Duan ^3,

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

^* Corresponding author

Received May 2017 Revised December 2017 Published January 2019

Fund Project: Supported by NNSF of China under the grant 11471240 and the Foundation Research Funds for the Central Universities under the grant 3122014K008

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 [4,6] that $ {\mathscr A} _\varepsilon $ inherits the natural Morse structure of $ {\mathscr A} $. In this present work, we introduce the notion of nonautonomous equilibria and give a more precise description on the Morse structure of $ {\mathscr A} _\varepsilon $ and the asymptotically synchronizing behavior of the perturbed system. Based on the above result we further prove that the sections of $ {\mathscr A} _\varepsilon $ depend on time symbol continuously in the sense of Hausdorff distance. Consequently, one concludes that $ {\mathscr A} _\varepsilon $ is a forward attractor of the perturbed nonautonomous system. It will also be shown that the perturbed system exhibits completely a global forward synchronizing behavior with the external force.

Keywords: Gradient-like system, nonautonomous perturbation, nonautonomous equilibrium, Morse structure, pullback attractor, forward attractor, synchronizing behavior.

Mathematics Subject Classification: 37B35, 37B55, 37E10, 34D06, 34D45, 34D06.

Citation: 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

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, 2^nd 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, 2^nd 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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