×

zbMATH — the first resource for mathematics

A note on phase synchronization in coupled chaotic fractional order systems. (English) Zbl 1238.34121
Summary: The dynamic behaviors of fractional order systems have received increasing attention in recent years. This paper addresses the reliable phase synchronization problem between two coupled chaotic fractional order systems. An active nonlinear feedback control scheme is constructed to achieve phase synchronization between two coupled chaotic fractional order systems. We investigated the necessary conditions for fractional order Lorenz, Lü and Rössler systems to exhibit chaotic attractor similar to their integer order counterpart. Then, based on the stability results of fractional order systems, sufficient conditions for phase synchronization of the fractional models of Lorenz, Lü and Rössler systems are derived. The synchronization scheme that is simple and global enables synchronization of fractional order chaotic systems to be achieved without the computation of the conditional Lyapunov exponents. Numerical simulations are performed to assess the performance of the presented analysis.

MSC:
34H10 Chaos control for problems involving ordinary differential equations
34A08 Fractional ordinary differential equations and fractional differential inclusions
34D06 Synchronization of solutions to ordinary differential equations
93B52 Feedback control
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Chen, G.; Yu, X., Chaos control: theory and applications, (2003), Springer-Verlag Berlin, Germany
[2] Yamada, T.; Fujisaka, H., Stability theory of synchronized motion in coupled-oscillator systems, Progr. theoret. phys., 70, 1240-1248, (1983) · Zbl 1171.70307
[3] Pecora, L.M.; Carrol, T.L., Synchronization in chaotic systems, Phys. rev. lett., 64, 821-824, (1990) · Zbl 0938.37019
[4] Ott, E.; Grebogi, C.; Yorke, J.A., Controlling chaos, Phys. rev. lett., 64, 1196-1199, (1990) · Zbl 0964.37501
[5] Aziz-Alaoui, M.A., Synchronization of chaos, Encyclopedia math. phys., 213-226, (2006)
[6] Carroll, T.L.; Heagy, J.F.; Pecora, L.M., Transforming signals with chaotic synchronization, Phys. rev. E, 54, 5, 4676-4680, (1996)
[7] Kocarev, L.; Parlitz, U., Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems, Phys. rev. E, 76, 11, 1816-1819, (1996)
[8] Rosenblum, M.G.; Pikovsky, A.S.; Kurths, J., From phase to lag synchronization in coupled chaotic oscillators, Phys. rev. lett., 78, 22, 4193-4196, (1996)
[9] Bai, E.; Lonngren, K., Synchronization and control of chaotic systems, Chaos solitons fractals, 10, 9, 1571-1575, (1999) · Zbl 0958.93513
[10] Liu, J.; Ye, C.; Zhang, S.; Song, W., Anti-phase synchronization in coupled map lattices, Phys. lett. A, 274, 1-2, 27-29, (2000) · Zbl 1050.37518
[11] Ho, M.; Hung, Y.; Chou, C., Phase and anti-phase synchronization of two chaotic systems by using active control, Phys. lett. A, 296, 1, 43-48, (2002) · Zbl 1098.37529
[12] Petráš, I., A note on the fractional-order chua’s system, Chaos solitons fractals, 38, 1, 140-147, (2008)
[13] Ge, Z.M.; Ou, C.Y., Chaos in a fractional order modified Duffing system, Chaos solitons fractals, 34, 2, 262-291, (2007) · Zbl 1132.37324
[14] Hartley, T.; Lorenzo, C.; Qammer, H., Chaos in a fractional order chua’s system, IEEE trans. circuits systems, 42, 485-490, (1995)
[15] Li, C.; Chen, G., Chaos and hyperchaos in fractional order Rössler equations, Physica A, 341, 55-61, (2004)
[16] Li, C.; Chen, G., Chaos in the fractional order Chen system and its control, Chaos solitons fractals, 22, 3, 549-554, (2004) · Zbl 1069.37025
[17] Lü, J.G.; Chen, G., A note on the fractional-order Chen system, Chaos solitons fractals, 27, 3, 685-688, (2006) · Zbl 1101.37307
[18] Grigorenko, I.; Grigorenko, E., Chaotic dynamics of the fractional Lorenz system, Phys. rev. lett., 91, 3, 034101, (2003)
[19] Arneodo, A.; Coullet, P.; Spiegel, E.; Tresser, C., Asymptotic chaos, Physica D, 14, 3, 327-347, (1985) · Zbl 0595.58030
[20] Deng, W.H.; Li, C.P., Chaos synchronization of the fractional Lü system, Physica A, 353, 61-72, (2005)
