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January  2014, 34(1): 229-248. doi: 10.3934/dcds.2014.34.229

## Uniform attractor of the non-autonomous discrete Selkov model

 1 Department of Mathematics and Information Science, Wenzhou University, Zhejiang Province, 325035, China, China 2 College of Teacher Education, Wenzhou University, Zhejiang Province, 325035, China

Received  November 2012 Revised  March 2013 Published  June 2013

This paper studies the asymptotic behavior of solutions for the non-autonomous lattice Selkov model. We prove the existence of a uniform attractor for the generated family of processes and obtain an upper bound of the Kolmogorov $\varepsilon$-entropy for it. Also we establish the upper semicontinuity of the uniform attractor when the infinite lattice systems are approximated by finite lattice systems.
Citation: Xiaolin Jia, Caidi Zhao, Juan Cao. Uniform attractor of the non-autonomous discrete Selkov model. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 229-248. doi: 10.3934/dcds.2014.34.229
##### References:
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Pilyugin, Attractors of reaction diffusion systems on infinite lattices, J. Dyn. Diff. Eqs., 15 (2003), 485-515. doi: 10.1023/B:JODY.0000009745.41889.30.  Google Scholar [7] H. Chate and M. Courbage, Lattice systems, Phys. D, 103 (1997), 1-612. doi: 10.1016/S0167-2789(96)00249-7.  Google Scholar [8] 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 [9] S.-N. Chow and J. M. Paret, Pattern formation and spatial chaos in lattice dynamical systems, IEEE Trans. Circuits Syst., 42 (1995), 746-751. doi: 10.1109/81.473583.  Google Scholar [10] 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 [11] S.-N. Chow, J. M. Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291. doi: 10.1006/jdeq.1998.3478.  Google Scholar [12] S.-N. Chow, J. M. Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Comp. Dyna., 4 (1996), 109-178.  Google Scholar [13] L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 40 (1993), 147-156. doi: 10.1109/81.222795.  Google Scholar [14] L. O. Chua and Y. Yang, Cellular neural networks: theory, IEEE Trans. Circuits Syst., 35 (1988), 1257-1272. doi: 10.1109/31.7600.  Google Scholar [15] V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," AMS Colloquium Publications, 49, AMS, Providence, RI, 2002.  Google Scholar [16] 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 [17] L. Fabiny, P. Colet and R. 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Sun, Dynamical behavior for stochastic lattice systems, Chaos Soli. Fract., 27 (2006), 1080-1090. doi: 10.1016/j.chaos.2005.04.089.  Google Scholar [28] P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonl. Analysis, 71 (2009), 3956-3963. doi: 10.1016/j.na.2009.02.065.  Google Scholar [29] E. E. Selkov, Self-oscillations in glycolysis: A simple kinetic model, Euorpean J. Bio., 4 (1968), 79-86. Google Scholar [30] G. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002.  Google Scholar [31] R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," Springer, Berlin, 1997.  Google Scholar [32] T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multipliative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693. doi: 10.1016/j.jde.2012.03.020.  Google Scholar [33] T. Caraballo, F. Morillas and J. Valero, Random attractors for sotchastic lattice systems with non-Lipschitz nonlinearity, J. Difference Equ. Appl., 17 (2011), 161-184. doi: 10.1080/10236198.2010.549010.  Google Scholar [34] T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335. doi: 10.1007/s11464-008-0028-7.  Google Scholar [35] E. V. Vlecka 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 [36] B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245. doi: 10.1016/j.jde.2005.01.003.  Google Scholar [37] B. Wang, Asymptotic behavior of non-autonomous lattice system, J. Math. Anal. Appl., 331 (2007), 121-136. doi: 10.1016/j.jmaa.2006.08.070.  Google Scholar [38] B. Wang, Uniform attractors of non-autonomous discret reaction-diffusion systems in weighted spaces, Inter. J. Bifur. Chaos, 18 (2008), 695-716. doi: 10.1142/S0218127408020598.  Google Scholar [39] R. L. Winaow, A. L. Kimball and A. Varghese, Simulating cardiac sinus and atrial network dynamics on connection machine, Phys. D, 64 (1993), 281-298. Google Scholar [40] Y. You, Asymptotic dynamics of Selkov equations, Disc. Cont. Dyn. Syst. (S), 2 (2009), 193-219. doi: 10.3934/dcdss.2009.2.193.  Google Scholar [41] 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 [42] S. Zhou, Attractors and approximations for lattice dynamincal systems, J. Differential Equations, 200 (2004), 342-368. doi: 10.1016/j.jde.2004.02.005.  Google Scholar [43] S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624. doi: 10.1006/jdeq.2001.4032.  Google Scholar [44] S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems, J. Differential Equations, 224 (2006), 172-204. doi: 10.1016/j.jde.2005.06.024.  Google Scholar [45] X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Disc. Cont. Dyn. Syst. (B), 9 (2008), 763-785. doi: 10.3934/dcdsb.2008.9.763.  Google Scholar [46] C. Zhao and S. Zhou, Compact kernel sections for nonautonomous Klein-Gordon-Schr\"odinger equations on infinite lattices, J. Math. Anal. Appl., 332 (2007), 32-56. doi: 10.1016/j.jmaa.2006.10.002.  Google Scholar [47] C. Zhao and S. Zhou, Compact kernel sections of long-wave-short-wave resonance equations on infinite lattices, Nonl. Analysis, 68 (2008), 652-670. doi: 10.1016/j.na.2006.11.027.  Google Scholar [48] C. Zhao and S. Zhou, Compact uniform attractors for dissipative lattice dynamical systems with delays, Disc. Cont. Dyn. Syst. (A), 21 (2008), 643-663. doi: 10.3934/dcds.2008.21.643.  Google Scholar [49] C. Zhao and S. Zhou, Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, J. Math. Anal. Appl., 354 (2009), 78-95. doi: 10.1016/j.jmaa.2008.12.036.  Google Scholar [50] S. Zhou, C. Zhao and X. Liao, 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 [51] S. Zhou, C. Zhao and Y. Wang, Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems, Disc. Cont. Dyn. Syst. (A), 21 (2008), 1259-1277. doi: 10.3934/dcds.2008.21.1259.  Google Scholar

