# American Institute of Mathematical Sciences

doi: 10.3934/era.2020112

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

 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

* Corresponding author: Yejuan Wang

Received  April 2020 Revised  August 2020 Published  October 2020

Fund Project: This work was supported by NSF of China (Grant Nos. 41875084, 11571153), the Fundamental Research Funds for the Central Universities under Grant Nos. lzujbky-2018-it58 and lzujbky-2018-ot03

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.

Citation: Meiyu Sui, Yejuan Wang, Peter E. Kloeden. Pullback attractors for stochastic recurrent neural networks with discrete and distributed delays. Electronic Research Archive, doi: 10.3934/era.2020112
##### References:
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Syed Ali, Stochastic stability of uncertain recurrent neural networks with Markovian jumping parameters, Acta Math. Sci. Ser. B, 35 (2015), 1122-1136.  doi: 10.1016/S0252-9602(15)30044-8.  Google Scholar [28] P. Venetianer and T. Roska, Image compression by delayed CNNs, IEEE Trans. Circuits Syst. I, 45 (1998), 205-215.   Google Scholar [29] C. Vidhya, S. Dharani and P. Balasubramaniam, Global asymptotic stability of stochastic reaction-diffusion recurrent neural networks with Markovian jumping parameters and mixed delays, J. Anal., 27 (2019), 277-292.  doi: 10.1007/s41478-018-0123-4.  Google Scholar [30] L. Wan and Q. Zhou, Almost sure exponential stability of stochastic recurrent neural networks with time-varying delays, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 539-544.  doi: 10.1142/S0218127410025594.  Google Scholar [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.  Google Scholar [32] Y. Wang and M. Sui, Finite lattice approximation of infinite lattice systems with delays and non-Lipschitz nonlinearities, Asymptot. Anal., 106 (2018), 169-203.  doi: 10.3233/ASY-171444.  Google Scholar [33] Y. Wang and J. Wang, Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, J. Differential Equations, 259 (2015), 728-776.  doi: 10.1016/j.jde.2015.02.026.  Google Scholar [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.  Google Scholar [35] S. Zhu, W. Luo and Y. Shen, Robustness analysis for connection weight matrices of global exponential stability of stochastic delayed recurrent neural networks, Circuits Systems Signal Process, 33 (2014), 2065-2083.  doi: 10.1007/s00034-013-9735-8.  Google Scholar

