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Projective synchronization of different chaotic time-delayed neural networks based on integral sliding mode controller. (English) Zbl 1200.65103
Summary: An integral sliding mode control approach is presented to study the projective synchronization for different chaotic time-delayed neural networks. A sliding mode surface is appropriately constructed and a sliding mode controller is synthesized to guarantee the reachability of the specified sliding surface. The global asymptotic stability of the error dynamical system in the specified switching surface is investigated with the Lyapunov-Krasovskii (L-K) functional method. A delay-dependent sufficient condition is derived and the maximum time-delay value is obtained by means of the linear matrix inequality (LMI) technique. A simulation example is finally exploited to illustrate the feasibility and effectiveness of the proposed approach, verify the conservativeness of L-K method and LMI technique, and exhibit the relationship between the convergence velocity of error system and the gain matrix.

65P20Numerical chaos
37D45Strange attractors, chaotic dynamics
65L03Functional-differential equations (numerical methods)
34K28Numerical approximation of solutions of functional-differential equations
37M05Simulation (dynamical systems)
65L20Stability and convergence of numerical methods for ODE
Full Text: DOI
[1] Pecora, L. M.; Carroll, T. L.: Synchronization in chaotic systems, Phys. rev. Lett. 64, 821-824 (1990) · Zbl 0938.37019
[2] Chen, G. R.; Dong, X.: From chaos to order: methodologies, perspectives, and applications, (1998) · Zbl 0908.93005
[3] Yang, T.; Chua, L. O.: Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication, IEEE trans. Circuits syst. I 44, 976-988 (1997)
[4] Ojalvo, J. G.; Roy, R.: Spatiotemporal communication with synchronized optical chaos, Phys. rev. Lett. 86, 5204-5207 (2001)
[5] Boccaletti, S.; Kurths, J.; Osipov, G.; Valladares, D. L.; Zhou, C. S.: The synchronization of chaotic systems, Phys. rep. 366, 1-101 (2002) · Zbl 0995.37022 · doi:10.1016/S0370-1573(02)00137-0
[6] Sprott, J. C.: Chaos and time-series analysis, (2003) · Zbl 1012.37001
[7] Ott, E.; Grebogi, C.; Yoke, J. A.: Controlling chaos, Phys. rev. Lett. 64, 1196-1199 (1990) · Zbl 0964.37501 · doi:10.1103/PhysRevLett.64.1196
[8] Schuster, H. G.: Handbook of chaos control, (1999) · Zbl 0997.93500
[9] Balasubramaniam, P.; Rakkiyappan, R.: Global asymptotic stability of stochastic recurrent neural networks with multiple discrete delays and unbounded distributed delays, Appl. math. Comput. 204, 680-686 (2008) · Zbl 1152.93049 · doi:10.1016/j.amc.2008.05.001
[10] Zhou, L. Q.; Hu, G. D.: Global exponential periodicity and stability of cellular neural networks with variable and distributed delays, Appl. math. Comput. 195, 402-411 (2008) · Zbl 1137.34350 · doi:10.1016/j.amc.2007.04.114
[11] Huang, H.; Cao, J. D.: On global asymptotic stability of recurrent neural networks with time-varying delays, Appl. math. Comput. 142, 143-154 (2003) · Zbl 1035.34081 · doi:10.1016/S0096-3003(02)00289-8
[12] Chen, Z.; Zhao, D. H.; Fu, X. L.: Discrete analogue of high-order periodic Cohen -- Grossberg neural networks with delay, Appl. math. Comput. 214, 210-217 (2009) · Zbl 1172.39020 · doi:10.1016/j.amc.2009.03.083
[13] Lu, H. T.: Chaotic attractors in delayed neural networks, Phys. lett. A 298, 109-116 (2002) · Zbl 0995.92004 · doi:10.1016/S0375-9601(02)00538-8
[14] 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
[15] Yu, P.; Yuan, Y.; Xu, J.: Study of double Hopf bifurcation and chaos for an oscillator with time-delayed feedback, Commun. nonlinear sci. Numer. simul. 7, 69-91 (2002) · Zbl 1010.34070 · doi:10.1016/S1007-5704(02)00007-2
[16] Huang, H.; Feng, G.: Synchronization of nonidentical chaotic neural networks with time delays, Neural networks 22, 869-874 (2009)
[17] Chee, C. Y.; Xu, D. L.: Control of the formation of projective synchronization in lower-dimensional discrete-time systems, Phys. lett. A 318, 112-118 (2003) · Zbl 1098.37512 · doi:10.1016/j.physleta.2003.09.024
[18] Wang, Q.; Chen, Y.: Generalized Q -- S (lag, anticipated and complete) synchronization in modified Chua’s circuit and hindmarsh -- rose systems, Appl. math. Comput. 181, 48-56 (2006) · Zbl 1145.37312 · doi:10.1016/j.amc.2006.01.017
[19] Wang, D. X.; Zhong, Y. L.; Chen, S. H.: Lag synchronizing chaotic system based on a single controller, Commun. nonlinear sci. Numer. simul. 13, 637-644 (2008) · Zbl 1130.34322 · doi:10.1016/j.cnsns.2006.05.005
[20] Ghosh, D.; Chowdhury, A. R.; Saha, P.: On the various kinds of synchronization in delayed Duffing -- van der Pol system, Commun. nonlinear sci. Numer. simul. 13, 790-803 (2008) · Zbl 1221.34196 · doi:10.1016/j.cnsns.2006.07.001
