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Chaos control and hybrid projective synchronization of several new chaotic systems. (English) Zbl 1248.93082

Summary: The problems on purposefully creating chaos and chaos controlare investigated. First, several new 4D chaotic systems are presented based on a 3D chaotic system and three dynamical characteristics are verified. Second, it is demonstrated that they can be stabilized to their equilibrium points by a same single scalar adaptive feedback control law. Third, two of the chaotic systems with constant parameters are in Hybrid Projective Synchronization (HPS) by presented adaptive feedback control strategies, which can be simplified into a scalar form for identical drive–response systems to achieve complete synchronization. Furthermore, HPS between two of the chaotic systems with unknown response system parameters is also considered and adaptive parameter update laws are developed. Numerical simulations show their characteristics and verify the effectiveness of the strategies.

MSC:

93C15 Control/observation systems governed by ordinary differential equations
93B52 Feedback control
93C40 Adaptive control/observation systems
37N35 Dynamical systems in control
34D06 Synchronization of solutions to ordinary differential equations
34H10 Chaos control for problems involving ordinary differential equations
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[1] Ueta, T.; Chen, G., Bifurcation analysis of Chen’s attractor, Int. J. Bifurcat. Chaos, 10, 1917-1931 (2000) · Zbl 1090.37531
[2] Lü, J. H.; Chen, G., A new chaotic attractor coined, Int. J. Bifurcat. Chaos, 12, 659-661 (2002) · Zbl 1063.34510
[3] Lü, J. H.; Chen, G.; Cheng, D.; Celikovsky, S., Bridge the gap between the Lorenz system and the Chen system, Int. J. Bifurcat. Chaos, 12, 2917-2926 (2002) · Zbl 1043.37026
[4] Tigan, G.; Opris, D., Analysis of a 3D chaotic system, Chaos Solitons Fract., 36, 1315-1319 (2008) · Zbl 1148.37027
[5] Chen, Z.; Yang, Y.; Qi, G.; Yuan, Z., A novel hyperchaos system only with one equilibrium, Phys. Lett. A, 360, 696-701 (2007) · Zbl 1236.37022
[6] T. Wang, K. Wang, N. Jia, Chaos control and hybrid projective synchronization of a novel chaotic system, Math. Probl. Eng., vol. 2011, Article ID 452671, 13p.; T. Wang, K. Wang, N. Jia, Chaos control and hybrid projective synchronization of a novel chaotic system, Math. Probl. Eng., vol. 2011, Article ID 452671, 13p. · Zbl 1213.34077
[7] Ott, E.; Grebogi, G.; York, J. A., Controlling chaos, Phys. Rev. Lett., 64, 1196-1199 (1990) · Zbl 0964.37501
[8] Pecora, L. M.; Carroll, T. L., Synchronization in chaotic system, Phys. Rev. Lett., 64, 821-825 (1990) · Zbl 0938.37019
[9] Lan, Y.; Li, Q., Chaos synchronization of a new hyperchaotic system, Appl. Math. Comput., 217, 2125-2132 (2010) · Zbl 1204.65144
[10] Nian, F.; Wang, X.; Niu, Y.; Lin, D., Module-phase synchronization in complex dynamic system, Appl. Math. Comput., 217, 2481-2489 (2010) · Zbl 1207.93047
[11] Shi, X. R.; Wang, Z. L., Adaptive added-order anti-synchronization of chaotic systems with fully unknown parameters, Appl. Math. Comput., 215, 1711-1717 (2009) · Zbl 1179.65158
[12] Zhang, D.; Xu, J., Projective synchronization of different chaotic time-delayed neural networks based on integral sliding mode controller, Appl. Math. Comput., 217, 164-171 (2010) · Zbl 1200.65103
[13] Zhao, X.; Li, Z.; Li, S., Synchronization of chaotic finance system, Appl. Math. Comput., 217, 6031-6039 (2011) · Zbl 1208.91175
[14] Park, J. H., Further results on functional projective synchronization of Genesio-Tesi chaotic system, Mod. Phys. Lett. B, 23, 1889-1895 (2009) · Zbl 1168.37311
[15] Park, J. H.; Ji, D. H.; Won, S. C.; Lee, S. M., Adaptive \(H_\infty\) synchronization of unified chaotic systems, Mod. Phys. Lett. B, 23, 1157-1169 (2009) · Zbl 1179.37123
[16] Hu, M.; Xu, Z.; Zhang, R.; Hu, A., Adaptive full state hybrid projective synchronization of chaotic systems with the same and different order, Phys. Lett. A, 365, 315-327 (2007)
[17] Park, J. H., Controlling chaotic systems via nonlinear feedback control, Chaos Solitons Fract., 23, 1049-1054 (2005) · Zbl 1061.93508
[18] Chen, H. K., Global chaos synchronization of new chaotic systems via nonlinear control, Chaos Solitons Fract., 23, 1245-1251 (2005) · Zbl 1102.37302
[19] Park, J. H., On synchronization of unified chaotic systems via nonlinear control, Chaos Solitons Fract., 25, 699-704 (2005) · Zbl 1125.93469
[20] Tang, R. A.; Liu, Y. L.; Xue, J. K., An extended active control for chaos synchronization, Phys. Lett. A, 373, 1449-1454 (2009) · Zbl 1228.34078
[21] Nana Nbendjo, B. R.; Tchoukuegno, R.; Woafo, P., Active control with delay of vibration and chaos in a double-well Duffing oscillator, Chaos Solitons Fract., 18, 345-353 (2003) · Zbl 1057.37081
[22] Njah, A. N., Synchronization via active control of identical and non-identical chaotic oscillators with external excitation, J. Sound Vib., 327, 322-332 (2009)
[23] Li, W., Adaptive chaos control and synchronization in only locally Lipschitz systems, Phys. Lett. A, 372, 3195-3200 (2008) · Zbl 1220.34080
[24] Jia, N.; Wang, T., Chaos control and hybrid projective synchronization for a class of new chaotic systems, Comput. Math. Appl., 62, 4783-4795 (2011) · Zbl 1236.93073
[25] Ge, Z. M.; Li, S. C.; Li, S. Y.; Chang, C. M., Pragmatical adaptive chaos control from a new double van der Pol system to a new double Duffing system, Appl. Math. Comput., 203, 513-522 (2008) · Zbl 1152.93037
[26] Salarieh, H.; Alasty, A., Adaptive chaos synchronization in Chua’s systems with noisy parameters, Math. Comput. Simulat., 79, 233-241 (2008) · Zbl 1166.34029
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