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Chaos synchronization of the fractional-order Chen’s system. (English) Zbl 1198.93206

Summary: Based on the stability theorem of linear fractional systems, a necessary condition is given to check the chaos synchronization of fractional systems with incommensurate order. Chaos synchronization is studied by utilizing the Pecora-Carroll (PC) method and the coupling method. The necessary condition can also be used as a tool to confirm results of a numerical simulation. Numerical simulation results show the effectiveness of the necessary condition.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

MSC:

93D99 Stability of control systems
34H10 Chaos control for problems involving ordinary differential equations
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[1] Gao, X.; Yu, J., Chaos in the fractional order periodically forced complex Duffing’s oscillators, Chaos, Solitons & Fractals, 26, 1125-1133 (2005)
[2] Hartley, T. T.; Lorenzo, C. F.; Qammer, H. K., Chaos on a fractional Chua’s system, IEEE Trans Circ Syst Theory Appl, 42, 485-490 (1995)
[3] Ivo, Petras, A note on the fractional-order Chua’s system, Chaos, Solitons & Fractals, 38, 140-147 (2008)
[4] Li, C. P.; Peng, G. J., Chaos in Chen’s system with a fractional order, Chaos, Solitons & Fractals, 20, 443-450 (2004) · Zbl 1060.37026
[5] Li, C.; Chen, G., Chaos in the fractional order Chen system and its control, Chaos, Solitons & Fractals, 22, 549-554 (2004) · Zbl 1069.37025
[6] deng, W. H.; Li, C. P., Chaos synchronization of the fractional Lü system, Physica A, 353, 61-72 (2005)
[7] Petras I. A note on the fractional-order cellular neural networks. In: Proceedings of the IEEE world congress on computational intelligence, International joint conference on neural networks, Vancouver, Canada, 2006, p.16-21.; Petras I. A note on the fractional-order cellular neural networks. In: Proceedings of the IEEE world congress on computational intelligence, International joint conference on neural networks, Vancouver, Canada, 2006, p.16-21.
[8] Shangbo, Zhou; Hua, Li.; Zhengzhou, Zhu, Chaos control and synchronization in a fractional neuron network system, Chaos, Solitons & Fractals, 36, 973-984 (2008) · Zbl 1139.93320
[9] Tavazoei, M. S., Limitations of frequency domain approximation for detecting chaos in fractional order systems, Nonlinear Anal, 69, 1299-1320 (2008) · Zbl 1148.65094
[10] Matignon D. Stability results for fractional differential equations with applications to control processing, In: Computational engineering in systems and application multi-conference, vol. 2, IMACS, IEEE-SMC Proceedings, Lille, France, July 1996, p. 963-8.; Matignon D. Stability results for fractional differential equations with applications to control processing, In: Computational engineering in systems and application multi-conference, vol. 2, IMACS, IEEE-SMC Proceedings, Lille, France, July 1996, p. 963-8.
[11] Tavazoei, M. S., A necessary condition for double-scroll attractor existence in fractional-order system, Phys Lett A, 367, 102-113 (2007) · Zbl 1209.37037
[12] Tavazoei, M. S., Chaotic attractors in incommensurate fractional order system, Physics D, 237, 2628-2637 (2008) · Zbl 1157.26310
[13] Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Phys Rev Lett, 64, 821-824 (1990) · Zbl 0938.37019
[14] Sira-Ramirez, H.; Cruz-Hernandez, C., Synchronization of chaotic systems: a generalized Hamiltonian systems approach, Int J Bifurcation Chaos, 11, 1381-1395 (2001) · Zbl 1206.37053
[15] Petras, I., Control of fractional-order Chua’s system, J Electron Eng, 53, 219-222 (2002)
[16] Lu, J. G., Chaotic dynamics and synchronization of fractional-order Chua’s circuits with a piecewise-linear nonlinearity, Int J Modern Phys B, 19, 3249-3259 (2005) · Zbl 1124.37309
[17] Li, C. P.; Deng, W. H.; Xu, D., Chaos synchronization of the Chua system with a fractional order, Physica A, 360, 171-185 (2006)
[18] Wang, Junwei; Xiong, Xiaohua; Zhang, Yanbin, Extending synchronization scheme to chaotic fractional-order Chen systems, Physica A, 370, 279-285 (2006)
[19] Li, Changpin; Yan, Jianping, The synchronization of three fractional differential systems, Chaos, Solitons & Fractals, 32, 751-757 (2007) · Zbl 1132.37308
[20] Butzer, P. L.; Westphal, U., An introduction to fractional calculus (2000), World Scientific: World Scientific Singapore · Zbl 0987.26005
[21] Kenneth, S. M.; Bertram, R., An introduction to the fractional calculus and fractional differential equations (1993), Wiley-Interscience: Wiley-Interscience USA · Zbl 0789.26002
[22] Hao, Zhu; Shangbo, Zhou, Chaos and synchronization of the fractional-order Chua’s system, Chaos, Solitons & Fractals, 39, 1595-1603 (2009) · Zbl 1197.94233
[23] Deng, W.; Li, C., Stability analysis of linear fractional differential system with multiple time delays, Nonlinear Dyn, 48, 409-416 (2007) · Zbl 1185.34115
[24] Chen, G.; Ueta, T., Yet another attractor, Internat J Bifurc Chaos, 9, 1465-1466 (1999) · Zbl 0962.37013
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