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Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families
Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts
Department of Mathematics, The University of Jordan, Amman 11942, Jordan |
For FitzHugh-Nagumo lattice dynamical systems (LDSs) many authors studied the existence of global attractors for deterministic systems [
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. 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. |
[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. 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. |
[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. |
[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. 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.
|
[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. 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.
|
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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.
|
[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. 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.
|
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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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