doi: 10.3934/dcdss.2021143
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Pullback and forward dynamics of nonautonomous Laplacian lattice systems on weighted spaces

1. 

221 Parker Hall, Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849 USA

2. 

Mathematisches Institut, Universität Tübingen, 72076 Tübingen, Germany

* Corresponding author: Peter E. Kloeden

Dedicated to Professor Georg Hetzer on the occasion of his 75th birthday

Received  July 2021 Revised  September 2021 Early access December 2021

A nonautonomous lattice system with discrete Laplacian operator is revisited in the weighted space of infinite sequences $ {{\ell_{\rho}^2}} $. First the existence of a pullback attractor in $ {{\ell_{\rho}^2}} $ is established by utilizing the dense inclusion of $ \ell^2 \subset {{\ell_{\rho}^2}} $. Moreover, the pullback attractor is shown to consist of a singleton trajectory when the lattice system is uniformly strictly contracting. Then forward dynamics is investigated in terms of the existence of a nonempty compact forward omega limit set. A general class of weights for the sequence space are adopted, instead of particular types of weights often used in the literature. The analysis presented in this work is more direct compare with previous studies.

Citation: Xiaoying Han, Peter E. Kloeden. Pullback and forward dynamics of nonautonomous Laplacian lattice systems on weighted spaces. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021143
References:
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W.-J. Beyn and S. Yu. Pilyugin, Attractors of reaction diffusion systems on infinite lattices, J. Dynam. Differential Equations, 15 (2003), 485-515.  doi: 10.1023/B:JODY.0000009745.41889.30.  Google Scholar

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X. Han and P. E. Kloeden, Asymptotic behavior of a neural field lattice model with a Heaviside operator, Phys. D, 389 (2019), 1-12.  doi: 10.1016/j.physd.2018.09.004.  Google Scholar

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X. Han and P. E. Kloeden, Lattice dynamical systems in the biological sciences., Modeling, Stochastic Control, Optimization, and Applications, 164 (2019), 201-233.   Google Scholar

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X. Han and P. E. Kloeden, Sigmoidal approximations of Heaviside functions in neural lattice models, J. Differential Equations, 268 (2020), 5283-5300.  doi: 10.1016/j.jde.2019.11.010.  Google Scholar

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[29]

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[33]

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[34]

P. E. KloedenT. Lorenz and M. Yang, Forward attractors in discrete time nonautonomous dynamical systems, Differential and Difference Equations with Applications, 164 (2016), 313-322.  doi: 10.1007/978-3-319-32857-7_29.  Google Scholar

[35]

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[36]

P. E. Kloeden and H. M. Rodrigues, Dynamics of a class of ODEs more general than almost periodic, Nonlinear Anal., 74 (2011), 2695-2719.  doi: 10.1016/j.na.2010.12.025.  Google Scholar

[37]

P. E. Kloeden and M. Yang, Forward attraction in nonautonomous difference equations, J. Difference Equ. Appl., 22 (2016), 513-525.  doi: 10.1080/10236198.2015.1107550.  Google Scholar

[38]

P. E. Kloeden and M. Yang, Forward attracting sets of reaction-diffusion equations on variable domains, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1259-1271.  doi: 10.3934/dcdsb.2019015.  Google Scholar

[39]

P. E. Kloeden and M. Yang, AN Introduction to Nonautonomous Dynamical Systems and Their Attractors, World Scientific Publishing Co. Inc, Singapore, 2021.  Google Scholar

[40]

N. Rashevsky, Mathematical Biophysics, Dover Publications, New York, 1960.  Google Scholar

[41]

A. C. Scott, Analysis of a myelinated nerve model, Bull. Math. Biophys., 26 (1964), 247-254.  doi: 10.1007/BF02479046.  Google Scholar

[42]

M. SuiY. WangX. Han and P. E. Kloeden, Random recurrent neural networks with delays, J. Differential Equations, 269 (2020), 8597-8639.  doi: 10.1016/j.jde.2020.06.008.  Google Scholar

[43]

E. V. Vleck 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

[44]

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

[45]

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

[46]

Y. Wang and K. Bai, Pullback attractors for a class of nonlinear lattices with delays, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1213-1230.  doi: 10.3934/dcdsb.2015.20.1213.  Google Scholar

[47]

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[48]

W. Zhao and Y. Zhang, Compactness and attracting of random attractors for non-autonomous stochastic lattice dynamical systems in weighted space, Appl. Math. Comput., 291 (2016), 226-243.  doi: 10.1016/j.amc.2016.06.045.  Google Scholar