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##### References:
 [1] J.-P. Aubin and H. Franskowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2. Birkhäuser Boston, Inc., Boston, MA, 1990.  Google Scholar [2] P. Balasubramaniam and R. Rakkiyappan, Global asymptotic stability of stochastic recurrent neural networks with multiple discrete delays and unbounded distributed delays, Appl. Math. Comput., 204 (2008), 680-686.  doi: 10.1016/j.amc.2008.05.001.  Google Scholar [3] T. Caraballo, M. J. Garrido-Atienza, B. Schmalfuss and J. Valero, Non-autonomous and random attractors for delay random semilinear equations without uniqueness, Discrete Contin. Dyn. Syst., 21 (2008), 415-443.  doi: 10.3934/dcds.2008.21.415.  Google Scholar [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.  Google Scholar [5] T. Caraballo, F. Morillas and J. Valero, On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems, Discrete Contin. Dyn. Syst., 34 (2014), 51-77.  doi: 10.3934/dcds.2014.34.51.  Google Scholar [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.  Google Scholar [7] T. Chen, Global exponential stability of delayed Hopfield neural networks, Neural Netw., 14 (2001), 977-980.   Google Scholar [8] G. Chen, D. Li, L. Shi, O. van Gaans and S. Verduyn Lunel, Stability results for stochastic delayed recurrent neural networks with discrete and distributed delays, J. Differential Equations, 264 (2018), 3864-3898.  doi: 10.1016/j.jde.2017.11.032.  Google Scholar [9] A. Cichocki and R. Unbehauen, Neural Networks for Optimization and Signal Processing, Wiley, Chichester, 1993. Google Scholar [10] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.  doi: 10.1007/BF01193705.  Google Scholar [11] F. Flandoli and B. Schmalfuss, Random attractors for the $3$D stochastic Navier-Stokes equation with multiplicative white noise, Stochast. Stochast. Rep., 59 (1996), 21-45.  doi: 10.1080/17442509608834083.  Google Scholar [12] Y. Guo, Mean square global asymptotic stability of stochastic recurrent neural networks with distributed delays, Appl. Math. Comput., 215 (2009), 791-795.  doi: 10.1016/j.amc.2009.06.002.  Google Scholar [13] 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 [14] S. Haykin, Neural Networks: A Comprehensive Foundation, Prentice-Hall, Englewood Cliffs, 1994. Google Scholar [15] J. Hu, S. Zhong and L. Liang, Exponential stability analysis of stochastic delayed cellular neural network, Chaos Solitons Fractals, 27 (2006), 1006-1010.  doi: 10.1016/j.chaos.2005.04.067.  Google Scholar [16] C. Huang, Y. He and H. Wang, Mean square exponential stability of stochastic recurrent neural networks with time-varying delays, Comput. Math. Appl., 56 (2008), 1773-1778.  doi: 10.1016/j.camwa.2008.04.004.  Google Scholar [17] P. E. Kloeden, Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112.  doi: 10.1142/S0219493703000632.  Google Scholar [18] X. Li and X. Fu, Stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks, J. Comput. Appl. Math., 234 (2010), 407-417.  doi: 10.1016/j.cam.2009.12.033.  Google Scholar [19] X. Li, F. Li, X. Zhang, C. Yang and W. Gui, Exponential stability analysis for delayed semi-Markovian recurrent neural networks: A homogeneous polynomial approach, IEEE Trans. Neural Netw. Learn. Syst., 29 (2018), 6374-6384.  doi: 10.1109/TNNLS.2018.2830789.  Google Scholar [20] G. Nagamani, S. Ramasamy and P. Balasubramaniam, Robust dissipativity and passivity analysis for discrete-time stochastic neural networks with time-varying delay, Complexity, 21 (2015), 47-58.  doi: 10.1002/cplx.21614.  Google Scholar [21] G. Peng and L. Huang, Exponential stability of hybrid stochastic recurrent neural networks with time-varying delays, Nonlinear Anal. Hybrid Syst., 2 (2008), 1198-1204.  doi: 10.1016/j.nahs.2008.09.012.  Google Scholar [22] T. Roska and L. O. Chua, Cellular neural networks with nonlinear and delay-type template elements and non-uniform grids, Int. J. Circuit Theory Appl., 20 (1992), 469-481.   Google Scholar [23] R. Sakthivel, R. Samidurai and S. M. Anthoni, Asymptotic stability of stochastic delayed recurrent neural networks with impulsive effects, J. Optim. Theory Appl., 147 (2010), 583-596.  doi: 10.1007/s10957-010-9728-8.  Google Scholar [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. Google Scholar [25] Y. Sun and J. Cao, $P$th moment exponential stability of stochastic recurrent neural networks with time-varying delays, Nonlinear Anal. Real World Appl., 8 (2007), 1171-1185.  doi: 10.1016/j.nonrwa.2006.06.009.  Google Scholar [26] M. Syed Ali and M. Marudai, Stochastic stability of discrete-time uncertain recurrent neural networks with Markovian jumping and time-varying delays, Math. Comput. Modelling, 54 (2011), 1979-1988.  doi: 10.1016/j.mcm.2011.05.004.  Google Scholar [27] M. Syed Ali, Stochastic stability of uncertain recurrent neural networks with Markovian jumping parameters, Acta Math. Sci. Ser. B, 35 (2015), 1122-1136.  doi: 10.1016/S0252-9602(15)30044-8.  Google Scholar [28] P. Venetianer and T. Roska, Image compression by delayed CNNs, IEEE Trans. Circuits Syst. I, 45 (1998), 205-215.   Google Scholar [29] C. Vidhya, S. Dharani and P. Balasubramaniam, Global asymptotic stability of stochastic reaction-diffusion recurrent neural networks with Markovian jumping parameters and mixed delays, J. Anal., 27 (2019), 277-292.  doi: 10.1007/s41478-018-0123-4.  Google Scholar [30] L. Wan and Q. Zhou, Almost sure exponential stability of stochastic recurrent neural networks with time-varying delays, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 539-544.  doi: 10.1142/S0218127410025594.  Google Scholar [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.  Google Scholar [32] Y. Wang and M. Sui, Finite lattice approximation of infinite lattice systems with delays and non-Lipschitz nonlinearities, Asymptot. Anal., 106 (2018), 169-203.  doi: 10.3233/ASY-171444.  Google Scholar [33] Y. Wang and J. Wang, Pullback attractors for multi-valued non-compact random dynamical systems generated by reaction-diffusion equations on an unbounded domain, J. Differential Equations, 259 (2015), 728-776.  doi: 10.1016/j.jde.2015.02.026.  Google Scholar [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.  Google Scholar [35] S. Zhu, W. Luo and Y. Shen, Robustness analysis for connection weight matrices of global exponential stability of stochastic delayed recurrent neural networks, Circuits Systems Signal Process, 33 (2014), 2065-2083.  doi: 10.1007/s00034-013-9735-8.  Google Scholar
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