March  2021, 26(3): 1549-1563. doi: 10.3934/dcdsb.2020172

Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts

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

Received  November 2019 Revised  March 2020 Published  March 2021 Early access  May 2020

For FitzHugh-Nagumo lattice dynamical systems (LDSs) many authors studied the existence of global attractors for deterministic systems [4,34,41,43] and the existence of global random attractors for stochastic systems [23,24,27,48,49], where for non-autonomous cases, the nonlinear parts are considered of the form $ f\left( u\right) $. Here we study the existence of the uniform global attractor for a new family of non-autonomous FitzHugh-Nagumo LDSs with nonlinear parts of the form $ f\left( u,t\right) $, where we introduce a suitable Banach space of functions $ W $ and we assume that $ f $ is an element of the hull of an almost periodic function $ f_{0}\left( \cdot ,t\right) $ with values in $ W $.

Citation: Amira M. Boughoufala, Ahmed Y. Abdallah. Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1549-1563. doi: 10.3934/dcdsb.2020172
References:
[1]

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

[2]

A. Y. Abdallah, Long-time behavior for second order lattice dynamical systems, Acta Appl. Math., 106 (2009), 47-59.  doi: 10.1007/s10440–008–9281–8.

[3]

A. Y. Abdallah, Uniform global attractors for first order non-autonomous lattice dynamical systems, Proc. Amer. Math. Soc., 138 (2010), 3219-3228.  doi: 10.1090/S0002–9939–10–10440–7.

[4]

A. Y. Abdallah, Upper semicontinuity of the attractor for lattice dynamical systems of partly dissipative reaction diffusion systems, J. Appl. Math., 2005 (2005), 273-288.  doi: 10.1155/JAM.2005.273.

[5]

A. Y. Abdallah and R. T. Wannan, Second order non-autonomous lattice systems and their uniform attractors, Commun. 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]

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.

[10]

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.

[11]

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

[12]

H. Chate and M. Courbage, Lattice systems, Phys. D, 103 (1997), 1-611. 

[13]

V. V. Chepyzhov and M. I. Vishik, Attractors of non-autonomous dynamical systems and their dimension, J. Math. Pures Appl. (9), 73 (1994), 279-333. 

[14]

S.-N. Chow, Lattice dynamical systems, in Dynamical Systems, Lecture Notes in Math., 1822, Springer, Berlin, 2003, 1–102. doi: 10.1007/978–3–540–45204–1_1.

[15]

S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems. Ⅰ, Ⅱ, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 746–751,752–756. doi: 10.1109/81.473583.

[16]

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. 

[17]

L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Systems I Fund. Theory Appl., 40 (1993), 147-156.  doi: 10.1109/81.222795.

[18]

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

[19]

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

[20]

C. E. Elmer and E. S. Van Vleck, Spatially discrete FitzHugh-Nagumo equations, SIAM J. Appl. Math., 65 (2005), 1153-1174.  doi: 10.1137/S003613990343687X.

[21]

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

[22]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6.

[23]

A. Gu and Y. Li, Singleton sets random attractor for stochastic FitzHugh-Nagumo lattice equations driven by fractional Brownian motions, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 3929-3937.  doi: 10.1016/j.cnsns.2014.04.005.

[24]

A. Gu, Y. Li and J. Li, Random attractors on lattice of stochastic FitzHugh-Nagumo systems driven by $\alpha$–stable Lévy noises, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24 (2014), 9pp. doi: 10.1142/S0218127414501235.

[25]

X. Han and P. E. Kloeden, Asymptotic behavior of a neural field lattice model with a Heaviside operator, Phys. D, 389 (2019), 1-12.  doi: 10.1016/j.physd.2018.09.004.

[26]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.

[27]

J. Huang, The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with white noises, Phys. D., 233 (2007), 83-94.  doi: 10.1016/j.physd.2007.06.008.

[28]

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

[29]

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

[30]

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

[31]

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.

[32]

J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium, J. Theor. Biol., 148 (1991), 49-82.  doi: 10.1016/S0022-5193(05)80465-5.

[33] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge-New York, 1982. 
[34]

X.-J. Li and D.-B. Wang, Attractors for partly dissipative lattice dynamic systems in weighted spaces, J. Math. Anal. Appl., 325 (2007), 141-156.  doi: 10.1016/j.jmaa.2006.01.054.

[35]

X. LiaoC. Zhao and S. Zhou, Compact uniform attractors for dissipative non-autonomous lattice dynamical systems, Commun. Pure Appl. Anal., 6 (2007), 1087-1111.  doi: 10.3934/cpaa.2007.6.1087.

