doi: 10.3934/dcdsb.2022006
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Attractors of the Klein-Gordon-Schrödinger lattice systems with almost periodic nonlinear part

Department of Mathematics, The University of Jordan, Amman 11942, Jordan

* Corresponding author: Ahmed Y. Abdallah

Received  April 2021 Revised  October 2021 Early access January 2022

We study the existence of the uniform global attractor for a family of Klein-Gordon-Schrödingernon-autonomous infinite dimensional lattice dynamical systems with nonlinear part of the form $ f\left( u, v, t\right) $, where we introduce a suitable Banach space of functions $ \mathcal{\mathcal{W}} $ and we assume that $ f\left( \cdot , \cdot , t\right) $ is an element of the hull of an almost periodic function $ f_{0}\left( \cdot , \cdot , t\right) $ with values in $ \mathcal{\mathcal{W}} $.

Citation: Ahmed Y. Abdallah, Taqwa M. Al-Khader, Heba N. Abu-Shaab. Attractors of the Klein-Gordon-Schrödinger lattice systems with almost periodic nonlinear part. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022006
References:
[1]

A. Y. Abdallah, Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems, Comm. Pure. Appl. Anal., 5 (2006), 55-69.  doi: 10.3934/cpaa.2006.5.55.

[2]

A. Y. Abdallah, Attractors for first order lattice systems with almost periodic nonlinear part, Disc. Cont. Dyn. Sys.-B, 25 (2020), 1241-1255.  doi: 10.3934/dcdsb.2019218.

[3]

A. Y. Abdallah, Dynamics of second order lattice systems with almost periodic nonlinear part, Qual. Theory Dyn. Syst., 20 (2021), Paper No. 58, 23 pp. doi: 10.1007/s12346-021-00497-3.

[4]

A. Y. Abdallah, Uniform exponential attractors for non-autonomous Klein—Gordon—Schrödinger lattice systems in weighted spaces, Nonlinear Anal., 127 (2015), 279-297.  doi: 10.1016/j.na.2015.07.013.

[5]

A. Y. Abdallah and R. T. Wannan, Second order non-autonomous lattice systems and their uniform attractors, Comm. Pure Appl. Anal., 18 (2019), 1827-1846.  doi: 10.3934/cpaa.2019085.

[6]

P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.

[7]

J. Bell, Some threshold results for models of myelinated nerves, Math. Biosci., 54 (1981), 181-190.  doi: 10.1016/0025-5564(81)90085-7.

[8]

J. Bell and C. Cosner, Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quart. Appl. Math., 42 (1984), 1-14.  doi: 10.1090/qam/736501.

[9]

V. Bellrti and V. Pata, Attractors for semilinear strongly damped wave equation on $\mathbb{R}^{3}$, Discrete Contin. Dynam. Systems, 7 (2001), 719-735.  doi: 10.3934/dcds.2001.7.719.

[10]

A. M. Boughoufala and A. Y. Abdallah, Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 1549-1563.  doi: 10.3934/dcdsb.2020172.

[11]

T. CaraballoF. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity, J. Difference Equ. Appl., 17 (2011), 161-184.  doi: 10.1080/10236198.2010.549010.

[12]

T. CaraballoF. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020.

[13]

H. Chate and M. Courbage (Eds.), Lattice systems, Phys. D, 103 (1997), 1-612. 

[14]

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

[15]

S.-N. Chow, Lattice dynamical systems, Dynamical System, Lecture Notes in Mathematics (Springer, Berlin), (2003), 1–102. doi: 10.1007/978-3-540-45204-1_1.

[16]

S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems: I, IEEE Trans. Circuits Systems, 42 (1995), 746-751.  doi: 10.1109/81.473583.

[17]

S.-N. ChowJ. Mallet-Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Comput. Dynam., 4 (1996), 109-178. 

[18]

S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems: II, IEEE Trans. Circuits Systems, 42 (1995), 752-756. 

[19]

L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Systems, 40 (1993), 147-156. 

[20]

L. O. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits Systems, 35 (1988), 1257-1272.  doi: 10.1109/31.7600.

[21]

L. O. Chua and L. Yang, Cellular neural networks: Applications, IEEE Trans. Circuits Systems, 35 (1988), 1273-1290.  doi: 10.1109/31.7601.

[22]

T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems, Physica D, 67 (1993), 237-244.  doi: 10.1016/0167-2789(93)90208-I.

[23]

J. Huang, X. Han and S. Zhou, Uniform attractors for non-autonomous Klein-Gordon-Schrödinger lattice systems, Appl. Math. Mech., Engl. Ed. 30 (2009), 1597–1607. doi: 10.1007/s10483-009-1211-z.

[24]

X. JiaC. Zhao and X. Yang, Global attractor and Kolmogorov entropy of three component reversible Gray—Scott model on infinite lattices, Appl. Math. Comp., 218 (2012), 9781-9789.  doi: 10.1016/j.amc.2012.03.036.

