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

Previous Article
The Poincaré bifurcation of a SD oscillator
DCDS-B Home
This Issue
Next Article
Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families

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 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. 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.
[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.
[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.

[1]	Wenqiang Zhao. Smoothing dynamics of the non-autonomous stochastic Fitzhugh-Nagumo system on $\mathbb{R}^N$ driven by multiplicative noises. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3453-3474. doi: 10.3934/dcdsb.2018251
[2]	Abiti Adili, Bixiang Wang. Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 643-666. doi: 10.3934/dcdsb.2013.18.643
[3]	Abiti Adili, Bixiang Wang. Random attractors for non-autonomous stochastic FitzHugh-Nagumo systems with multiplicative noise. Conference Publications, 2013, 2013 (special) : 1-10. doi: 10.3934/proc.2013.2013.1
[4]	Chao Xing, Zhigang Pan, Quan Wang. Stabilities and dynamic transitions of the Fitzhugh-Nagumo system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 775-794. doi: 10.3934/dcdsb.2020134
[5]	Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087
[6]	Zhen Zhang, Jianhua Huang, Xueke Pu. Pullback attractors of FitzHugh-Nagumo system on the time-varying domains. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3691-3706. doi: 10.3934/dcdsb.2017150
[7]	Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809
[8]	Michael Zgurovsky, Mark Gluzman, Nataliia Gorban, Pavlo Kasyanov, Liliia Paliichuk, Olha Khomenko. Uniform global attractors for non-autonomous dissipative dynamical systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 2053-2065. doi: 10.3934/dcdsb.2017120
[9]	B. Ambrosio, M. A. Aziz-Alaoui, V. L. E. Phan. Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3787-3797. doi: 10.3934/dcdsb.2018077
[10]	Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106
[11]	Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101
[12]	Bao Quoc Tang. Regularity of pullback random attractors for stochastic FitzHugh-Nagumo system on unbounded domains. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 441-466. doi: 10.3934/dcds.2015.35.441
[13]	Matthieu Alfaro, Hiroshi Matano. On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1639-1649. doi: 10.3934/dcdsb.2012.17.1639
[14]	Ahmed Y. Abdallah, Rania T. Wannan. Second order non-autonomous lattice systems and their uniform attractors. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1827-1846. doi: 10.3934/cpaa.2019085
[15]	Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743
[16]	Xiaolin Jia, Caidi Zhao, Juan Cao. Uniform attractor of the non-autonomous discrete Selkov model. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 229-248. doi: 10.3934/dcds.2014.34.229
[17]	Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639
[18]	Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure and Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281
[19]	Xiaoyue Li, Xuerong Mao. Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation. Discrete and Continuous Dynamical Systems, 2009, 24 (2) : 523-545. doi: 10.3934/dcds.2009.24.523
[20]	Ahmed Y. Abdallah. Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 899-916. doi: 10.3934/dcdsb.2005.5.899

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Amira M. Boughoufala Ahmed Y. Abdallah
on Google Scholar
Amira M. Boughoufala Ahmed Y. Abdallah

2021 Impact Factor: 1.497

Tools

Article outline

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Amira M. Boughoufala Ahmed Y. Abdallah
on Google Scholar
Amira M. Boughoufala Ahmed Y. Abdallah

[Back to Top]