Pullback attractors for stochastic recurrent neural networks with discrete and distributed delays

Meiyu Sui; Yejuan Wang; Peter E. Kloeden; Meiyu Sui; Yejuan Wang; Peter E. Kloeden

doi:10.3934/era.2020112

Electronic Research Archive

2021, Volume 29, Issue 2: 2187-2221. doi: 10.3934/era.2020112

Previous Article Next Article

Special Issues

Pullback attractors for stochastic recurrent neural networks with discrete and distributed delays

Meiyu Sui ^{1
,},
Yejuan Wang ^{1
,
,},
Peter E. Kloeden ^{2
,}

1.
School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou 730000, China
2.
Mathematisches Institut, Universität Tübingen, Tübingen 72076, Germany

Received: 01 April 2020 Revised: 01 August 2020 Published: 19 October 2020
Primary: 34A33; Secondary: 34D45, 34F05

In this paper, we investigate a class of stochastic recurrent neural networks with discrete and distributed delays for both biological and mathematical interests. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth conditions so that the uniqueness of the Cauchy problem fails to be true. Moreover, the existence of pullback attractors with or without periodicity is presented for the multi-valued noncompact random dynamical system. In particular, a new method for checking the asymptotical compactness of solutions to the class of nonautonomous stochastic lattice systems with infinite delay is used.
- Pullback attractor,
- multi-valued dynamical system,
- stochastic recurrent neural networks,
- variable delays,
- unbounded delays
Citation: Meiyu Sui, Yejuan Wang, Peter E. Kloeden. Pullback attractors for stochastic recurrent neural networks with discrete and distributed delays[J]. Electronic Research Archive, 2021, 29(2): 2187-2221. doi: 10.3934/era.2020112

Related Papers:

Abstract

In this paper, we investigate a class of stochastic recurrent neural networks with discrete and distributed delays for both biological and mathematical interests. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth conditions so that the uniqueness of the Cauchy problem fails to be true. Moreover, the existence of pullback attractors with or without periodicity is presented for the multi-valued noncompact random dynamical system. In particular, a new method for checking the asymptotical compactness of solutions to the class of nonautonomous stochastic lattice systems with infinite delay is used.