[25]

R. Kapral, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.  doi: 10.1007/BF01192578.

[26]

J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572.  doi: 10.1137/0147038.

[27]

J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium, J. Theor. Biol., 148 (1991), 49-82. 

[28] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, Cambridge, 1982. 
[29]

C. Li, M. Zhao and C. Zhao, Pullback exponential attractors for nonautonomous Klein—Gordon—Schrö dinger equations on infinite lattices, Abstr. Appl. Anal., 2013 (2013), Art. ID 809476, 9 pp. doi: 10.1155/2013/809476.

[30]

H. Li and L. Sun, Upper semicontinuity of attractors for small perturbations of Klein-Gordon-Schrödinger lattice system, Adv. Difference Equ., 2014 (2014), 300, 16 pp. doi: 10.1186/1687-1847-2014-300.

[31]

J. C. OliveiraJ. M. Pereira and G. Perla Menzala, Attractors for second order periodic lattices with nonlinear damping, J. Difference Equ. Appl., 14 (2008), 899-921.  doi: 10.1080/10236190701859211.

[32]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[33]

L. M. Pecora and T. L. Carrol, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824.  doi: 10.1103/PhysRevLett.64.821.

[34]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Appl. Math. Sci., 68, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[35]

B. Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl., 331 (2007), 121-136.  doi: 10.1016/j.jmaa.2006.08.070.

[36]

X. YangC. Zhao and J. Cao, Dynamics of the discrete coupled nonlinear Schrödinger—Boussinesq equations, Appl. Math. Comp., 219 (2013), 8508-8524.  doi: 10.1016/j.amc.2013.01.053.

[37]

F. Yin, S. Zhou, C. Yin and C. Xiao, Global attractor for Klein-Gordon-Schrödinger lattice system, Appl. Math. Mech., -Engl. Ed., 28 (2007), 695–706. doi: 10.1007/s10483-007-0514-y.

[38]

C. ZhaoG. Xue and G. Lukaszewicz, Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4021-4044.  doi: 10.3934/dcdsb.2018122.

[39]

C. Zhao and S. Zhou, Compact uniform attractors for dissipative lattice dynamical systems with delays, Discrete Contin. Dyn. Syst., 21 (2008), 643-663.  doi: 10.3934/dcds.2008.21.643.

[40]

S. Zhou, Attractors for first order dissipative lattice dynamical systems, Physica D, 178 (2003), 51-61.  doi: 10.1016/S0167-2789(02)00807-2.

[41]

S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368.  doi: 10.1016/j.jde.2004.02.005.

[42]

S. Zhou and X. Han, Uniform exponential attractors for non-autonomous KGS and Zakharov lattice systems with quasiperiodic external forces, Nonlinear Anal., 78 (2013), 141-155.  doi: 10.1016/j.na.2012.10.001.

[43]

S. Zhou and M. Zhao, Uniform exponential attractor for second order lattice system with quasi-periodic external forces in weighted space, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24 (2014), 1450006, 9 pp. doi: 10.1142/S0218127414500060.

show all references

References:
[1]

A. Y. Abdallah, Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems, Comm. Pure. Appl. Anal., 5 (2006), 55-69.  doi: 10.3934/cpaa.2006.5.55.

[2]

A. Y. Abdallah, Attractors for first order lattice systems with almost periodic nonlinear part, Disc. Cont. Dyn. Sys.-B, 25 (2020), 1241-1255.  doi: 10.3934/dcdsb.2019218.

[3]

A. Y. Abdallah, Dynamics of second order lattice systems with almost periodic nonlinear part, Qual. Theory Dyn. Syst., 20 (2021), Paper No. 58, 23 pp. doi: 10.1007/s12346-021-00497-3.

[4]

A. Y. Abdallah, Uniform exponential attractors for non-autonomous Klein—Gordon—Schrödinger lattice systems in weighted spaces, Nonlinear Anal., 127 (2015), 279-297.  doi: 10.1016/j.na.2015.07.013.

[5]

A. Y. Abdallah and R. T. Wannan, Second order non-autonomous lattice systems and their uniform attractors, Comm. Pure Appl. Anal., 18 (2019), 1827-1846.  doi: 10.3934/cpaa.2019085.

[6]

P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.

[7]

J. Bell, Some threshold results for models of myelinated nerves, Math. Biosci., 54 (1981), 181-190.  doi: 10.1016/0025-5564(81)90085-7.

[8]

J. Bell and C. Cosner, Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quart. Appl. Math., 42 (1984), 1-14.  doi: 10.1090/qam/736501.

[9]

V. Bellrti and V. Pata, Attractors for semilinear strongly damped wave equation on $\mathbb{R}^{3}$, Discrete Contin. Dynam. Systems, 7 (2001), 719-735.  doi: 10.3934/dcds.2001.7.719.

[10]

A. M. Boughoufala and A. Y. Abdallah, Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 1549-1563.  doi: 10.3934/dcdsb.2020172.