[21] Li, C.; Zhou, T., Synchronization in fractional-order differential systems, Physica D, 212, 1-2, 111-125, (2005) · Zbl 1094.34034
[22] Zhou, S.; Li, H.; Zhu, Z., Chaos control and synchronization in a fractional neuron network system, Chaos solitons fractals, 36, 4, 973-984, (2008) · Zbl 1139.93320
[23] Peng, G., Synchronization of fractional order chaotic systems, Phys. lett. A, 363, 5-6, 426-432, (2007) · Zbl 1197.37040
[24] Sheu, L.J.; Chen, H.K.; Chen, J.H.; Tam, L.M., Chaos in a new system with fractional order, Chaos solitons fractals, 31, 5, 1203-1212, (2007)
[25] Yan, J.; Li, C., On chaos synchronization of fractional differential equations, Chaos solitons fractals, 32, 2, 725-735, (2007) · Zbl 1132.37308
[26] Li, C.; Yan, J., The synchronization of three fractional differential systems, Chaos solitons fractals, 32, 2, 751-757, (2007)
[27] Wang, J.; Xiong, X.; Zhang, Y., Extending synchronization scheme to chaotic fractional-order Chen systems, Physica A, 370, 2, 279-285, (2006)
[28] Li, C.P.; Deng, W.H.; Xu, D., Chaos synchronization of the Chua system with a fractional order, Physica A, 360, 2, 171-185, (2006)
[29] Zhu, H.; Zhou, S.; Zhang, J., Chaos and synchronization of the fractional-order chua’s system, Chaos solitons fractals, 39, 4, 1595-1603, (2009) · Zbl 1197.94233
[30] Wu, X.; Wang, H., A new chaotic system with fractional order and its projective synchronization, Nonlinear dynam., 61, 3, 407-417, (2010) · Zbl 1204.37035
[31] Odibat, Z., Adaptive feedback control and synchronization of non-identical chaotic fractional order systems, Nonlinear dynam., 60, 4, 479-487, (2010) · Zbl 1194.93105
[32] Odibat, Z.; Corson, N.; Aziz-Alaoui, M.A.; Bertelle, C., Synchronization of chaotic fractional-order systems via linear control, Internat. J. bifur. chaos, 20, 1, 81-97, (2010) · Zbl 1183.34095
[33] Li, C., Phase and lag synchronization in coupled fractional order chaotic oscillators, Internat. J. modern phys. B, 21, 30, 5159-5166, (2007)
[34] Erjaee, G.H.; Momani, S., Phase synchronization in fractional differential chaotic systems, Phys. lett. A, 372, 14, 2350-2354, (2008) · Zbl 1220.34004
[35] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York · Zbl 0428.26004
[36] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[37] Hilfer, R., Applications of fractional calculus in physics, (2000), World Scientific New Jersey · Zbl 0998.26002
[38] R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, in: Carpinteri and Mainardi (Ed.), Fractals and Fractional Calculus, New York, 1997. · Zbl 1030.26004
[39] Caputo, M., Linear models of dissipation whose \(Q\) is almost frequency independent, part II, J. R. astron. soc., 13, 529-539, (1967)
[40] D. Matignon, Stability results of fractional differential equations with applications to control processing, in: Proceeding of IMACS, IEEE-SMC, Lille, France, 1996, pp. 963-968.
[41] Deng, W.; Li, C.; Lü, J., Stability analysis of linear fractional differential system with multiple time delays, Nonlinear dynam., 48, 409-416, (2007) · Zbl 1185.34115
[42] Ahmed, E.; El-Sayed, A.M.; El-Saka, H., Equilibrium points, stability and numerical solutions of fractional-order predator – prey and rabies models, J. math. anal. appl., 325, 1, 542-553, (2007) · Zbl 1105.65122
[43] Tavazoei, M.S.; Haeri, M., A note on the stability of fractional order systems, Math. comput. simulation, 79, 5, 1566-1576, (2009) · Zbl 1168.34036
[44] Lorenz, E., Deterministic nonperiodic flow, J. atmos. sci., 20, 130-141, (1963) · Zbl 1417.37129
[45] Tavazoei, M.S.; Haeri, M., Chaotic attractors in incommensurate fractional order systems, Physica D, 237, 2628-2637, (2008) · Zbl 1157.26310
[46] Lü, J.; Chen, G., A new chaotic attractor coined, Internat. J. bifur. chaos, 12, 3, 659-661, (2002) · Zbl 1063.34510
[47] Yongguang, Y.; Suochun, Z., Controlling uncertain Lü system using backstepping design, Chaos solitons fractals, 15, 5, 897-902, (2003) · Zbl 1033.37050
[48] Rossler, O.E., An equation for continuous chaos, Phys. lett. A, 57, 5, 397-398, (1976) · Zbl 1371.37062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.