Uniform attractor of the non-autonomous discrete Selkov model

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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

Xiaolin Jia ^1, , Caidi Zhao ^1, and Juan Cao ^2,

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

Abstract
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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.

Keywords: Upper semicontinuity., Uniform attractor, Kolmogorov $\varepsilon$-entropy, Selkov model, Non-autonomous lattice dynamical system.

Mathematics Subject Classification: 37L30, 35B40, 35B4.

Citation: 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

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.
[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.
[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.
[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.
[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.
[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.
[7]	H. Chate and M. Courbage, Lattice systems, Phys. D, 103 (1997), 1-612. doi: 10.1016/S0167-2789(96)00249-7.
[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.
[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.
[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.
[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.
[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.
[13]	L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 40 (1993), 147-156. doi: 10.1109/81.222795.
[14]	L. O. Chua and Y. Yang, Cellular neural networks: theory, IEEE Trans. Circuits Syst., 35 (1988), 1257-1272. doi: 10.1109/31.7600.
[15]	V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," AMS Colloquium Publications, 49, AMS, Providence, RI, 2002.
[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.
[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.
[18]	J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Math. Surveys and Monographs, AMS, Providence, RI, 1988.
[19]	M. Hillert, A solid-solution model for inhomogeneous systems, Acta Metall., 9 (1961), 525-535. doi: 10.1016/0001-6160(61)90155-9.
[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.
[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.
[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.
[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.
[24]	R. Kapval, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163. doi: 10.1007/BF01192578.
[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.
[26]	O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations," Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.
[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.
[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.
[29]	E. E. Selkov, Self-oscillations in glycolysis: A simple kinetic model, Euorpean J. Bio., 4 (1968), 79-86.
[30]	G. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002.
[31]	R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," Springer, Berlin, 1997.
[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.
[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.
[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.
[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.
[36]	B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245. doi: 10.1016/j.jde.2005.01.003.
[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.
[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.
[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.
[40]	Y. You, Asymptotic dynamics of Selkov equations, Disc. Cont. Dyn. Syst. (S), 2 (2009), 193-219. doi: 10.3934/dcdss.2009.2.193.
[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.
[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.
[43]	S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624. doi: 10.1006/jdeq.2001.4032.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

show all references

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.
[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.
[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.
[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.
[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.
[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.
[7]	H. Chate and M. Courbage, Lattice systems, Phys. D, 103 (1997), 1-612. doi: 10.1016/S0167-2789(96)00249-7.
[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.
[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.
[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.
[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.
[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.
[13]	L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 40 (1993), 147-156. doi: 10.1109/81.222795.
[14]	L. O. Chua and Y. Yang, Cellular neural networks: theory, IEEE Trans. Circuits Syst., 35 (1988), 1257-1272. doi: 10.1109/31.7600.
[15]	V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," AMS Colloquium Publications, 49, AMS, Providence, RI, 2002.
[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.
[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.
[18]	J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Math. Surveys and Monographs, AMS, Providence, RI, 1988.
[19]	M. Hillert, A solid-solution model for inhomogeneous systems, Acta Metall., 9 (1961), 525-535. doi: 10.1016/0001-6160(61)90155-9.
[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.
[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.
[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.
[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.
[24]	R. Kapval, Discrete models for chemically reacting systems, J. Math. Chem., 6 (1991), 113-163. doi: 10.1007/BF01192578.
[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.
[26]	O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations," Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.
[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.
[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.
[29]	E. E. Selkov, Self-oscillations in glycolysis: A simple kinetic model, Euorpean J. Bio., 4 (1968), 79-86.
[30]	G. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002.
[31]	R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics," Springer, Berlin, 1997.
[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.
[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.
[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.
[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.
[36]	B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245. doi: 10.1016/j.jde.2005.01.003.
[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.
[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.
[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.
[40]	Y. You, Asymptotic dynamics of Selkov equations, Disc. Cont. Dyn. Syst. (S), 2 (2009), 193-219. doi: 10.3934/dcdss.2009.2.193.
[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.
[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.
[43]	S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2002), 605-624. doi: 10.1006/jdeq.2001.4032.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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