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Jia, Nuo; Wang, Tao

Generation and modified projective synchronization for a class of new hyperchaotic systems. (English) Zbl 1417.37128

Abstr. Appl. Anal. 2013, Article ID 804964, 11 p. (2013).
Summary: A class of new hyperchaotic systems with different nonlinear terms is proposed, and the existence of hyperchaos is exhibited by calculating their Lyapunov exponent spectrums. Then the universal theories on modified projective synchronization (MPS) of the systems with general form which linearly depends on unknown parameters or time-varying parameters, are investigated by presenting an adaptive control strategy together with parameter update laws and a nonlinear control scheme based on Lyapunov stability theory. Subsequently, the presented control methods are applied to achieve MPS of the new hyperchaotic systems, and their effectiveness is illustrated by numerical simulations.

Cited in 2 Documents

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34D06 Synchronization of solutions to ordinary differential equations
70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics

Keywords:

hyperchaotic systems; Lyapunov exponent; modified projective synchronization
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References:

[1] Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Physical Review Letters, 64, 8, 821-824 (1990) · Zbl 0938.37019 · doi:10.1103/PhysRevLett.64.821
[2] Boccaletti, S.; Kurths, J.; Osipov, G.; Valladares, D. L.; Zhou, C. S., The synchronization of chaotic systems, Physics Reports, 366, 1-2, 1-101 (2002) · Zbl 0995.37022 · doi:10.1016/S0370-1573(02)00137-0
[3] Pikovsky, A. S.; Rosenblum, M. G.; Osipov, G. V.; Kurths, J., Phase synchronization of chaotic oscillators by external driving, Physica D, 104, 3-4, 219-238 (1997) · Zbl 0898.70015 · doi:10.1016/S0167-2789(96)00301-6
[4] Rosenblum, M. G.; Pikovsky, A. S.; Kurths, J., From phase to lag synchronization in coupled chaotic oscillators, Physical Review Letters, 78, 22, 4193-4196 (1997)
[5] Rulkov, N. F.; Sushchik, M. M.; Tsimring, L. S.; Abarbanel, H. D. I., Generalized synchronization of chaos in directionally coupled chaotic systems, Physical Review E, 51, 2, 980-994 (1995) · doi:10.1103/PhysRevE.51.980
[6] Mainieri, R.; Rehacek, J., Projective synchronization in three-dimensional chaotic systems, Physical Review Letters, 82, 15, 3042-3045 (1999)
[7] Li, G.-H., Generalized projective synchronization of two chaotic systems by using active control, Chaos, Solitons and Fractals, 30, 1, 77-82 (2006) · Zbl 1144.37372 · doi:10.1016/j.chaos.2005.08.130
[8] Li, G.-H., Modified projective synchronization of chaotic system, Chaos, Solitons and Fractals, 32, 5, 1786-1790 (2007) · Zbl 1134.37331 · doi:10.1016/j.chaos.2005.12.009
[9] Rössler, O. E., An equation for hyperchaos, Physics Letters A, 71, 2-3, 155-157 (1979) · Zbl 0996.37502 · doi:10.1016/0375-9601(79)90150-6
[10] Wang, X.; Wang, M., A hyperchaos generated from Lorenz system, Physica A, 387, 14, 3751-3758 (2008) · doi:10.1016/j.physa.2008.02.020
[11] Gao, T.; Chen, G.; Chen, Z.; Cang, S., The generation and circuit implementation of a new hyper-chaos based upon Lorenz system, Physics Letters A, 361, 1-2, 78-86 (2007) · Zbl 1170.37308 · doi:10.1016/j.physleta.2006.09.042
[12] Li, Y.; Tang, W. K. S.; Chen, G., Generating hyperchaos via state feedback control, International Journal of Bifurcation and Chaos, 15, 10, 3367-3375 (2005) · doi:10.1142/S0218127405013988
[13] Chen, A.; Lu, J.; Lü, J.; Yu, S., Generating hyperchaotic Lü attractor via state feedback control, Physica A, 364, 103-110 (2006) · doi:10.1016/j.physa.2005.09.039
[14] Yassen, M. T., Synchronization hyperchaos of hyperchaotic systems, Chaos, Solitons and Fractals, 37, 2, 465-475 (2008) · doi:10.1016/j.chaos.2006.09.045
[15] 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 Analysis: Real World Applications, 10, 4, 2088-2096 (2009) · Zbl 1163.65337 · doi:10.1016/j.nonrwa.2008.03.015
[16] Huang, J., Chaos synchronization between two novel different hyperchaotic systems with unknown parameters, Nonlinear Analysis: Theory, Methods and Applications, 69, 11, 4174-4181 (2008) · Zbl 1161.34338 · doi:10.1016/j.na.2007.10.045
[17] Al-Sawalha, M. M.; Noorani, M. S. M., Adaptive anti-synchronization of two identical and different hyperchaotic systems with uncertain parameters, Communications in Nonlinear Science and Numerical Simulation, 15, 4, 1036-1047 (2010) · Zbl 1221.93123 · doi:10.1016/j.cnsns.2009.05.037
[18] Zhu, C., Adaptive synchronization of two novel different hyperchaotic systems with partly uncertain parameters, Applied Mathematics and Computation, 215, 2, 557-561 (2009) · Zbl 1182.37028 · doi:10.1016/j.amc.2009.05.026