[49]

X.-Q. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 763-785.  doi: 10.3934/dcdsb.2008.9.763.  Google Scholar

[50]

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

[51]

S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 2002 (2004), 342-368.  doi: 10.1016/j.jde.2004.02.005.  Google Scholar

[52]

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

[53]

S. Zhou and X. Han, Pullback exponential attractors for non-autonomous lattice systems, J. Dynam. Differential Equations, 24 (2012), 601-631.  doi: 10.1007/s10884-012-9260-7.  Google Scholar

show all references

References:
[1]

A. Y. Abdallah, Uniform exponential attractors 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]

A. Y. Abdallah, Upper semicontinuity of the attractor for lattice dynamical systems of partly dissipative reaction-diffusion systems, J. Appl. Math., 2005 (2005), 273-288.  doi: 10.1155/JAM.2005.273.  Google Scholar

[3]

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

[4]

P. W. BatesH. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21.  doi: 10.1142/S0219493706001621.  Google Scholar

[5]

P. W. BatesK. Lu and B. Wang, Attractors for lattice dynamical systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 143-153.  doi: 10.1142/S0218127401002031.  Google Scholar

[6]

P. W. BatesK. Lu and B. Wang, Attractors of non-autonomous stochastic lattice systems in weighted spaces, Phys. D, 289 (2014), 32-50.  doi: 10.1016/j.physd.2014.08.004.  Google Scholar

[7]

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

[8]

J. Bell and C. Cosner, Threshold behaviour 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

[9]

W.-J. Beyn and S. Yu. Pilyugin, Attractors of reaction diffusion systems on infinite lattices, J. Dynam. Differential Equations, 15 (2003), 485-515.  doi: 10.1023/B:JODY.0000009745.41889.30.  Google Scholar

[10]

T. Caraballo and X. Han, Applied Nonautonomous and Random Dynamical Systems, Applied dynamical systems. SpringerBriefs in Mathematics. Springer, Cham, 2016.  Google Scholar

[11]

T. CaraballoX. HanB. Schmalfuss and J. Valero, Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise, Nonlinear Anal., 130 (2016), 255-278.  doi: 10.1016/j.na.2015.09.025.  Google Scholar

[12]

T. CaraballoP. E. Kloeden and B. Schmalfuß, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183-207.  doi: 10.1007/s00245-004-0802-1.  Google Scholar

[13]

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

[14]

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

[15]

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

[16]

T. CaraballoF. Morillas and J. Valero, Attractors for non-autonomous retarded lattice dynamical systems, Nonauton. Dyn. Syst., 2 (2015), 31-51.  doi: 10.1515/msds-2015-0003.  Google Scholar

[17]

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

[18]

L. O. Chua and T. Roska, The CNN paradigm, IEEE Trans. Circuits Syst., 40 (1993), 147-156.  doi: 10.1109/81.222795.  Google Scholar

[19]

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

[20]

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

[21]

H. Cui and P. E. Kloeden, Invariant forward attractors of non-autonomous random dynamical systems, J. Differential Equations, 265 (2018), 6166-6186.  doi: 10.1016/j.jde.2018.07.028.  Google Scholar

[22]

X. Han, Random attractors for second order stochastic lattice dynamical systems with multiplicative noise in weighted spaces, Stoch. Dyn., 12 (2012), 1150024, 20pp. doi: 10.1142/S0219493711500249.  Google Scholar

[23]

X. Han and P. E. Kloeden, Non-autonomous lattice systems with switching effects and delayed recovery, J. Differential Equations, 261 (2016), 2986-3009.  doi: 10.1016/j.jde.2016.05.015.  Google Scholar

[24]

X. Han and P. E. Kloeden, Asymptotic behavior of a neural field lattice model with a Heaviside operator, Phys. D, 389 (2019), 1-12.  doi: 10.1016/j.physd.2018.09.004.  Google Scholar

[25]

X. Han and P. E. Kloeden, Lattice dynamical systems in the biological sciences., Modeling, Stochastic Control, Optimization, and Applications, 164 (2019), 201-233.   Google Scholar

[26]

X. Han and P. E. Kloeden, Sigmoidal approximations of Heaviside functions in neural lattice models, J. Differential Equations, 268 (2020), 5283-5300.  doi: 10.1016/j.jde.2019.11.010.  Google Scholar

[27]

