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April  2020, 25(4): 1241-1255. doi: 10.3934/dcdsb.2019218

Attractors for first order lattice systems with almost periodic nonlinear part

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

Received  February 2019 Revised  June 2019 Published  April 2020 Early access  September 2019

We study the existence of the uniform global attractor for a family of infinite dimensional first order non-autonomous lattice dynamical systems of the following form:
$ \begin{equation*} \overset{.}{u}+Au+\alpha u+f\left( u,t\right) = g\left( t\right) ,\,\,\left( g,f\right) \in \mathcal{H}\left( \left( g_{0},f_{0}\right) \right) ,t>\tau ,\tau \in \mathbb{R}, \end{equation*} $
with initial data
$ \begin{equation*} u\left( \tau \right) = u_{\tau }. \end{equation*} $
The nonlinear part of the system
$ f\left( u,t\right) $
presents the main difficultly of this work. To overcome this difficulty we introduce a suitable Banach space
$ W $
of functions satisfying (3)-(7) with norm (8) such that
$ f_{0}\left( \cdot ,t\right) $
is an almost periodic function of
$ t $
with values in
$ W $
and
$ \left( g,f\right) \in \mathcal{H}\left( \left( g_{0},f_{0}\right) \right) $
.
Citation: Ahmed Y. Abdallah. Attractors for first order lattice systems with almost periodic nonlinear part. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1241-1255. doi: 10.3934/dcdsb.2019218
References:
[1]

A. Y. Abdallah, Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems, Comm. Pure. Appl. Anal., 5$\left(2006\right) $, 55–69. doi: 10.3934/cpaa.2006.5.55.

[2]

A. Y. Abdallah, Global attractor for the lattice dynamical system of a nonlinear Boussinesq equation, Abst. Appl. Anal., 2005 (2005), 655-671.  doi: 10.1155/AAA.2005.655.

[3]

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.

[4]

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

[5]

A. Y. Abdallah, Upper semicontinuity of the attractor for a second order lattice dynamical system, Disc. Cont. Dyn. Sys-B, 5 $\left(2005\right) $, 899–916. doi: 10.3934/dcdsb.2005.5.899.

[6]

A. Y. Abdallah and R. T. Wannan, Second order non-autonomous lattice systems and their uniform attractors, Comm. Pure Appl. Anal., 18 (2019), 1827-1846.  doi: 10.3934/cpaa.2019085.

[7]

V. S. Afraimovich and V. I. Nekorkin, Chaos of traveling waves in a discrete chain of diusively coupled maps, Int. J. Bifurc. Chaos, 4 (1994), 631-637.  doi: 10.1142/S0218127494000459.

[8]

P. W. BatesX. Chen and A. Chmaj, Traveling waves of bistable dynamics of a lattice, SIAM J. Math. Anal., 35 (2003), 520-546.  doi: 10.1137/S0036141000374002.

[9]

P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, Int. J. Bifurc. Chaos, 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.

[10]

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

[11]

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.

[12]

T. CaraballoF. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Diff. Eqs., 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020.

[13]

T. CaraballoF. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity, J. Diff. Eqs. Appl., 17 (2011), 161-184.  doi: 10.1080/10236198.2010.549010.

[14]

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

[15]

H. Chate and M. Courbage (Eds.), Lattice systems, Physica D, 103 (1997), 1-612.

[16]

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

[17]

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

[18]

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

[19]

S. N. ChowJ. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Diff. Eqs, 149 (1998), 248-291.  doi: 10.1006/jdeq.1998.3478.

[20]

S. N. ChowJ. Mallet-Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Rand. Comput. Dyn., 4 (1996), 109-178. 

[21]

L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Systems, 40 (1993), 147-156. 

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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.

[29]

J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium, J. Theor. Biol., 148 (1991), 49-82. 

[30] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, Cambridge, 1982. 
[31]

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

[32]

Q. F. MaS. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractor for semigroup and application, Indiana Univ. Math. J., 51 (2002), 1541-1559.  doi: 10.1512/iumj.2002.51.2255.

[33]

J. Mallet-Paret and S. N. Chow, Pattern formation and spatial chaos in lattice dynamical systems: Ⅱ, IEEE Trans. Circuits Systems, 42 (1995), 752-756. 

[34]

J. OliveiraJ. Pereira and M. Perla, Attractors for second order periodic lattices with nonlinear damping, J. Diff. Eqs. Appl., 14 (2008), 899-921.  doi: 10.1080/10236190701859211.

[35]

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

[36]

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.

[37]

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

[38]

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

[39]

C. Zhao and S. Zhou, Compact uniform attractors for dissipative lattice dynamical systems with delays,, Disc. Cont. Dyn. Sys, 21 (2008), 643-663.  doi: 10.3934/dcds.2008.21.643.

