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More bijections for Entringer and Arnold families
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 |
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.
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J.-P. Aubin and H. Franskowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2. Birkhäuser Boston, Inc., Boston, MA, 1990. |
[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. |
[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. |
[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] |
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. |
[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] |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
P. E. Kloeden,
Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112.
doi: 10.1142/S0219493703000632. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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] |
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. |
[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. |
[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] |
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. |
show all references
References:
[1] |
J.-P. Aubin and H. Franskowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2. Birkhäuser Boston, Inc., Boston, MA, 1990. |
[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. |
[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. |
[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] |
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. |
[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] |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
P. E. Kloeden,
Pullback attractors of nonautonomous semidynamical systems, Stoch. Dyn., 3 (2003), 101-112.
doi: 10.1142/S0219493703000632. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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] |
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. |
[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. |
[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] |
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. |
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