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  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 & 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.  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. 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.  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. 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.  Google Scholar

[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.  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

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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.   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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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

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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

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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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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. 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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  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. 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.  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. 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.  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. 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.  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]

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.  Google Scholar

[53]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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. 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.  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. 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.  Google Scholar

[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.  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. 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.   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. 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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  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. 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.  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. 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.  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. 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.  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]

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.  Google Scholar

[53]

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

[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.  Google Scholar

[55]

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