Adaptive projective synchronization between two different fractional-order chaotic systems with fully unknown parameters. (English) Zbl 1264.34103

Summary: Projective synchronization between two different fractional-order chaotic systems with fully unknown parameters for drive and response systems is investigated. On the basis of the stability theory of fractional-order differential equations, a suitable and effective adaptive control law and a parameter update rule for unknown parameters are designed, such that projective synchronization between the fractional-order chaotic Chen system and the fractional-order chaotic Lü system with unknown parameters is achieved. Theoretical analysis and numerical simulations are presented to demonstrate the validity and feasibility of the proposed method.


34D06 Synchronization of solutions to ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
Full Text: DOI


[1] G. Chen and X. Dong, From Chaos to Order-Perspectives, Methodologies and Applications, World Scientific, Singapore, 1998. · Zbl 1217.62104
[2] T. Yang and L. O. Chua, “Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication,” IEEE Transactions on Circuits and Systems I, vol. 44, no. 10, pp. 976-988, 1997.
[3] S. K. Dana, P. K. Roy, and J. Kurths, Complex Dynamics in Physiological Systems: From Heart to Brain, Springer, NewYork, NY, USA, 2009. · Zbl 1170.92014
[4] J. García-Ojalvo and R. Roy, “Spatiotemporal communication with synchronized optical chaos,” Physical Review Letters, vol. 86, no. 22, pp. 5204-5207, 2001.
[5] J. C. Sprott, Chaos and Time-Series Analysis, Oxford University Press, Oxford, UK, 2003. · Zbl 1012.37001
[6] G. M. Mahmoud and E. E. Mahmoud, “Complete synchronization of chaotic complex nonlinear systems with uncertain parameters,” Nonlinear Dynamics, vol. 62, no. 4, pp. 875-882, 2010. · Zbl 1215.93114
[7] L. Wang, Z. Yuan, X. Chen, and Z. Zhou, “Lag synchronization of chaotic systems with parameter mismatches,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 2, pp. 987-992, 2011. · Zbl 1221.37226
[8] G. H. Erjaee and S. Momani, “Phase synchronization in fractional differential chaotic systems,” Physics Letters A, vol. 372, no. 14, pp. 2350-2354, 2008. · Zbl 1220.34004
[9] X. R. Shi and Z. L. Wang, “Adaptive added-order anti-synchronization of chaotic systems with fully unknown parameters,” Applied Mathematics and Computation, vol. 215, no. 5, pp. 1711-1717, 2009. · Zbl 1179.65158
[10] J. W. Wang and A. M. Chen, “Partial synchronization in coupled chemical chaotic oscillators,” Journal of Computational and Applied Mathematics, vol. 233, no. 8, pp. 1897-1904, 2010. · Zbl 1194.34095
[11] S. Y. Li and Z. M. Ge, “Generalized synchronization of chaotic systems with different orders by fuzzy logic constant controller,” Expert Systems with Applications, vol. 38, no. 3, pp. 2302-2310, 2011.
[12] X. Liu, “Impulsive synchronization of chaotic systems subject to time delay,” Nonlinear Analysis, Theory, Methods and Applications, vol. 71, no. 12, pp. e1320-e1327, 2009. · Zbl 1238.34120
[13] J. M. González-Miranda, “Synchronization of symmetric chaotic systems,” Physical Review E, vol. 53, no. 6, pp. 5656-5669, 1996.
[14] R. Mainieri and J. Rehacek, “Projective synchronization in three-dimensional chaotic systems,” Physical Review Letters, vol. 82, no. 15, pp. 3042-3045, 1999.
[15] D. Ghosh, “Generalized projective synchronization in time-delayed systems: nonlinear observer approach,” Chaos, vol. 19, no. 1, Article ID 013102, 2009. · Zbl 1311.34111
[16] G. H. Li, “Modified projective synchronization of chaotic system,” Chaos, Solitons and Fractals, vol. 32, no. 5, pp. 1786-1790, 2007. · Zbl 1134.37331
[17] Y. Chen and X. Li, “Function projective synchronization between two identical chaotic systems,” International Journal of Modern Physics C, vol. 18, no. 5, pp. 883-888, 2007. · Zbl 1139.37301
[18] H. Du, Q. Zeng, and C. Wang, “Modified function projective synchronization of chaotic system,” Chaos, Solitons and Fractals, vol. 42, no. 4, pp. 2399-2404, 2009. · Zbl 1198.93011
[19] C. Y. Chee and D. Xu, “Chaos-based M-ary digital communication technique using controlled projective synchronisation,” IEE Proceedings: Circuits, Devices and Systems, vol. 153, no. 4, pp. 357-360, 2006.
[20] Y. Yu and H. X. Li, “Adaptive hybrid projective synchronization of uncertain chaotic systems based on backstepping design,” Nonlinear Analysis, vol. 12, no. 1, pp. 388-393, 2011. · Zbl 1214.34042
[21] R. Z. Luo, S. C. Deng, and Z. M. Wei, “Modified projective synchronization between two different hyperchaotic systems with unknown or/and uncertain parameters,” Physica Scripta, vol. 81, no. 1, Article ID 015006, 2010. · Zbl 1198.37055
[22] H. Du, Q. Zeng, and N. Lü, “A general method for modified function projective lag synchronization in chaotic systems,” Physics Letters A, vol. 374, no. 13-14, pp. 1493-1496, 2010. · Zbl 1236.34068
