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Synchronization of chaotic delayed neural networks with impulsive and stochastic perturbations. (English) Zbl 1221.93225
Summary: We study the exponential synchronization problem of a class of chaotic delayed neural networks with impulsive and stochastic perturbations. The involved time delays include time-varying delays and unbounded distributed delays. Employing the method of impulsive delay differential inequality, several new sufficient conditions ensuring the exponential synchronization are obtained, which can be easily checked by the {\tt LMI Control Toolbox} in {\tt Matlab}. Compared with the previous methods, our method does not resort to complicated Lyapunov--Krasovkii, and the results derived are independent of the time-varying delays and do not require the differentiability of delay functions and the monotony of the activation functions. Finally, a numerical example and its simulation is given to show the effectiveness of the obtained results in this paper.

93D15Stabilization of systems by feedback
34K50Stochastic functional-differential equations
60G35Signal detection and filtering (stochastic processes)
92B20General theory of neural networks (mathematical biology)
Full Text: DOI
[1] Milanovic, V.; Zaghloul, M.: Synchronization of chaotic neural networks and applications to communications, Int J bifurcation chaos 6, 2571-2585 (1996) · Zbl 1298.94005
[2] Liao, T.; Tsai, S.: Adaptive synchronization of chaotic systems and its application to secure communications, Chaos solitons fract 11, 1387-1396 (2001) · Zbl 0967.93059 · doi:10.1016/S0960-0779(99)00051-X
[3] Perez-Munuzuri, V.; Perez-Villar, V.; Chua, L.: Autowaves for image processing on a two-dimensional CNN array of excitable nonlinear circuits: flat and wrinkled labyrinths, IEEE trans circuits syst I 40, 174-181 (1993) · Zbl 0800.92038 · doi:10.1109/81.222798
[4] Fox, J.; Jayaprakash, C.; Wang, D.; Campbell, S.: Synchronization in relaxation oscillator networks with conduction delays, Neuralcomputing 13, 1003-1021 (2001) · Zbl 0979.92007 · doi:10.1162/08997660151134307
[5] Skarda, A.; Freeman, W.: How brains make chaos in order to make sense of the world, Brain behav sci 10, 161-195 (1987)
[6] Steinmetz, P.; Roy, A.; Fitzgerald, P.; Hsiao, S.; Johnson, K.; Niebur, E.: Attention modulates synchronized neuronal firing in primate somatosensory cortex, Nature 404, 187-190 (2000)
[7] Lu, H.: Chaotic attractors in delayed neural networks, Phys lett A 298, 109-116 (2002) · Zbl 0995.92004 · doi:10.1016/S0375-9601(02)00538-8
[8] Zou, F.; Nossek, J.: Bifurcation and chaos in cellular neural networks, IEEE trans on circuits syst I 40, 166-173 (1993) · Zbl 0782.92003 · doi:10.1109/81.222797
[9] Gilli, M.: Strange attractors in delayed cellular neural networks, IEEE trans circuits syst I 40, 849-853 (1993) · Zbl 0844.58056 · doi:10.1109/81.251826
[10] Zhu, Q.; Cao, J.: Stochastic stability of neural networks with both Markovian jump parameters and continuously distributed delays, Discrete dyn nature soc 2009, 1-20 (2009) · Zbl 1185.93147 · doi:10.1155/2009/490515
[11] Xia, Y.; Huang, Z.; Han, M.: Exponential p-stability of delayed Cohen -- Grossberg-type BAM neural networks with impulses, Chaos solitons fract 38, 806-818 (2008) · Zbl 1146.34329 · doi:10.1016/j.chaos.2007.01.009
[12] Liao, X.; Li, C.: An LMI approach to asymptotical stability of multi-delayed neural networks, Physica D 200, 139-155 (2005) · Zbl 1078.34052 · doi:10.1016/j.physd.2004.10.009
[13] Li, X.: Global exponential stability for a class of neural networks, Appl math lett 22, 1235-1239 (2009) · Zbl 1173.34345 · doi:10.1016/j.aml.2009.01.036
[14] Xia, Y.; Wong, P.: Global exponential stability of a class of retarded impulsive differential equations with applications, Chaos solitons fract 39, 440-453 (2009) · Zbl 1197.34146 · doi:10.1016/j.chaos.2007.04.005
[15] Ahmada, S.; Stamova, I.: Global exponential stability for impulsive cellular neural networks with time-varying delays, Nonlinear anal 69, 786-795 (2008) · Zbl 1151.34061 · doi:10.1016/j.na.2008.02.067
[16] Song, Q.: Synchronization analysis of coupled connected neural networks with mixed time delays, Neurocomputing 72, 3907-3914 (2009)