[40]

C. ZhaoG. Xue and G. L ukaszewicz, Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations, Discrete Cont. Dyn. Syst.-B, 23 (2018), 4021-4044. 

[41]

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

[42]

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

[43]

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

[44]

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), 1450006, 9 pp. doi: 10.1142/S0218127414500060.

[45]

B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Diff. Eqs., 96 (1992), 1-27.  doi: 10.1016/0022-0396(92)90142-A.

show all references

References:
[1]

A. Y. Abdallah, Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems, Comm. Pure. Appl. Anal., 5$\left(2006\right) $, 55–69. doi: 10.3934/cpaa.2006.5.55.

[2]

A. Y. Abdallah, Global attractor for the lattice dynamical system of a nonlinear Boussinesq equation, Abst. Appl. Anal., 2005 (2005), 655-671.  doi: 10.1155/AAA.2005.655.

[3]

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.

[4]

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

[5]

A. Y. Abdallah, Upper semicontinuity of the attractor for a second order lattice dynamical system, Disc. Cont. Dyn. Sys-B, 5 $\left(2005\right) $, 899–916. doi: 10.3934/dcdsb.2005.5.899.

[6]

A. Y. Abdallah and R. T. Wannan, Second order non-autonomous lattice systems and their uniform attractors, Comm. Pure Appl. Anal., 18 (2019), 1827-1846.  doi: 10.3934/cpaa.2019085.

[7]

V. S. Afraimovich and V. I. Nekorkin, Chaos of traveling waves in a discrete chain of diusively coupled maps, Int. J. Bifurc. Chaos, 4 (1994), 631-637.  doi: 10.1142/S0218127494000459.

[8]

P. W. BatesX. Chen and A. Chmaj, Traveling waves of bistable dynamics of a lattice, SIAM J. Math. Anal., 35 (2003), 520-546.  doi: 10.1137/S0036141000374002.

[9]

P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, Int. J. Bifurc. Chaos, 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.

[10]

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

[11]

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.

[12]

T. CaraballoF. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, J. Diff. Eqs., 253 (2012), 667-693.  doi: 10.1016/j.jde.2012.03.020.

[13]

T. CaraballoF. Morillas and J. Valero, Random attractors for stochastic lattice systems with non-Lipschitz nonlinearity, J. Diff. Eqs. Appl., 17 (2011), 161-184.  doi: 10.1080/10236198.2010.549010.

[14]

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

[15]

H. Chate and M. Courbage (Eds.), Lattice systems, Physica D, 103 (1997), 1-612.

[16]

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

[17]

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

[18]

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

[19]

S. N. ChowJ. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Diff. Eqs, 149 (1998), 248-291.  doi: 10.1006/jdeq.1998.3478.

[20]

S. N. ChowJ. Mallet-Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Rand. Comput. Dyn., 4 (1996), 109-178. 

[21]

L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Systems, 40 (1993), 147-156. 

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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.

[29]

J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium, J. Theor. Biol., 148 (1991), 49-82. 

[30] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge Univ. Press, Cambridge, 1982. 
[31]

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

[32]

Q. F. MaS. H. Wang and C. K. Zhong, Necessary and sufficient conditions for the existence of global attractor for semigroup and application, Indiana Univ. Math. J., 51 (2002), 1541-1559.  doi: 10.1512/iumj.2002.51.2255.

[33]

J. Mallet-Paret and S. N. Chow, Pattern formation and spatial chaos in lattice dynamical systems: Ⅱ, IEEE Trans. Circuits Systems, 42 (1995), 752-756. 

[34]

J. OliveiraJ. Pereira and M. Perla, Attractors for second order periodic lattices with nonlinear damping, J. Diff. Eqs. Appl., 14 (2008), 899-921.  doi: 10.1080/10236190701859211.

[35]

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

[36]

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.

[37]

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

[38]

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

[39]

C. Zhao and S. Zhou, Compact uniform attractors for dissipative lattice dynamical systems with delays,, Disc. Cont. Dyn. Sys, 21 (2008), 643-663.  doi: 10.3934/dcds.2008.21.643.

[40]

C. ZhaoG. Xue and G. L ukaszewicz, Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations, Discrete Cont. Dyn. Syst.-B, 23 (2018), 4021-4044. 

[41]

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

[42]

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

[43]

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

[44]

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), 1450006, 9 pp. doi: 10.1142/S0218127414500060.

[45]

B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Diff. Eqs., 96 (1992), 1-27.  doi: 10.1016/0022-0396(92)90142-A.

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