[21] Sawalha, M. M. A.; Noorani, M. S. M.: Anti-synchronization of two hyperchaotic systems via nonlinear control, Commun. nonlinear sci. Numer. simul. 14, 3402-3411 (2009) · Zbl 1221.37210 · doi:10.1016/j.cnsns.2008.12.021
[22] Huang, X.; Cao, J. D.: Generalized synchronization for delayed chaotic neural networks: a novel coupling scheme, Nonlinearity 19, 2797-2811 (2006) · Zbl 1111.37022 · doi:10.1088/0951-7715/19/12/004
[23] Ghosh, D.: Generalized projective synchronization in time-delayed systems: nonlinear observer approach, Chaos 19, 013102 (2009) · Zbl 1311.34111
[24] Gonzalez-Miranda, J. M.: Amplification and displacement of chaotic attractors by means of unidirectional chaotic driving, Phys. rev. E 57, 7321-7324 (1998)
[25] Mainieri, R.; Rehacek, J.: Projective synchronization in three-dimensional chaotic systems, Phys. rev. Lett. 82, 3042-3045 (1999)
[26] Yan, J. P.; Li, C. P.: Generalized projective synchronization of a unified chaotic system, Chaos solitons fract. 26, 1119-1124 (2005) · Zbl 1073.65147 · doi:10.1016/j.chaos.2005.02.034
[27] Li, G. H.: Generalized projective synchronization of two chaotic systems by using active control, Chaos solitons fract. 30, 77-82 (2006) · Zbl 1144.37372 · doi:10.1016/j.chaos.2005.08.130
[28] Astakhov, V. V.; Anishchenko, V. S.; Kapitaniak, T.; Shabunin, A. V.: Synchronization of chaotic oscillators by periodic parametric perturbations, Phys. D 109, 11-16 (1997) · Zbl 0925.58055 · doi:10.1016/S0167-2789(97)00153-X
[29] Yang, X. S.; Duan, C. K.; Liao, X. X.: A note on mathematical aspects of drive -- response type synchronization, Chaos solitons fract. 10, 1457-1462 (1999) · Zbl 0955.37020 · doi:10.1016/S0960-0779(98)00123-4
[30] Zhang, H. G.; Xie, Y. H.; Wang, Z. L.; Zheng, C. D.: Adaptive synchronization between two different chaotic neural networks with time delay, IEEE trans. Neural networks 18, 1841-1845 (2007)
[31] Han, X. R.; Fridman, E.; Spurgeon, S. K.; Edwards, C.: On the design of sliding mode static-output-feedback controllers for systems with state-delay, IEEE trans. Ind. electron. 56, 3656-3664 (2009)
[32] Roopaei, M.; Jahromi, M. Z.; John, R.; Lin, T. C.: Unknown nonlinear chaotic gyros synchronization using adaptive fuzzy sliding mode control with unknown dead-zone input, Commun. nonlinear sci. Numer. simul. 15, 2536-2545 (2010) · Zbl 1222.93123 · doi:10.1016/j.cnsns.2009.09.022
[33] Cai, N.; Jing, Y. W.; Zhang, S. Y.: Modified projective synchronization of chaotic systems with disturbances via active sliding mode control, Commun. nonlinear sci. Numer. simul. 15, 1613-1620 (2010) · Zbl 1221.37211 · doi:10.1016/j.cnsns.2009.06.012
[34] Li, G. H.; Zhou, S. P.; Yang, K.: Generalized projective synchronization between two different chaotic systems using active backstepping control, Phys. lett. A 355, 326-330 (2006)
[35] Roopaei, M.; Jahromi, M. Z.: Synchronization of two different chaotic systems using novel adaptive fuzzy sliding mode control, Chaos 18, 033133 (2008) · Zbl 1309.34075
[36] Yang, Y. Q.; Cao, J. D.: Exponential lag synchronization of a class of chaotic delayed neural networks with impulsive effects, Phys. A 386, 492-502 (2007)
[37] Utkin, V. I.: Variable structure systems with sliding modes, IEEE trans. Autom. control 22, 212-222 (1977) · Zbl 0382.93036 · doi:10.1109/TAC.1977.1101446
[38] Meng, J.; Wang, X. Y.: Generalized projective synchronization of a class of delayed neural networks, Mod. phys. Lett. B 22, 181-190 (2008) · Zbl 1158.93026 · doi:10.1142/S0217984908014596
[39] Feng, C. F.; Zhang, Y.; Sun, J. T.; Qi, W.; Wang, Y. H.: Generalized projective synchronization in time-delayed chaotic systems, Chaos solitons fract. 38, 743-747 (2008) · Zbl 1146.37318 · doi:10.1016/j.chaos.2007.01.037
[40] Zhu, S. Q.; Zhang, C. H.; Cheng, Z. L.; Feng, J.: Delay-dependent robust stability criteria for two classes of uncertain singular time-delay systems, IEEE trans. Autom. control 52, 880-885 (2007)
[41] Fridman, E.: Effects of small delays on stability of singularly perturbed systems, Automatica 38, 897-902 (2002) · Zbl 1014.93025 · doi:10.1016/S0005-1098(01)00265-5
[42] Fridman, E.; Shaked, U.: H-infinity control of linear state-delay descriptor systems: an LMI approach, Linear algebra appl. 351 -- 352, 271-302 (2002) · Zbl 1006.93021 · doi:10.1016/S0024-3795(01)00563-8
[43] Fridman, E.: Stability of linear descriptor systems with delay: A Lyapunov-based approach, J. math. Anal. appl. 273, 24-44 (2002) · Zbl 1032.34069 · doi:10.1016/S0022-247X(02)00202-0
[44] Utkin, V. I.: Sliding modes in control and optimization, (1992) · Zbl 0748.93044
[45] Hale, J.; Lunel, S.: Introduction to the theory of functional differential equations, (1991) · Zbl 0725.34071
[46] Olgac, N.; Iragavarapu, V. R.: Sliding mode control with backlash and saturation laws, Int. J. Robot autom. 10, 49-55 (1995)