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##### References:
 [1] Ahmed Y. Abdallah, Uniform exponential attractor for first order non-autonomous lattice dynamical systems, J. Differential Equations, 251 (2011), 1489-1504. doi: 10.1016/j.jde.2011.05.030.  Google Scholar [2] Ahmed Y. Abdallah, Exponential attractors for first-order lattice dynamical systems, J. Math. Anal. Appl., 339 (2008), 217-224. doi: 10.1016/j.jmaa.2007.06.054.  Google Scholar [3] P. W. Bates, X. Chen and A. Chmaj, Travelling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546. doi: 10.1137/S0036141000374002.  Google Scholar [4] P. W. Bates, H. Lisei and K. Lu, Attrators for stochastic lattice dynamical systems, Stoch. Dyna., 6 (2006), 1-21. doi: 10.1142/S0219493706001621.  Google Scholar [5] P. W. Bates, K. Lu and B. Wang, Attractors for lattice dynamical systems, Inter. J. Bifur. Chaos, 11 (2001), 143-153. doi: 10.1142/S0218127401002031.  Google Scholar [6] W.-J. Beyn and S. Yu. Pilyugin, Attractors of reaction diffusion systems on infinite lattices, J. Dyn. Diff. Eqs., 15 (2003), 485-515. doi: 10.1023/B:JODY.0000009745.41889.30.  Google Scholar [7] H. Chate and M. Courbage, Lattice systems, Phys. D, 103 (1997), 1-612. doi: 10.1016/S0167-2789(96)00249-7.  Google Scholar [8] 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 [9] S.-N. Chow and J. M. Paret, Pattern formation and spatial chaos in lattice dynamical systems, IEEE Trans. Circuits Syst., 42 (1995), 746-751. doi: 10.1109/81.473583.  Google Scholar [10] 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 [11] S.-N. Chow, J. M. Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291. doi: 10.1006/jdeq.1998.3478.  Google Scholar [12] S.-N. Chow, J. M. Paret and E. S. Van Vleck, Pattern formation and spatial chaos in spatially discrete evolution equations, Random Comp. Dyna., 4 (1996), 109-178.  Google Scholar [13] L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 40 (1993), 147-156. doi: 10.1109/81.222795.  Google Scholar [14] L. O. Chua and Y. Yang, Cellular neural networks: theory, IEEE Trans. Circuits Syst., 35 (1988), 1257-1272. doi: 10.1109/31.7600.  Google Scholar [15] V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," AMS Colloquium Publications, 49, AMS, Providence, RI, 2002.  Google Scholar [16] 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 [17] L. Fabiny, P. Colet and R. Roy, Coherence and phase dynamics of spatially coupled solid-state lasers, Phys. Rev. A, 47 (1993), 4287-4296. doi: 10.1103/PhysRevA.47.4287.  Google Scholar [18] J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Math. Surveys and Monographs, AMS, Providence, RI, 1988.  Google Scholar [19] M. Hillert, A solid-solution model for inhomogeneous systems, Acta Metall., 9 (1961), 525-535. doi: 10.1016/0001-6160(61)90155-9.  Google Scholar [20] X. Han, Random attractors for second order stochastic lattice dynamical systems with multiplicative noise in weighted spaces, Stoch. Dyn., 12 (2012), 1150024, 20 pp. doi: 10.1142/S0219493711500249.  Google Scholar [21] 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.  Google Scholar [22] X. Jia, C. 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 [23] J. P. Keener, Propagation and its failure in coupled systems of discret excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572. doi: 10.1137/0147038.  Google Scholar [24] R. Kapval, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163. doi: 10.1007/BF01192578.  Google Scholar [25] N. I. Karachalios and A. N. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrödinger equation, J. Differntial Equations, 217 (2005), 88-123. doi: 10.1016/j.jde.2005.06.002.  Google Scholar [26] O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations," Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.  Google Scholar [27] Y. Lv and J. H. Sun, Dynamical behavior for stochastic lattice systems, Chaos Soli. Fract., 27 (2006), 1080-1090. doi: 10.1016/j.chaos.2005.04.089.  