[36]

Q. MaS. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559.  doi: 10.1512/iumj.2002.51.2255.

[37]

J. Mallet-Paret and S.-N. Chow, Pattern formation and spatial chaos in lattice dynamical systems. Ⅱ, Ⅱ, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 752-756.  doi: 10.1109/81.473583.

[38]

J. NagumoS. Arimoto and S. Yosimzawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1964), 2061-2070.  doi: 10.1109/JRPROC.1962.288235.

[39]

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.

[40]

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

[41]

E. Van Vleck and B. Wang, Attractors for lattice FitzHugh-Nagumo systems, Phys. D, 212 (2005), 317-336.  doi: 10.1016/j.physd.2005.10.006.

[42]

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.

[43]

B. Wang, Dynamical behavior of the almost-periodic discrete FitzHugh-Nagumo systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 1673-1685.  doi: 10.1142/S0218127407017987.

[44]

B. Wang, Dynamics of stochastic reaction-diffusion lattice system driven by nonlinear noise, J. Math. Anal. Appl., 477 (2019), 104-132.  doi: 10.1016/j.jmaa.2019.04.015.

[45]

C. WangG. Xue and C. Zhao, Invariant Borel probability measures for discrete long-wave-short-wave resonance equations, Appl. Math. Comput., 339 (2018), 853-865.  doi: 10.1016/j.amc.2018.06.059.

[46]

R. Wang and Y. Li, Regularity and backward compactness of attractors for non-autonomous lattice systems with random coefficients, Appl. Math. Comput., 354 (2019), 86-102.  doi: 10.1016/j.amc.2019.02.036.

[47]

R. Wang and B. Wang, Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2461-2493.  doi: 10.3934/dcdsb.2020019.

[48]

Y. WangY. Liu and Z. Wang, Random attractors for partly dissipative stochastic lattice dynamical systems, J. Difference. Equ. Appl., 14 (2008), 799-817.  doi: 10.1080/10236190701859542.

[49]

Z. Wang and S. Zhou, Random attractors for non-autonomous stochastic lattice FitzHugh-Nagumo systems with random coupled coefficients, Taiwanese J. Math., 20 (2016), 589-616.  doi: 10.11650/tjm.20.2016.6699.

[50]

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

[51]

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.

[52]

C. ZhaoG. Xue and G. Łukaszewicz, 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.

[53]

S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624.  doi: 10.1006/jdeq.2001.4032.

[54]

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

[55]

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

[56]

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), 9pp. doi: 10.1142/S0218127414500060.

show all references

References:
[1]

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

[2]

A. Y. Abdallah, Long-time behavior for second order lattice dynamical systems, Acta Appl. Math., 106 (2009), 47-59.  doi: 10.1007/s10440–008–9281–8.

[3]

A. Y. Abdallah, Uniform global attractors for first order non-autonomous lattice dynamical systems, Proc. Amer. Math. Soc., 138 (2010), 3219-3228.  doi: 10.1090/S0002–9939–10–10440–7.

[4]

A. Y. Abdallah, Upper semicontinuity of the attractor for lattice dynamical systems of partly dissipative reaction diffusion systems, J. Appl. Math., 2005 (2005), 273-288.  doi: 10.1155/JAM.2005.273.

[5]

A. Y. Abdallah and R. T. Wannan, Second order non-autonomous lattice systems and their uniform attractors, Commun. 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]

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.

[10]

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.

[11]

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

[12]

H. Chate and M. Courbage, Lattice systems, Phys. D, 103 (1997), 1-611. 

[13]

V. V. Chepyzhov and M. I. Vishik, Attractors of non-autonomous dynamical systems and their dimension, J. Math. Pures Appl. (9), 73 (1994), 279-333. 

[14]

S.-N. Chow, Lattice dynamical systems, in Dynamical Systems, Lecture Notes in Math., 1822, Springer, Berlin, 2003, 1–102. doi: 10.1007/978–3–540–45204–1_1.

[15]

S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems. Ⅰ, Ⅱ, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 746–751,752–756. doi: 10.1109/81.473583.

[16]

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. 

[17]

L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Systems I Fund. Theory Appl., 40 (1993), 147-156.  doi: 10.1109/81.222795.

[18]

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

[19]

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

[20]

C. E. Elmer and E. S. Van Vleck, Spatially discrete FitzHugh-Nagumo equations, SIAM J. Appl. Math., 65 (2005), 1153-1174.  doi: 10.1137/S003613990343687X.

[21]

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

[22]

R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.  doi: 10.1016/S0006-3495(61)86902-6.