References

[1]	J.-P. Aubin and H. Franskowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2. Birkhäuser Boston, Inc., Boston, MA, 1990.
[2]	Global asymptotic stability of stochastic recurrent neural networks with multiple discrete delays and unbounded distributed delays. Appl. Math. Comput. (2008) 204: 680-686.
[3]	Non-autonomous and random attractors for delay random semilinear equations without uniqueness. Discrete Contin. Dyn. Syst. (2008) 21: 415-443.
[4]	T. Caraballo, F. Morillas and J. Valero, Pullback attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities, Differential and Difference Equations with Applications, Springer Proc. Math. Stat., Springer, New York, 47 (2013), 341-349. doi: 10.1007/978-1-4614-7333-6_27
[5]	On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems. Discrete Contin. Dyn. Syst. (2014) 34: 51-77.
[6]	D. N. Cheban, Global Attractors of Non-Autonomous Dissipative Dynamical Systems: Interdisciplinary Mathematical Sciences, 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2004. doi: 10.1142/9789812563088
[7]	Global exponential stability of delayed Hopfield neural networks. Neural Netw. (2001) 14: 977-980.
[8]	Stability results for stochastic delayed recurrent neural networks with discrete and distributed delays. J. Differential Equations (2018) 264: 3864-3898.
[9]	A. Cichocki and R. Unbehauen, Neural Networks for Optimization and Signal Processing, Wiley, Chichester, 1993.
[10]	Attractors for random dynamical systems. Probab. Theory Related Fields (1994) 100: 365-393.
[11]	Random attractors for the $3$D stochastic Navier-Stokes equation with multiplicative white noise. Stochast. Stochast. Rep. (1996) 59: 21-45.
[12]	Mean square global asymptotic stability of stochastic recurrent neural networks with distributed delays. Appl. Math. Comput. (2009) 215: 791-795.
[13]	Non-autonomous lattice systems with switching effects and delayed recovery. J. Differential Equations (2016) 261: 2986-3009.
[14]	S. Haykin, Neural Networks: A Comprehensive Foundation, Prentice-Hall, Englewood Cliffs, 1994.
[15]	Exponential stability analysis of stochastic delayed cellular neural network. Chaos Solitons Fractals (2006) 27: 1006-1010.
[16]	Mean square exponential stability of stochastic recurrent neural networks with time-varying delays. Comput. Math. Appl. (2008) 56: 1773-1778.
[17]	Pullback attractors of nonautonomous semidynamical systems. Stoch. Dyn. (2003) 3: 101-112.
[18]	Stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks. J. Comput. Appl. Math. (2010) 234: 407-417.
[19]	Exponential stability analysis for delayed semi-Markovian recurrent neural networks: A homogeneous polynomial approach. IEEE Trans. Neural Netw. Learn. Syst. (2018) 29: 6374-6384.
[20]	Robust dissipativity and passivity analysis for discrete-time stochastic neural networks with time-varying delay. Complexity (2015) 21: 47-58.
[21]	Exponential stability of hybrid stochastic recurrent neural networks with time-varying delays. Nonlinear Anal. Hybrid Syst. (2008) 2: 1198-1204.
[22]	Cellular neural networks with nonlinear and delay-type template elements and non-uniform grids. Int. J. Circuit Theory Appl. (1992) 20: 469-481.
[23]	Asymptotic stability of stochastic delayed recurrent neural networks with impulsive effects. J. Optim. Theory Appl. (2010) 147: 583-596.
[24]	B. Schmalfuss, Measure Attractors of the Stochastic Navier-Stokes Equation, Report 258, Universität Bremen, Fachbereiche Mathematik/Informatik, Elektrotechnik/Physik, Forschungsschwerpunkt Dynamische Systeme, Bremen, 1991.
[25]	$P$th moment exponential stability of stochastic recurrent neural networks with time-varying delays. Nonlinear Anal. Real World Appl. (2007) 8: 1171-1185.
[26]	Stochastic stability of discrete-time uncertain recurrent neural networks with Markovian jumping and time-varying delays. Math. Comput. Modelling (2011) 54: 1979-1988.
[27]	Stochastic stability of uncertain recurrent neural networks with Markovian jumping parameters. Acta Math. Sci. Ser. B (2015) 35: 1122-1136.
[28]	Image compression by delayed CNNs. IEEE Trans. Circuits Syst. I (1998) 45: 205-215.
[29]	Global asymptotic stability of stochastic reaction-diffusion recurrent neural networks with Markovian jumping parameters and mixed delays. J. Anal. (2019) 27: 277-292.
[30]	Almost sure exponential stability of stochastic recurrent neural networks with time-varying delays. Internat. J. Bifur. Chaos Appl. Sci. Engrg. (2010) 20: 539-544.
[31]	B. Wang, Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems, J. Differential Equations, 253 (2012), 1544-1583. doi: 10.1016/j.jde.2012.05.015
[32]	Finite lattice approximation of infinite lattice systems with delays and non-Lipschitz nonlinearities. Asymptot. Anal. (2018) 106: 169-203.
[33]	Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain. J. Differential Equations (2015) 259: 728-776.
[34]	J. Wang, Y. Wang and D. Zhao, Pullback attractors for multi-valued non-compact random dynamical systems generated by semi-linear degenerate parabolic equations with unbounded delays, Stoch. Dyn., 16 (2016), 1750001, 49 pp. doi: 10.1142/S0219493717500010
[35]	Robustness analysis for connection weight matrices of global exponential stability of stochastic delayed recurrent neural networks. Circuits Systems Signal Process (2014) 33: 2065-2083.

Reader Comments

your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)