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Chaos control and hybrid projective synchronization of a novel chaotic system. (English) Zbl 1213.34077

Summary: Adaptive feedback controllers based on Lyapunov’s direct method for chaos control and hybrid projective synchronization (HPS) of a novel 3D chaotic system are proposed. Especially, the controller can be simplified ulteriorly into a single scalar one to achieve complete synchronization. The HPS between two nearly identical chaotic systems with unknown parameters is also studied, and adaptive parameter update laws are developed. Numerical simulations are demonstrated to verify the effectiveness of the control strategies.

MSC:

34H10 Chaos control for problems involving ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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[1] T. Ueta and G. Chen, “Bifurcation analysis of Chen’s equation,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 10, no. 8, pp. 1917-1931, 2000. · Zbl 1090.37531
[2] J. Lü and G. Chen, “A new chaotic attractor coined,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 12, no. 3, pp. 659-661, 2002. · Zbl 1063.34510
[3] J. Lü, G. Chen, D. Cheng, and S. Celikovsky, “Bridge the gap between the Lorenz system and the Chen system,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 12, no. 12, pp. 2917-2926, 2002. · Zbl 1043.37026
[4] G. Y. Qi, G. R. Chen, S. Z. Du, Z. Q. Chen, and Z. Z. Yuan, “Analysis of a new chaotic attractor,” Physica A, vol. 352, no. 2-4, pp. 295-308, 2005.
[5] G. Tigan and D. Opri\cs, “Analysis of a 3D chaotic system,” Chaos, Solitons and Fractals, vol. 36, no. 5, pp. 1315-1319, 2008. · Zbl 1148.37027
[6] W. Zhou, Y. Xu, H. Lu, and L. Pan, “On dynamics analysis of a new chaotic attractor,” Physics Letters A, vol. 372, no. 36, pp. 5773-5777, 2008. · Zbl 1223.37045
[7] W. Liu and G. Chen, “A new chaotic system and its generation,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 13, no. 1, pp. 261-267, 2003. · Zbl 1078.37504
[8] X. Xiong and J. Wang, “Conjugate Lorenz-type chaotic attractors,” Chaos, Solitons and Fractals, vol. 40, no. 2, pp. 923-929, 2009. · Zbl 1197.37047
[9] L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821-824, 1990. · Zbl 0938.37019
[10] A. E. Matouk, “Chaos synchronization between two different fractional systems of Lorenz family,” Mathematical Problems in Engineering, vol. 2009, Article ID 572724, 11 pages, 2009. · Zbl 1181.37047
[11] U. E. Vincent, A. N. Njah, O. Akinlade, and A. R. T. Solarin, “Phase synchronization in bi-directionally coupled chaotic ratchets,” Physica A, vol. 360, no. 2, pp. 186-196, 2006. · Zbl 1080.37039
[12] R. Follmann, E. E. N. Macau, and E. Rosa Jr., “Detecting phase synchronization between coupled non-phase-coherent oscillators,” Physics Letters A, vol. 373, no. 25, pp. 2146-2153, 2009. · Zbl 1229.34079
[13] S. Yanchuk, Y. Maistrenko, and E. Mosekilde, “Partial synchronization and clustering in a system of diffusively coupled chaotic oscillators,” Mathematics and Computers in Simulation, vol. 54, no. 6, pp. 491-508, 2001. · Zbl 0987.68759
[14] Z.-M. Ge and C.-M. Chang, “Generalized synchronization of chaotic systems by pure error dynamics and elaborate Lyapunov function,” Nonlinear Analysis: Theory, Methods and Applications, vol. 71, no. 11, pp. 5301-5312, 2009. · Zbl 1197.37134
[15] D. Xu and C. Y. Chee, “Controlling the ultimate state of projective synchronization in chaotic systems of arbitrary dimension,” Physical Review E, vol. 66, no. 4, Article ID 046218, 5 pages, 2002.
[16] G. L. Wen and D. Xu, “Observer-based control for full-state projective synchronization of a general class of chaotic maps in any dimension,” Physics Letters A, vol. 333, no. 5-6, pp. 420-425, 2004. · Zbl 1123.37326
[17] Z. Li and D. Xu, “A secure communication scheme using projective chaos synchronization,” Chaos, Solitons and Fractals, vol. 22, no. 2, pp. 477-481, 2004. · Zbl 1060.93530
[18] J. H. Park, “Controlling chaotic systems via nonlinear feedback control,” Chaos, Solitons and Fractals, vol. 23, no. 3, pp. 1049-1054, 2005. · Zbl 1061.93508
[19] H. K. Chen, “Global chaos synchronization of new chaotic systems via nonlinear control,” Chaos, Solitons and Fractals, vol. 23, no. 4, pp. 1245-1251, 2005. · Zbl 1102.37302
[20] M. Salah Abd-Elouahab, N.-E. Hamri, and J. Wang, “Chaos control of a fractional-order financial system,” Mathematical Problems in Engineering, vol. 2010, Article ID 270646, 18 pages, 2010. · Zbl 1195.91185
[21] R.-A. Tang, Y.-L. Liu, and J.-K. Xue, “An extended active control for chaos synchronization,” Physics Letters A, vol. 373, no. 16, pp. 1449-1454, 2009. · Zbl 1228.34078
[22] B. R. Nana Nbendjo, R. Tchoukuegno, and P. Woafo, “Active control with delay of vibration and chaos in a double-well Duffing oscillator,” Chaos, Solitons and Fractals, vol. 18, no. 2, pp. 345-353, 2003. · Zbl 1057.37081
[23] A. N. Njah, “Synchronization via active control of identical and non-identical chaotic oscillators with external excitation,” Journal of Sound and Vibration, vol. 327, no. 3-5, pp. 322-332, 2009.
[24] W. Lin, “Adaptive chaos control and synchronization in only locally Lipschitz systems,” Physics Letters A, vol. 372, no. 18, pp. 3195-3200, 2008. · Zbl 1220.34080
[25] H. Salarieh and A. Alasty, “Adaptive chaos synchronization in Chua’s systems with noisy parameters,” Mathematics and Computers in Simulation, vol. 79, no. 3, pp. 233-241, 2008. · Zbl 1166.34029
[26] Z.-M. Ge, S.-C. Li, S.-Y. Li, and C.-M. Chang, “Pragmatical adaptive chaos control from a new double van der Pol system to a new double Duffing system,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 513-522, 2008. · Zbl 1152.93037
[27] H.-K. Chen and C.-I. Lee, “Anti-control of chaos in rigid body motion,” Chaos, Solitons and Fractals, vol. 21, no. 4, pp. 957-965, 2004. · Zbl 1046.70005
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