[17] Xia, Y.; Yang, Z.; Han, M.: Synchronization schemes for coupled identical Yang -- Yang type fuzzy cellular neural networks, Commun nonlinear sci numer simul 14, 3645-3659 (2009) · Zbl 1221.37227 · doi:10.1016/j.cnsns.2009.01.028
[18] Gao, X.; Zhong, S.; Gao, F.: Exponential synchronization of neural networks with time-varying delays, Nonlinear anal: theory methods appl 71, 2003-2011 (2009) · Zbl 1173.34349 · doi:10.1016/j.na.2009.01.243
[19] Yu, W.; Cao, J.: Synchronization control of stochastic delayed neural networks, Physica A 373, 252-260 (2007)
[20] Tang, Y.; Qiu, R.; Fang, J.; Miao, Q.; Xia, M.: Adaptive lag synchronization in unknown stochastic chaotic neural networks with discrete and distributed time-varying delays, Phys lett A 372, 4425-4433 (2008) · Zbl 1221.82078 · doi:10.1016/j.physleta.2008.04.032
[21] Li, X.; Cao, J.: Adaptive synchronization for delayed neural networks with stochastic perturbation, J franklin inst 345, 779-791 (2008) · Zbl 1169.93350 · doi:10.1016/j.jfranklin.2008.04.012
[22] Sheng, L.; Yang, H.: Exponential synchronization of a class of neural networks with mixed time-varying delays and impulsive effects, Neurocomputing 71, 3666-3674 (2008)
[23] Li, T.; Fei, S.; Zhu, Q.; Cong, S.: Exponential synchronization of chaotic neural networks with mixed delays, Neurocomputing 71, 3005-3019 (2008)
[24] Li, T.; Song, A.; Fei, S.; Guo, Y.: Synchronization control of chaotic neural networks with time-varying and distributed delays, Nonlinear anal: theory, methods appl 71, 2372-2384 (2009) · Zbl 1171.34049 · doi:10.1016/j.na.2009.01.079
[25] Ding, W.; Han, M.; Li, M.: Exponential lag synchronization of delayed fuzzy cellular neural networks with impulses, Phys lett A 373, 832-837 (2009) · Zbl 1228.34075 · doi:10.1016/j.physleta.2008.12.049
[26] Haykin, S.: Neural networks, (1994) · Zbl 0828.68103
[27] Niculescu, S.: Delay effects on stability: a robust control approach, (2001) · Zbl 0997.93001
[28] Song, Q.: Design of controller on synchronization of chaotic neural networks with mixed time-varying delays, Neurocomputing 72, 3288-3295 (2009)
[29] Sun, Y.; Cao, J.; Wang, Z.: Exponential synchronization of stochastic perturbed chaotic delayed neural networks, Neurocomputing 70, 2477-2485 (2007)
[30] Yang, X.; Cao, J.: Stochastic synchronization of coupled neural networks with intermittent control, Phys lett A 373, 3259-3272 (2009) · Zbl 1233.34020 · doi:10.1016/j.physleta.2009.07.013
[31] Cheng, C.; Liao, T.; Hwang, C.: Exponential synchronization of a class of chaotic neural networks, Chaos solitons fract 24, 197-206 (2005) · Zbl 1060.93519 · doi:10.1016/j.chaos.2004.09.022
[32] Cao, J.; Yuan, K.; Li, H.: Global asymptotic stability of recurrent neural networks with multiple discrete delays and distributed delays, IEEE trans neural networks 17, 1646-1651 (2006)
[33] Li, X.: Existence and global exponential stability of periodic solution for impulsive Cohen -- Grossberg-type BAM neural networks with continuously distributed delays, Appl math comput 215, 292-307 (2009) · Zbl 1190.34093 · doi:10.1016/j.amc.2009.05.005
[34] Liu, Z.; L., S.; Zhong, S.; Ye, M.: Pth moment exponential synchronization analysis for a class of stochastic neural networks with mixed delays, Commun nonlinear sci numer simul 15, 1899-1909 (2010) · Zbl 1222.65012 · doi:10.1016/j.cnsns.2009.07.018
[35] Sun, Y.; Cao, J.: Adaptive lag synchronization of unknown chaotic delayed neural networks with noise perturbation, Phys lett A 364, 277-285 (2007) · Zbl 1203.93110 · doi:10.1016/j.physleta.2006.12.019
[36] Xia, Y.; Yang, Z.; Han, M.: Lag synchronization of chaotic delayed Yang -- Yang type fuzzy neural networks with noise perturbation based on adaptive control and parameter identification, IEEE trans neural networks 20, 1165-1180 (2009)
[37] Song, Q.; Cao, J.: Impulsive effects on stability of fuzzy Cohen -- Grossberg neural networks with time-varying delays, IEEE trans on systems man cybern -- part B 37, 733-741 (2007)
[38] Ignatyev, A.: On the stability of invariant sets of systems with impulse effect, Nonlinear anal: theory methods appl 69, 53-72 (2008) · Zbl 1145.34032 · doi:10.1016/j.na.2007.04.040
[39] Lakshmikantham, V.; Bainov, D.; Simeonov, P.: Theory of impulsive differential equations, (1989) · Zbl 0718.34011