X. HanP. E. Kloden and B. Usman, Upper semi-continuous convergence of attractors for a Hopfield-type lattice model, Nonlinearity, 33 (2020), 1881-1906.  doi: 10.1088/1361-6544/ab6813.  Google Scholar

[28]

X. HanP. E. Kloeden and B. Usman, Long term behavior of a random Hopfield neural lattice model, Commun. Pure Appl. Anal., 18 (2019), 809-824.  doi: 10.3934/cpaa.2019039.  Google Scholar

[29]

X. HanW. 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

[30]

N. I. Karachalios and A. N. Yannacopoulos, Global existence and compact attractors for the discrete nonlinear Schrödinger equation, J. Differential Equations, 217 (2005), 88-123.  doi: 10.1016/j.jde.2005.06.002.  Google Scholar

[31]

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

[32]

J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium, J. Theor. Biol., 148 (1991), 49-82.  doi: 10.1016/S0022-5193(05)80465-5.  Google Scholar

[33]

P. E. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proc. Amer. Math. Soc., 144 (2016), 259-268.  doi: 10.1090/proc/12735.  Google Scholar

[34]

P. E. KloedenT. Lorenz and M. Yang, Forward attractors in discrete time nonautonomous dynamical systems, Differential and Difference Equations with Applications, 164 (2016), 313-322.  doi: 10.1007/978-3-319-32857-7_29.  Google Scholar

[35]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, volume 176 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2011.  Google Scholar

[36]

P. E. Kloeden and H. M. Rodrigues, Dynamics of a class of ODEs more general than almost periodic, Nonlinear Anal., 74 (2011), 2695-2719.  doi: 10.1016/j.na.2010.12.025.  Google Scholar

[37]

P. E. Kloeden and M. Yang, Forward attraction in nonautonomous difference equations, J. Difference Equ. Appl., 22 (2016), 513-525.  doi: 10.1080/10236198.2015.1107550.  Google Scholar

[38]

P. E. Kloeden and M. Yang, Forward attracting sets of reaction-diffusion equations on variable domains, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1259-1271.  doi: 10.3934/dcdsb.2019015.  Google Scholar

[39]

P. E. Kloeden and M. Yang, AN Introduction to Nonautonomous Dynamical Systems and Their Attractors, World Scientific Publishing Co. Inc, Singapore, 2021.  Google Scholar

[40]

N. Rashevsky, Mathematical Biophysics, Dover Publications, New York, 1960.  Google Scholar

[41]

A. C. Scott, Analysis of a myelinated nerve model, Bull. Math. Biophys., 26 (1964), 247-254.  doi: 10.1007/BF02479046.  Google Scholar

[42]

M. SuiY. WangX. Han and P. E. Kloeden, Random recurrent neural networks with delays, J. Differential Equations, 269 (2020), 8597-8639.  doi: 10.1016/j.jde.2020.06.008.  Google Scholar

[43]

E. V. Vleck 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

[44]

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

[45]

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

[46]

Y. Wang and K. Bai, Pullback attractors for a class of nonlinear lattices with delays, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1213-1230.  doi: 10.3934/dcdsb.2015.20.1213.  Google Scholar

[47]

S. V. Zelik, The attractor for a nonlinear reaction-diffusion system in the unbounded domain and Kolmogorov's $\epsilon$-entropy, Math. Nachr., 232 (2001), 129-179.  doi: 10.1002/1522-2616(200112)232:1<129::AID-MANA129>3.0.CO;2-T.  Google Scholar

[48]

W. Zhao and Y. Zhang, Compactness and attracting of random attractors for non-autonomous stochastic lattice dynamical systems in weighted space, Appl. Math. Comput., 291 (2016), 226-243.  doi: 10.1016/j.amc.2016.06.045.  Google Scholar

[49]

X.-Q. Zhao and S. Zhou, Kernel sections for processes and nonautonomous lattice systems, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 763-785.  doi: 10.3934/dcdsb.2008.9.763.  Google Scholar

[50]

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

[51]

S. Zhou, Attractors and approximations for lattice dynamical systems, J. Differential Equations, 2002 (2004), 342-368.  doi: 10.1016/j.jde.2004.02.005.  Google Scholar

[52]

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

[53]

S. Zhou and X. Han, Pullback exponential attractors for non-autonomous lattice systems, J. Dynam. Differential Equations, 24 (2012), 601-631.  doi: 10.1007/s10884-012-9260-7.  Google Scholar

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