American Institute of Mathematical Sciences

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

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

Amira M. Boughoufala and  Ahmed Y. Abdallah

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 $.

Keywords: Non-autonomous lattice dynamical system, FitzHugh-Nagumo system, uniform global attractor, almost periodic symbol.
Mathematics Subject Classification: Primary: 37L30; Secondary: 37L60.
Citation: Amira M. Boughoufala, Ahmed Y. Abdallah. Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1549-1563. doi: 10.3934/dcdsb.2020172
References:
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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[6]

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

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

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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.  Google Scholar

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T. Caraballo, F. 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.  Google Scholar

[10]

T. Caraballo, F. 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.  Google Scholar

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

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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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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X. Han, W. 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.  Google Scholar

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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.  Google Scholar

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X. Jia, C. 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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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

[35]

X. Liao, C. 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.  Google Scholar

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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.  Google Scholar

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

[39]

J. C. Oliveira, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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C. Wang, G. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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Y. Wang, Y. 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.  Google Scholar

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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.  Google Scholar

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X. Yang, C. 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.  Google Scholar

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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.  Google Scholar

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C. Zhao, G. 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.  Google Scholar

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

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

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

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

T. Caraballo, F. 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.  Google Scholar

[10]

T. Caraballo, F. 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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

X. Han, W. 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.  Google Scholar

[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.  Google Scholar

[28]

J. Huang, X. 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.  Google Scholar

[29]

X. Jia, C. 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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

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

[35]

X. Liao, C. 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.  Google Scholar

[36]

Q. Ma, S. 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.  Google Scholar

[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.  Google Scholar

[38]

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

[39]

J. C. Oliveira, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

C. Wang, G. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48]

Y. Wang, Y. 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.  Google Scholar

[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.  Google Scholar

[50]

X. Yang, C. 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.  Google Scholar

[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.  Google Scholar

[52]

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