[23] K. S. Sudheer and M. Sabir, “Function projective synchronization in chaotic and hyperchaotic systems through open-plus-closed-loop coupling,” Chaos, vol. 20, no. 1, Article ID 013115, pp. 1-5, 2010. · Zbl 1311.34100
[24] R. C. Koeller, “Polynomial operators, Stieltjes convolution, and fractional calculus in hereditary mechanics,” Acta Mechanica, vol. 58, no. 3-4, pp. 251-264, 1986. · Zbl 0578.73040
[25] R. C. Koeller, “Applications of fractional calculus to the theory of viscoelasticity,” Journal of Applied Mechanics, vol. 51, no. 2, pp. 299-307, 1984. · Zbl 0544.73052
[26] O. Heaviside, Electromagnetic Theory, Chelsea, New York, NY, USA, 1971. · JFM 30.0801.03
[27] T. T. Hartley, C. F. Lorenzo, and H. K. Qammer, “Chaos in a fractional order Chua’s system,” IEEE Transactions on Circuits and Systems I, vol. 42, no. 8, pp. 485-490, 1995.
[28] C. Li and G. Chen, “Chaos and hyperchaos in the fractional-order Rössler equations,” Physica A, vol. 341, no. 1-4, pp. 55-61, 2004.
[29] C. Li and G. Peng, “Chaos in Chen’s system with a fractional order,” Chaos, Solitons and Fractals, vol. 22, no. 2, pp. 443-450, 2004. · Zbl 1060.37026
[30] W. H. Deng and C. P. Li, “Chaos synchronization of the fractional Lü system,” Physica A, vol. 353, no. 1-4, pp. 61-72, 2005.
[31] Z. M. Ge and C. Y. Ou, “Chaos in a fractional order modified Duffing system,” Chaos, Solitons and Fractals, vol. 34, no. 2, pp. 262-291, 2007. · Zbl 1132.37324
[32] X. Wang and Y. He, “Chaotic synchronization of fractional-order modified coupled dynamos system,” International Journal of Modern Physics B, vol. 23, no. 31, pp. 5769-5777, 2009. · Zbl 1186.78028
[33] M. M. Asheghan, M. T. H. Beheshti, and M. S. Tavazoei, “Robust synchronization of perturbed Chen’s fractional order chaotic systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 2, pp. 1044-1051, 2011. · Zbl 1221.34007
[34] D. Cafagna and G. Grassi, “Observer-based synchronization for a class of fractional chaotic systems via a scalar signal: results involving the exact solution of the error dynamics,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 21, no. 3, pp. 955-962, 2011. · Zbl 1215.34072
[35] S. Wang, Y. Yu, and M. Diao, “Hybrid projective synchronization of chaotic fractional order systems with different dimensions,” Physica A, vol. 389, no. 21, pp. 4981-4988, 2010.
[36] X. Wu and H. Wang, “A new chaotic system with fractional order and its projective synchronization,” Nonlinear Dynamics, vol. 61, no. 3, pp. 407-417, 2010. · Zbl 1204.37035
[37] G. Peng, Y. Jiang, and F. Chen, “Generalized projective synchronization of fractional order chaotic systems,” Physica A, vol. 387, no. 14, pp. 3738-3746, 2008. · Zbl 1220.34060
[38] D. Cafagna and G. Grassi, “Observer-based projective synchronization of fractional systems via a scalar signal: application to hyperchaotic Rössler systems,” Nonlinear Dynamics. In press. · Zbl 1243.93047
[39] L. D. Zhao, J. B. Hu, and X. H. Liu, “Adaptive tracking control and synchronization of fractional hyper-chaotic Lorenz system with unknown parameters,” Acta Physica Sinica, vol. 59, no. 4, pp. 2305-2309, 2010.
[40] D. Cafagna, “Past and present-Fractional calculus: a mathematical tool from the past for present engineers,” IEEE Industrial Electronics Magazine, vol. 1, no. 2, pp. 35-40, 2007.
[41] R. Gorenflo and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order: Fractal and Fractional Calculus in Continuum Mechanics, Springer, Wien, Germany, 1997.
[42] K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,” Nonlinear Dynamics, vol. 29, no. 1-4, pp. 3-22, 2002. · Zbl 1009.65049
[43] K. Diethelm, N. J. Ford, and A. D. Freed, “Detailed error analysis for a fractional Adams method,” Numerical Algorithms, vol. 36, no. 1, pp. 31-52, 2004. · Zbl 1055.65098
[44] K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229-248, 2002. · Zbl 1014.34003
[45] C. Li and G. Chen, “Chaos in the fractional order Chen system and its control,” Chaos, Solitons and Fractals, vol. 22, no. 3, pp. 549-554, 2004. · Zbl 1069.37025
[46] J. Wang, X. Xiong, and Y. Zhang, “Extending synchronization scheme to chaotic fractional-order Chen systems,” Physica A, vol. 370, no. 2, pp. 279-285, 2006.
[47] H. Zhu, S. Zhou, and Z. He, “Chaos synchronization of the fractional-order Chen’s system,” Chaos, Solitons and Fractals, vol. 41, no. 5, pp. 2733-2740, 2009. · Zbl 1198.93206
[48] J. G. Lu, “Chaotic dynamics of the fractional-order Lü system and its synchronization,” Physics Letters, Section A, vol. 354, no. 4, pp. 305-311, 2006.
[49] D. Matignon, “Stability results of fractional differential equations with applications to control processing,” in Proceedings of the IEEE-SMC International Association for Mathematics and Computers in Simulation (IMACS ’96), pp. 963-968, Lille, France, 1996.
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