Google Scholar [28] P. Marín-Rubio and J. Real, On the relation between two different concepts of pullback attractors for non-autonomous dynamical systems, Nonl. Analysis, 71 (2009), 3956-3963. doi: 10.1016/j.na.2009.02.065.  Google Scholar [29] E. E. Selkov, Self-oscillations in glycolysis: A simple kinetic model, Euorpean J. Bio., 4 (1968), 79-86. Google Scholar [30] G. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002.  Google Scholar [31] R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," Springer, Berlin, 1997.  Google Scholar [32] T. Caraballo, F. Morillas and J. Valero, Attractors of stochastic lattice dynamical systems with a multipliative noise and non-Lipschitz nonlinearities, J. Differential Equations, 253 (2012), 667-693. doi: 10.1016/j.jde.2012.03.020.  Google Scholar [33] T. Caraballo, F. Morillas and J. Valero, Random attractors for sotchastic lattice systems with non-Lipschitz nonlinearity, J. Difference Equ. Appl., 17 (2011), 161-184. doi: 10.1080/10236198.2010.549010.  Google Scholar [34] T. Caraballo and K. Lu, Attractors for stochastic lattice dynamical systems with a multiplicative noise, Front. Math. China, 3 (2008), 317-335. doi: 10.1007/s11464-008-0028-7.  Google Scholar [35] E. V. Vlecka 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 [36] B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245. doi: 10.1016/j.jde.2005.01.003.  Google Scholar [37] B. Wang, Asymptotic behavior of non-autonomous lattice system, J. Math. Anal. Appl., 331 (2007), 121-136. doi: 10.1016/j.jmaa.2006.08.070.  Google Scholar [38] B. Wang, Uniform attractors of non-autonomous discret reaction-diffusion systems in weighted spaces, Inter. J. Bifur. Chaos, 18 (2008), 695-716. doi: 10.1142/S0218127408020598.  Google Scholar [39] R. L. Winaow, A. L. Kimball and A. Varghese, Simulating cardiac sinus and atrial network dynamics on connection machine, Phys. D, 64 (1993), 281-298. Google Scholar [40] Y. You, Asymptotic dynamics of Selkov equations, Disc. Cont. Dyn. Syst. (S), 2 (2009), 193-219. doi: 10.3934/dcdss.2009.2.193.  Google Scholar [41] 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 [42] S. Zhou, Attractors and approximations for lattice dynamincal systems, J. Differential Equations, 200 (2004), 342-368. doi: 10.1016/j.jde.2004.02.005.  Google Scholar [43] S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624. doi: 10.1006/jdeq.2001.4032.  Google Scholar [44] S. Zhou and W. Shi, Attractors and dimension of dissipative lattice systems, J. Differential Equations, 224 (2006), 172-204. doi: 10.1016/j.jde.2005.06.024.  Google Scholar [45] X. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Disc. Cont. Dyn. Syst. (B), 9 (2008), 763-785. doi: 10.3934/dcdsb.2008.9.763.  Google Scholar [46] C. Zhao and S. Zhou, Compact kernel sections for nonautonomous Klein-Gordon-Schr\"odinger equations on infinite lattices, J. Math. Anal. Appl., 332 (2007), 32-56. doi: 10.1016/j.jmaa.2006.10.002.  Google Scholar [47] C. Zhao and S. Zhou, Compact kernel sections of long-wave-short-wave resonance equations on infinite lattices, Nonl. Analysis, 68 (2008), 652-670. doi: 10.1016/j.na.2006.11.027.  Google Scholar [48] C. Zhao and S. Zhou, Compact uniform attractors for dissipative lattice dynamical systems with delays, Disc. Cont. Dyn. Syst. (A), 21 (2008), 643-663. doi: 10.3934/dcds.2008.21.643.  Google Scholar [49] C. Zhao and S. Zhou, Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications, J. Math. Anal. Appl., 354 (2009), 78-95. doi: 10.1016/j.jmaa.2008.12.036.  Google Scholar [50] S. Zhou, C. Zhao and X. Liao, 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 [51] S. Zhou, C. Zhao and Y. Wang, Finite dimensionality and upper semicontinuity of compact kernel sections of non-autonomous lattice systems, Disc. Cont. Dyn. Syst. (A), 21 (2008), 1259-1277. doi: 10.3934/dcds.2008.21.1259.  Google Scholar
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