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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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[29]

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

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[41]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[21]

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

[22]

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

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[29]

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

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[41]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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