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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 and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021143
References:
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[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

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

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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.

[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.

[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.

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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.

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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.

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[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[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.

[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.

[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.

[39]

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

[40]

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

[41]

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

[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.

[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.

[44]

B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[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.

[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.

[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.

[39]

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

[40]

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

[41]

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

[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.

[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.

[44]

B. Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.  doi: 10.1016/j.jde.2005.01.003.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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