Modified generalized projective synchronization of a new fractional-order hyperchaotic system and its application to secure communication. (English) Zbl 1239.94003

Summary: This paper presents a new fractional-order hyperchaotic system. The chaotic behaviors of this system in phase portraits are analyzed by the fractional calculus theory and computer simulations. Numerical results have revealed that hyperchaos does exist in the new fractional-order four-dimensional system with order less than 4 and the lowest order to have hyperchaos in this system is 3.664. The existence of two positive Lyapunov exponents further verifies our results. Furthermore, a novel modified generalized projective synchronization (MGPS) for the fractional-order chaotic systems is proposed based on the stability theory of the fractional-order system, where the states of the drive and response systems are asymptotically synchronized up to a desired scaling matrix. The unpredictability of the scaling factors in projective synchronization can additionally enhance the security of communication. Thus MGPS of the new fractional-order hyperchaotic system is applied to secure communication. Computer simulations are done to verify the proposed methods and the numerical results show that the obtained theoretic results are feasible and efficient.


94A05 Communication theory
34A08 Fractional ordinary differential equations
34H10 Chaos control for problems involving ordinary differential equations
93A13 Hierarchical systems
93B52 Feedback control
Full Text: DOI


[1] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press New York · Zbl 0918.34010
[2] Butzer, P. L.; Westphal, U., An Introduction to Fractional Calculus (2000), World Scientific: World Scientific Singapore · Zbl 0987.26005
[3] Hifer, R., Applications of Fractional Calculus in Physics (2001), World Scientific: World Scientific New Jersey
[4] Samko, S. G.; Klibas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordan and Breach: Gordan and Breach Amsterdam
[5] Bagley, R. L.; Calico, R. A., Fractional order state equations for the control of viscoelastically damped structures, J. Guid. Control Dyn., 14, 304-311 (1991)
[7] Sun, H. H.; Abdelwahad, A. A.; Onaral, B., Linear approximation of transfer function with a pole of fractional order, IEEE Trans. Automat. Control, 29, 441-444 (1984) · Zbl 0532.93025
[8] Laskin, N., Fractional market dynamics, Physica A, 287, 482-492 (2000)
[9] Kunsezov, D.; Bulagc, A.; Dang, G. D., Quantum Levy processes and fractional kinetics, Phys. Rev. Lett., 82, 1136-1139 (1999)
[10] Li, C. G.; Chen, G., Chaos and hyperchaos in fractional order Rössler equations, Physica A, 341, 55-61 (2004)
[11] Li, C. G.; Chen, G., Chaos in the fractional order Chen system and its control, Chaos Solitons Fractals, 22, 549-554 (2004) · Zbl 1069.37025
[12] Lu, J. G., Chaotic dynamics of the fractional-order Lü system and its synchronization, Phys. Lett. A, 354, 305-311 (2006)
[13] Wu, X.; Li, J.; Chen, G., Chaos in the fractional order unified system and its synchronization, J. Franklin Inst., 345, 392-401 (2008) · Zbl 1166.34030
[14] Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Phys. Rev. Lett., 64, 821-825 (1990) · Zbl 0938.37019
[15] Zhu, H.; Zhou, S.; Zhang, J., Chaos and synchronization of the fractional-order Chua’s system, Chaos Solitons Fractals, 39, 1595-1603 (2009) · Zbl 1197.94233
[16] Wu, X.; Lu, H.; Shen, S., Synchronization of a new fractional-order hyperchaotic system, Phys. Lett. A, 373, 2329-2337 (2009) · Zbl 1231.34091
[17] Breve, F. A.; Zhao, L.; Quiles, M. G.; Macau, E. E.N., Chaotic phase synchronization and desynchronization in an oscillator network for object selection, Neural Netw., 22, 728-737 (2009)
[18] El-Dessoky, M. M., Anti-synchronization of four scroll attractor with fully unknown parameters, Nonlinear Anal. RWA, 11, 778-783 (2010) · Zbl 1181.37040
[19] Mainieri, R.; Rehacek, J., Projective synchronization in three-dimensional chaotic systems, Phys. Rev. Lett., 82, 3042-3045 (1999)
[20] Peng, G.; Jiang, Y.; Chen, F., Projective synchronization of Chua’s chaotic systems with dead-zone in the control input, Math. Comput. Simulation, 77, 374-382 (2008) · Zbl 1139.65082
[21] Wu, X.; Lu, Y., Generalized projective synchronization of the fractional-order Chen hyperchaotic system, Nonlinear Dynam., 57, 25-53 (2009) · Zbl 1176.70029
[22] Li, Z.; Xu, D., A secure communication scheme using projective chaos synchronization, Chaos Solitons Fractals, 22, 477-481 (2004) · Zbl 1060.93530
[23] Chee, C. Y.; Xu, D., Secure digital communication using controlled projective synchronization of chaos, Chaos Solitons Fractals, 23, 1063-1070 (2005) · Zbl 1068.94010
[24] Wang, X.; He, Y., Projective synchronization of fractional order chaotic system based on linear separation, Phys. Lett. A, 372, 435-441 (2008) · Zbl 1217.37035
[25] Ghosh, D.; Banerjee, S.; Chowdhury, A. R., Generalized and projective synchronization in modulated time-delayed systems, Phys. Lett. A, 374, 2143-2149 (2010) · Zbl 1237.34100
[26] Chen, C.-H.; Sheu, L.-J.; Chen, H.-K.; Chen, J.-H.; Wang, H.-C.; Chao, Y.-C.; Lin, Y.-K., A new hyper-chaotic system and its synchronization, Nonlinear Anal. RWA, 10, 2088-2096 (2009) · Zbl 1163.65337
[27] Zhou, P.; Zhu, W., Function projective synchronization for fractional-order chaotic systems, Nonlinear Anal. RWA, 12, 811-816 (2011) · Zbl 1209.34065
[28] Caputo, M., Linear models of dissipation whose \(Q\) is almost frequency independent-II, Geophys. J. R. Astron. Soc., 13, 529-539 (1967)
[29] Keil, F.; Mackens, W.; Werther, J., Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties (1999), Springer-Verlag: Springer-Verlag Heidelberg
[30] Diethelm, K.; Ford, N. J.; Freed, A. D., A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29, 3-22 (2002) · Zbl 1009.65049
[31] Deng, W., Numerical algorithm for the time fractional Fokker-Planck equation, J. Comput. Phys., 227, 1510-1522 (2007) · Zbl 1388.35095
[32] Diethelm, K.; Ford, N. J., Analysis of fractional differential equations, J. Math. Anal. Appl., 265, 229-248 (2002) · Zbl 1014.34003
[33] Gao, T.; Chen, G.; Chen, Z.; Cang, S., The generation and circuit implementation of a new hyper-chaos based upon Lorenz system, Phys. Lett. A, 361, 78-86 (2007) · Zbl 1170.37308
[34] Člikovský, S.; Chen, G., On the generalized Lorenz canonical form, Chaos Solitons Fractals, 26, 1271-1276 (2005) · Zbl 1100.37016
[35] Chen, C. T., Linear System Theory and Design (1984), Rinehart and Winston: Rinehart and Winston Holt
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.