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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.  Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonmous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.   Google Scholar

[13]

V. V. Chepyzhov and M. I. Vishik, Attractors of Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002.  Google Scholar

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H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

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J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr. 25, Amer. Math. Soc., R.I., 1988.  Google Scholar

[16]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, 1981.  Google Scholar

[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.  Google Scholar

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P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268.  doi: 10.1090/proc/12735.  Google Scholar

[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.  Google Scholar

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O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge Univ. Press, Cambridge, New-York, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

D. S. Li, Morse theory of attractors via Lyapunov functions, preprint, arXiv: 1003.0305v1. Google Scholar

[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.  Google Scholar

[25]

J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

M. I. Vishik, Asymptotic Behavior of Solutions of Evlutionary Equations, Cambridge Univ. Press, Cambriage, England, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

V. V. Chepyzhov and M. I. Vishik, Attractors of nonautonmous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333.   Google Scholar

[13]

V. V. Chepyzhov and M. I. Vishik, Attractors of Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002.  Google Scholar

[14]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.  doi: 10.1007/BF02219225.  Google Scholar

[15]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr. 25, Amer. Math. Soc., R.I., 1988.  Google Scholar

[16]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

O. A. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge Univ. Press, Cambridge, New-York, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

D. S. Li, Morse theory of attractors via Lyapunov functions, preprint, arXiv: 1003.0305v1. Google Scholar

[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.  Google Scholar

[25]

J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, 2001. doi: 10.1007/978-94-010-0732-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

M. I. Vishik, Asymptotic Behavior of Solutions of Evlutionary Equations, Cambridge Univ. Press, Cambriage, England, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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