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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.

MSC:

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