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March  2019, 18(2): 809-824. doi: 10.3934/cpaa.2019039

Long term behavior of a random Hopfield neural lattice model

Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA

* Corresponding author

Received  March 2018 Revised  May 2018 Published  October 2018

Fund Project: This work was partially supported by Simons Foundation, USA (Collaboration Grants for Mathematicians No. 429717) and NSF of China (Grant No. 11571125).

A Hopfield neural lattice model is developed as the infinite dimensional extension of the classical finite dimensional Hopfield model. In addition, random external inputs are considered to incorporate environmental noise. The resulting random lattice dynamical system is first formulated as a random ordinary differential equation on the space of square summable bi-infinite sequences. Then the existence and uniqueness of solutions, as well as long term dynamics of solutions are investigated.

Citation: Xiaoying Han, Peter E. Kloeden, Basiru Usman. Long term behavior of a random Hopfield neural lattice model. Communications on Pure and Applied Analysis, 2019, 18 (2) : 809-824. doi: 10.3934/cpaa.2019039
References:
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L. Arnold, Random Dynamical System, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

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

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

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T. CaraballoG. Lukaszewicz and J. Real, Pullback attractor for asymptotically compact nonautonomous dynamical systems, Nonlinear Analysis TMA, 6 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

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S. N. Chow, Lattice Dynamical Systems, in: Dynamical System, in Lecture notes in Math. Vol. 1822, Springer-Verlag, Berlin, pp. 1-102, 2003. doi: 10.1007/978-3-540-45204-1_1.

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S. N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems, IEEE Trans. Circuits Syst., 42 (1995), 746-751.  doi: 10.1109/81.473583.

[7]

I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, Heidelberg, 2002. doi: 10.1007/b83277.

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H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[9]

R. D. Dony and S. Haykin, Neural networks approaches to image compression, IEEE, 83 (1995), 288-303. 

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H. Engler, Global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity, Mathematische Zeitschrift, 202 (1989), 251-259.  doi: 10.1007/BF01215257.

[11]

F. Fandoli and B. Schmalfuss, Random attractor for the 3d stochastic navier-stokes equation with multiplicative noise, Stochastics Stochastics, 59 (1996), 21-45.  doi: 10.1080/17442509608834083.

[12]

F. Favata and R. Walker, A study of the application of kohonen-type neural networks to the traveling salesman problem, Springger-Verlag: Biological Cybernetics, 64 (1991), 463-468. 

[13]

E. GrusenG. Kayakutlu and T. U. Daim, Using artificial neural network models in stock market index prediction, Elsevier: Expert System with Application, 38 (2011), 10389-10397. 

[14]

X. Han and P. E. Kloeden, Random Ordinary Differential Equations and Their Numerical Solution, Springer, Singapore, 2017. doi: 10.1007/978-981-10-6265-0.

[15]

J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-stage neurons, Proc. Nat. Acad. Sci. U.S.A, 81 (1984), 3088-3092. 

[16]

T. Kimoto, K. Asakawa, M. Yoda and M. Takeoka, Stock market prediction system with modular neural networks, IEE Xplore, 1-6.

[17]

N. Mani and B. Srinivasan, Application of artificial neural network model for optical character recognition, IEEE Xplore, 100 (1997), 2517-2520. 

[18]

G. D. Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[19]

W. Shen, Lifted lattice, hyperbolic structure, and topological disorder in coupled map lattices, SIAM J. Appl. Math., 56 (1996), 1379-1399.  doi: 10.1137/S0036139995282670.

[20]

E. V. Vlerk and B. Wang, Attractors for lattice fitzhugh-nagumo systems, Phys. D, 221 (2005), 317-336.  doi: 10.1016/j.physd.2005.10.006.

[21]

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

[22]

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.

[23]

S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2007), 605-624.  doi: 10.1006/jdeq.2001.4032.

show all references

References:
[1]

L. Arnold, Random Dynamical System, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

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

[3]

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

[4]

T. CaraballoG. Lukaszewicz and J. Real, Pullback attractor for asymptotically compact nonautonomous dynamical systems, Nonlinear Analysis TMA, 6 (2006), 484-498.  doi: 10.1016/j.na.2005.03.111.

[5]

S. N. Chow, Lattice Dynamical Systems, in: Dynamical System, in Lecture notes in Math. Vol. 1822, Springer-Verlag, Berlin, pp. 1-102, 2003. doi: 10.1007/978-3-540-45204-1_1.

[6]

S. N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems, IEEE Trans. Circuits Syst., 42 (1995), 746-751.  doi: 10.1109/81.473583.

[7]

I. Chueshov, Monotone Random Systems Theory and Applications, Springer-Verlag, Berlin, Heidelberg, 2002. doi: 10.1007/b83277.

[8]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.

[9]

R. D. Dony and S. Haykin, Neural networks approaches to image compression, IEEE, 83 (1995), 288-303. 

[10]

H. Engler, Global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity, Mathematische Zeitschrift, 202 (1989), 251-259.  doi: 10.1007/BF01215257.

[11]

F. Fandoli and B. Schmalfuss, Random attractor for the 3d stochastic navier-stokes equation with multiplicative noise, Stochastics Stochastics, 59 (1996), 21-45.  doi: 10.1080/17442509608834083.

[12]

F. Favata and R. Walker, A study of the application of kohonen-type neural networks to the traveling salesman problem, Springger-Verlag: Biological Cybernetics, 64 (1991), 463-468. 

[13]

E. GrusenG. Kayakutlu and T. U. Daim, Using artificial neural network models in stock market index prediction, Elsevier: Expert System with Application, 38 (2011), 10389-10397. 

[14]

X. Han and P. E. Kloeden, Random Ordinary Differential Equations and Their Numerical Solution, Springer, Singapore, 2017. doi: 10.1007/978-981-10-6265-0.

[15]

J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-stage neurons, Proc. Nat. Acad. Sci. U.S.A, 81 (1984), 3088-3092. 

[16]

T. Kimoto, K. Asakawa, M. Yoda and M. Takeoka, Stock market prediction system with modular neural networks, IEE Xplore, 1-6.

[17]

N. Mani and B. Srinivasan, Application of artificial neural network model for optical character recognition, IEEE Xplore, 100 (1997), 2517-2520. 

[18]

G. D. Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[19]

W. Shen, Lifted lattice, hyperbolic structure, and topological disorder in coupled map lattices, SIAM J. Appl. Math., 56 (1996), 1379-1399.  doi: 10.1137/S0036139995282670.

[20]

E. V. Vlerk and B. Wang, Attractors for lattice fitzhugh-nagumo systems, Phys. D, 221 (2005), 317-336.  doi: 10.1016/j.physd.2005.10.006.

[21]

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

[22]

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.

[23]

S. Zhou, Attractors for second order lattice dynamical systems, J. Differential Equations, 179 (2007), 605-624.  doi: 10.1006/jdeq.2001.4032.

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