[23]

A. Gu and Y. Li, Singleton sets random attractor for stochastic FitzHugh-Nagumo lattice equations driven by fractional Brownian motions, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 3929-3937.  doi: 10.1016/j.cnsns.2014.04.005.

[24]

A. Gu, Y. Li and J. Li, Random attractors on lattice of stochastic FitzHugh-Nagumo systems driven by $\alpha$–stable Lévy noises, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24 (2014), 9pp. doi: 10.1142/S0218127414501235.

[25]

X. Han and P. E. Kloeden, Asymptotic behavior of a neural field lattice model with a Heaviside operator, Phys. D, 389 (2019), 1-12.  doi: 10.1016/j.physd.2018.09.004.

[26]

X. HanW. Shen and S. Zhou, Random attractors for stochastic lattice dynamical systems in weighted spaces, J. Differential Equations, 250 (2011), 1235-1266.  doi: 10.1016/j.jde.2010.10.018.

[27]

J. Huang, The random attractor of stochastic FitzHugh-Nagumo equations in an infinite lattice with white noises, Phys. D., 233 (2007), 83-94.  doi: 10.1016/j.physd.2007.06.008.

[28]

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

[29]

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

[30]

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

[31]

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.

[32]

J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium, J. Theor. Biol., 148 (1991), 49-82.  doi: 10.1016/S0022-5193(05)80465-5.

[33] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge-New York, 1982. 
[34]

X.-J. Li and D.-B. Wang, Attractors for partly dissipative lattice dynamic systems in weighted spaces, J. Math. Anal. Appl., 325 (2007), 141-156.  doi: 10.1016/j.jmaa.2006.01.054.

[35]

X. LiaoC. Zhao and S. Zhou, Compact uniform attractors for dissipative non-autonomous lattice dynamical systems, Commun. Pure Appl. Anal., 6 (2007), 1087-1111.  doi: 10.3934/cpaa.2007.6.1087.

[36]

Q. MaS. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559.  doi: 10.1512/iumj.2002.51.2255.

[37]

J. Mallet-Paret and S.-N. Chow, Pattern formation and spatial chaos in lattice dynamical systems. Ⅱ, Ⅱ, IEEE Trans. Circuits Systems I Fund. Theory Appl., 42 (1995), 752-756.  doi: 10.1109/81.473583.

[38]

J. NagumoS. Arimoto and S. Yosimzawa, An active pulse transmission line simulating nerve axon, Proc. IRE, 50 (1964), 2061-2070.  doi: 10.1109/JRPROC.1962.288235.

[39]

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.

[40]

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

[41]

E. Van Vleck and B. Wang, Attractors for lattice FitzHugh-Nagumo systems, Phys. D, 212 (2005), 317-336.  doi: 10.1016/j.physd.2005.10.006.

[42]

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.

[43]

B. Wang, Dynamical behavior of the almost-periodic discrete FitzHugh-Nagumo systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 1673-1685.  doi: 10.1142/S0218127407017987.

[44]

B. Wang, Dynamics of stochastic reaction-diffusion lattice system driven by nonlinear noise, J. Math. Anal. Appl., 477 (2019), 104-132.  doi: 10.1016/j.jmaa.2019.04.015.

[45]

C. WangG. Xue and C. Zhao, Invariant Borel probability measures for discrete long-wave-short-wave resonance equations, Appl. Math. Comput., 339 (2018), 853-865.  doi: 10.1016/j.amc.2018.06.059.

[46]

R. Wang and Y. Li, Regularity and backward compactness of attractors for non-autonomous lattice systems with random coefficients, Appl. Math. Comput., 354 (2019), 86-102.  doi: 10.1016/j.amc.2019.02.036.

[47]

R. Wang and B. Wang, Random dynamics of lattice wave equations driven by infinite-dimensional nonlinear noise, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2461-2493.  doi: 10.3934/dcdsb.2020019.

[48]

Y. WangY. Liu and Z. Wang, Random attractors for partly dissipative stochastic lattice dynamical systems, J. Difference. Equ. Appl., 14 (2008), 799-817.  doi: 10.1080/10236190701859542.

[49]

Z. Wang and S. Zhou, Random attractors for non-autonomous stochastic lattice FitzHugh-Nagumo systems with random coupled coefficients, Taiwanese J. Math., 20 (2016), 589-616.  doi: 10.11650/tjm.20.2016.6699.

[50]

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

[51]

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.

[52]

C. ZhaoG. Xue and G. Łukaszewicz, 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.

[53]

S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624.  doi: 10.1006/jdeq.2001.4032.

[54]

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

[55]

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

[56]

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), 9pp. doi: 10.1142/S0218127414500060.

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