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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

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 & Continuous Dynamical Systems - A, 2014, 34 (1) : 229-248. doi: 10.3934/dcds.2014.34.229
References:
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Ahmed Y. Abdallah, Uniform exponential attractor for first order non-autonomous lattice dynamical systems,, J. Differential Equations, 251 (2011), 1489.  doi: 10.1016/j.jde.2011.05.030.  Google Scholar

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Ahmed Y. Abdallah, Exponential attractors for first-order lattice dynamical systems,, J. Math. Anal. Appl., 339 (2008), 217.  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.  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.  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.  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.  doi: 10.1023/B:JODY.0000009745.41889.30.  Google Scholar

[7]

H. Chate and M. Courbage, Lattice systems,, Phys. D, 103 (1997), 1.  doi: 10.1016/S0167-2789(96)00249-7.  Google Scholar

[8]

S.-N. Chow, Lattice dynamical systems,, in, 1822 (2003), 1.  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.  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.  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.  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.   Google Scholar

[13]

L. O. Chua and T. Roska, The CNN paradigm,, IEEE Trans. Circuits Syst., 40 (1993), 147.  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.  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 (2002).   Google Scholar

[16]

T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems,, Phys. D, 67 (1993), 237.  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.  doi: 10.1103/PhysRevA.47.4287.  Google Scholar

[18]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", Math. Surveys and Monographs, (1988).   Google Scholar

[19]

M. Hillert, A solid-solution model for inhomogeneous systems,, Acta Metall., 9 (1961), 525.  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).  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.  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.  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.  doi: 10.1137/0147038.  Google Scholar

[24]

R. Kapval, Discrete models for chemically reacting systems,, J. Math. Chem., 6 (1991), 113.  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.  doi: 10.1016/j.jde.2005.06.002.  Google Scholar

[26]

O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,", Cambridge University Press, (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.  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.  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.   Google Scholar

[30]

G. Sell and Y. You, "Dynamics of Evolutionary Equations,", Applied Mathematical Sciences, 143 (2002).   Google Scholar

[31]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,", Springer, (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.  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.  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.  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.  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.  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.  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.  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.   Google Scholar

[40]

Y. You, Asymptotic dynamics of Selkov equations,, Disc. Cont. Dyn. Syst. (S), 2 (2009), 193.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  doi: 10.3934/dcds.2008.21.1259.  Google Scholar

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.  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.  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.  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.  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.  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.  doi: 10.1023/B:JODY.0000009745.41889.30.  Google Scholar

[7]

H. Chate and M. Courbage, Lattice systems,, Phys. D, 103 (1997), 1.  doi: 10.1016/S0167-2789(96)00249-7.  Google Scholar

[8]

S.-N. Chow, Lattice dynamical systems,, in, 1822 (2003), 1.  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.  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.  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.  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.   Google Scholar

[13]

L. O. Chua and T. Roska, The CNN paradigm,, IEEE Trans. Circuits Syst., 40 (1993), 147.  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.  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 (2002).   Google Scholar

[16]

T. Erneux and G. Nicolis, Propagating waves in discrete bistable reaction diffusion systems,, Phys. D, 67 (1993), 237.  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.  doi: 10.1103/PhysRevA.47.4287.  Google Scholar

[18]

J. K. Hale, "Asymptotic Behavior of Dissipative Systems,", Math. Surveys and Monographs, (1988).   Google Scholar

[19]

M. Hillert, A solid-solution model for inhomogeneous systems,, Acta Metall., 9 (1961), 525.  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).  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.  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.  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.  doi: 10.1137/0147038.  Google Scholar

[24]

R. Kapval, Discrete models for chemically reacting systems,, J. Math. Chem., 6 (1991), 113.  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.  doi: 10.1016/j.jde.2005.06.002.  Google Scholar

[26]

O. Ladyzhenskaya, "Attractors for Semigroups and Evolution Equations,", Cambridge University Press, (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.  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.  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.   Google Scholar

[30]

G. Sell and Y. You, "Dynamics of Evolutionary Equations,", Applied Mathematical Sciences, 143 (2002).   Google Scholar

[31]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,", Springer, (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.  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.  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.  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.  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.  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.  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.  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.   Google Scholar

[40]

Y. You, Asymptotic dynamics of Selkov equations,, Disc. Cont. Dyn. Syst. (S), 2 (2009), 193.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  doi: 10.3934/dcds.2008.21.1259.  Google Scholar

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