Sliding mode synchronization of an uncertain fractional order chaotic system. (English) Zbl 1189.34011

Summary: Synchronization of chaotic and uncertain Duffing-Holmes system has been done using the sliding mode control strategy. Regarding the synchronization task as a control problem, fractional order mathematics is used to express the system and sliding mode for synchronization. It has been shown that, not only the performance of the proposed method is satisfying with an acceptable level of control signal, but also a rather simple stability analysis is performed. The latter is usually a complicated task for uncertain nonlinear chaotic systems.


34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
93B52 Feedback control
37N35 Dynamical systems in control
34D06 Synchronization of solutions to ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
Full Text: DOI


[1] Yang, T., A survey of chaotic secure communication systems, International journal of computational cognition, 2, 2, 81-130, (2004)
[2] Hartley, T.T.; Lorenzo, C.F.; Qammer, H.K., Chaos in a fractional order chua’s system, IEEE transactions CAS-I, 42, 485-490, (1995)
[3] P. Arena, R. Caponetto, L. Fortuna, D. Porto, Chaos in a fractional order Duffing system, in: Proceedings ECCTD, Budapest, 1997, pp. 1259-1262
[4] Ahmad, W.M.; Sprott, J.C., Chaos in fractional-order autonomous nonlinear systems, Chaos, solitons fractals, 16, 339-351, (2003) · Zbl 1033.37019
[5] Lu, J.G.; Chen, G., A note on the fractional-order Chen system, Chaos, solitons fractals, 27, 3, 685-688, (2006) · Zbl 1101.37307
[6] Lu, J.G., Chaotic dynamics of the fractional-order Lü system and its synchronization, Physics letter A, 354, 4, 305-311, (2006)
[7] Li, C.; Chen, G., Chaos and hyperchaos in the fractional-order Rössler equations, Physica A: statistical mechanics and its applications, 341, 55-61, (2004)
[8] Lu, J.G., Chaotic dynamics and synchronization of fractional-order arneodo’s systems, Chaos, solitons fractals, 26, 4, 1125-1133, (2005) · Zbl 1074.65146
[9] Sheu, L.J.; Chen, H.K.; Chen, J.H.; Tam, L.M.; Chen, W.C.; Lin, K.T.; Kang, Y., Chaos in the Newton-leipnik system with fractional order, Chaos, solitons fractals, 36, 98-103, (2005)
[10] Ott, E.; Grebogi, C.; Yorke, J.A., Controlling chaos, Physic review letter, 64, 1196-1199, (1990) · Zbl 0964.37501
[11] Deng, W.; Li, C., Chaos synchronization of the fractional Lu system, Physica A, 353, 61-72, (2005)
[12] Utkin, V.I., Variable structure systems with sliding mode, IEEE transaction in automatic control, 22, 212-222, (1977) · Zbl 0382.93036
[13] J. Phuah, J. Lu, T. Yahagi, Chattering free sliding mode control in magnetic levitation system, IEEJ Transaction EIS, 125(4) 600-606
[14] Ertugrul, M.; Kaynak, O., Neuro-sliding mode control of robotic manipulators, Mechatronics, 10, 239-263, (2000)
[15] Slotine, J.E.; Sastry, S.S., Tracking control of nonlinear systems using sliding surface with application to robotic manipulators, International journal of control, 38, 465-492, (1983) · Zbl 0519.93036
[16] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[17] Ahmed, E.; El-Sayed, A.M.A.; El-Saka, H.A.A., Equilibrium points, stability and numerical solutions of fractional order predator – prey and rabies models, Journal of mathematical analysis and applications, 325, 1, 542-553, (2007) · Zbl 1105.65122
[18] Matignon, D., Stability results for fractional differential equations with applications to control processing, (), 963-968
[19] Slotine, J.J.E., Applied nonlinear control, (1991), Prentice Hall Englewood Cliffs, New Jersy, 07632
[20] Chang, W.-D.; Yan, J.-J., Adaptive robust PID controller design based on a sliding mode for uncertain chaotic systems, Chaos, solitons & fractals, 26, 1, 167-175, (2005) · Zbl 1122.93340
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.