[11]

T. CaraballoF. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity, J. Difference Equ. Appl., 17 (2011), 161-184.  doi: 10.1080/10236198.2010.549010.

[12]

T. CaraballoF. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020.

[13]

H. Chate and M. Courbage (Eds.), Lattice systems, Phys. D, 103 (1997), 1-612. 

[14]

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

[15]

S.-N. Chow, Lattice dynamical systems, Dynamical System, Lecture Notes in Mathematics (Springer, Berlin), (2003), 1–102. doi: 10.1007/978-3-540-45204-1_1.

[16]

S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems: I, IEEE Trans. Circuits Systems, 42 (1995), 746-751.  doi: 10.1109/81.473583.

[17]

S.-N. ChowJ. Mallet-Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Comput. Dynam., 4 (1996), 109-178. 

[18]

S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems: II, IEEE Trans. Circuits Systems, 42 (1995), 752-756. 

[19]

L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Systems, 40 (1993), 147-156. 

[20]

L. O. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits Systems, 35 (1988), 1257-1272.  doi: 10.1109/31.7600.

[21]

L. O. Chua and L. Yang, Cellular neural networks: Applications, IEEE Trans. Circuits Systems, 35 (1988), 1273-1290.  doi: 10.1109/31.7601.

[22]

T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems, Physica D, 67 (1993), 237-244.  doi: 10.1016/0167-2789(93)90208-I.

[23]

J. Huang, X. Han and S. Zhou, Uniform attractors for non-autonomous Klein-Gordon-Schrödinger lattice systems, Appl. Math. Mech., Engl. Ed. 30 (2009), 1597–1607. doi: 10.1007/s10483-009-1211-z.

[24]

X. JiaC. Zhao and X. Yang, Global attractor and Kolmogorov entropy of three component reversible Gray—Scott model on infinite lattices, Appl. Math. Comp., 218 (2012), 9781-9789.  doi: 10.1016/j.amc.2012.03.036.

[25]

R. Kapral, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163.  doi: 10.1007/BF01192578.

[26]

J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572.  doi: 10.1137/0147038.

[27]

J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium, J. Theor. Biol., 148 (1991), 49-82. 

[28] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, Cambridge, 1982. 
[29]

C. Li, M. Zhao and C. Zhao, Pullback exponential attractors for nonautonomous Klein—Gordon—Schrö dinger equations on infinite lattices, Abstr. Appl. Anal., 2013 (2013), Art. ID 809476, 9 pp. doi: 10.1155/2013/809476.

[30]

H. Li and L. Sun, Upper semicontinuity of attractors for small perturbations of Klein-Gordon-Schrödinger lattice system, Adv. Difference Equ., 2014 (2014), 300, 16 pp. doi: 10.1186/1687-1847-2014-300.

[31]

J. C. OliveiraJ. M. Pereira and G. Perla Menzala, Attractors for second order periodic lattices with nonlinear damping, J. Difference Equ. Appl., 14 (2008), 899-921.  doi: 10.1080/10236190701859211.

[32]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[33]

L. M. Pecora and T. L. Carrol, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 821-824.  doi: 10.1103/PhysRevLett.64.821.

[34]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Appl. Math. Sci., 68, Springer, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[35]

B. Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl., 331 (2007), 121-136.  doi: 10.1016/j.jmaa.2006.08.070.

[36]

X. YangC. Zhao and J. Cao, Dynamics of the discrete coupled nonlinear Schrödinger—Boussinesq equations, Appl. Math. Comp., 219 (2013), 8508-8524.  doi: 10.1016/j.amc.2013.01.053.

[37]

F. Yin, S. Zhou, C. Yin and C. Xiao, Global attractor for Klein-Gordon-Schrödinger lattice system, Appl. Math. Mech., -Engl. Ed., 28 (2007), 695–706. doi: 10.1007/s10483-007-0514-y.

[38]

C. ZhaoG. Xue and G. Lukaszewicz, Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4021-4044.  doi: 10.3934/dcdsb.2018122.

[39]

C. Zhao and S. Zhou, Compact uniform attractors for dissipative lattice dynamical systems with delays, Discrete Contin. Dyn. Syst., 21 (2008), 643-663.  doi: 10.3934/dcds.2008.21.643.

[40]

S. Zhou, Attractors for first order dissipative lattice dynamical systems, Physica D, 178 (2003), 51-61.  doi: 10.1016/S0167-2789(02)00807-2.

[41]

S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 200 (2004), 342-368.  doi: 10.1016/j.jde.2004.02.005.

[42]

S. Zhou and X. Han, Uniform exponential attractors for non-autonomous KGS and Zakharov lattice systems with quasiperiodic external forces, Nonlinear Anal., 78 (2013), 141-155.  doi: 10.1016/j.na.2012.10.001.

[43]

S. Zhou and M. Zhao, Uniform exponential attractor for second order lattice system with quasi-periodic external forces in weighted space, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24 (2014), 1450006, 9 pp. doi: 10.1142/S0218127414500060.

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