[19] Mahmoud, E. E., Dynamics and synchronization of new hyperchaotic complex Lorenz system, Mathematical and Computer Modelling, 55, 7-8, 1951-1962 (2012) · Zbl 1255.93102 · doi:10.1016/j.mcm.2011.11.053
[20] Fu, G., Robust adaptive modified function projective synchronization of different hyperchaotic systems subject to external disturbance, Communications in Nonlinear Science and Numerical Simulation, 17, 6, 2602-2608 (2012) · Zbl 1248.93089 · doi:10.1016/j.cnsns.2011.09.033
[21] Chen, Y.; Wu, X.; Gui, Z., Global synchronization criteria for a class of third-order non-autonomous chaotic systems via linear state error feedback control, Applied Mathematical Modelling, 34, 12, 4161-4170 (2010) · Zbl 1201.93045 · doi:10.1016/j.apm.2010.04.013
[22] Du, H.; Zeng, Q.; Wang, C.; Ling, M., Function projective synchronization in coupled chaotic systems, Nonlinear Analysis: Real World Applications, 11, 2, 705-712 (2010) · Zbl 1181.37039 · doi:10.1016/j.nonrwa.2009.01.016
[23] Chen, H.-H.; Sheu, G.-J.; Lin, Y.-L.; Chen, C.-S., Chaos synchronization between two different chaotic systems via nonlinear feedback control, Nonlinear Analysis: Theory, Methods and Applications, 70, 12, 4393-4401 (2009) · Zbl 1171.34324 · doi:10.1016/j.na.2008.10.069
[24] Park, J. H., Further results on functional projective synchronization of Genesiotesi chaotic system, Modern Physics Letters B, 23, 15, 1889-1895 (2009) · Zbl 1168.37311 · doi:10.1142/S0217984909020059
[25] Lee, T. H.; Park, J. H., Adaptive functional projective lag synchronization of a hyperchaotic Rössler system, Chinese Physics Letters, 26, 9 (2009) · doi:10.1088/0256-307X/26/9/090507
[26] Zheng, J., A simple universal adaptive feedback controller for chaos and hyperchaos control, Computers and Mathematics with Applications, 61, 8, 2000-2004 (2011) · Zbl 1219.93111 · doi:10.1016/j.camwa.2010.08.050
[27] Park, J. H., Adaptive controller design for modified projective synchronization of Genesio-Tesi chaotic system with uncertain parameters, Chaos, Solitons and Fractals, 34, 4, 1154-1159 (2007) · Zbl 1142.93428 · doi:10.1016/j.chaos.2006.04.053
[28] Park, J. H., Adaptive control for modified projective synchronization of a four-dimensional chaotic system with uncertain parameters, Journal of Computational and Applied Mathematics, 213, 1, 288-293 (2008) · Zbl 1137.93035 · doi:10.1016/j.cam.2006.12.003
[29] Jia, N.; Wang, T., Chaos control and hybrid projective synchronization for a class of new chaotic systems, Computers and Mathematics with Applications, 62, 12, 4783-4795 (2011) · Zbl 1236.93073 · doi:10.1016/j.camwa.2011.10.069
[30] Mu, X.; Pei, L., Synchronization of the near-identical chaotic systems with the unknown parameters, Applied Mathematical Modelling, 34, 7, 1788-1797 (2010) · Zbl 1193.37046 · doi:10.1016/j.apm.2009.09.023
[31] Zheng, S., Adaptive modified function projective synchronization of unknown chaotic systems with different order, Applied Mathematics and Computation, 218, 10, 5891-5899 (2012) · Zbl 1246.65243 · doi:10.1016/j.amc.2011.11.034
[32] Wei, Z., Delayed feedback on the 3-D chaotic system only with two stable node-foci, Computers and Mathematics with Applications, 63, 3, 728-738 (2012) · Zbl 1238.34122 · doi:10.1016/j.camwa.2011.11.037
[33] Botmart, T.; Niamsup, P.; Liu, X., Synchronization of non-autonomous chaotic systems with time-varying delay via delayed feedback control, Communications in Nonlinear Science and Numerical Simulation, 17, 4, 1894-1907 (2012) · Zbl 1239.93043 · doi:10.1016/j.cnsns.2011.07.038
[34] Yu, Y.; Li, H.-X., Adaptive hybrid projective synchronization of uncertain chaotic systems based on backstepping design, Nonlinear Analysis: Real World Applications, 12, 1, 388-393 (2011) · Zbl 1214.34042 · doi:10.1016/j.nonrwa.2010.06.024
[35] Chen, F.; Chen, L.; Zhang, W., Stabilization of parameters perturbation chaotic system via adaptive backstepping technique, Applied Mathematics and Computation, 200, 1, 101-109 (2008) · Zbl 1153.65376 · doi:10.1016/j.amc.2007.10.051
[36] Aghababa, M. P.; Heydari, A., Chaos synchronization between two different chaotic systems with uncertainties, external disturbances, unknown parameters and input nonlinearities, Applied Mathematical Modelling, 36, 4, 1639-1652 (2012) · Zbl 1243.93071 · doi:10.1016/j.apm.2011.09.023
[37] Hosseinnia, S. H.; Ghaderi, R.; Ranjbar N., A.; Mahmoudian, M.; Momani, S., Sliding mode synchronization of an uncertain fractional order chaotic system, Computers and Mathematics with Applications, 59, 5, 1637-1643 (2010) · Zbl 1189.34011 · doi:10.1016/j.camwa.2009.08.021
[38] Hu, M.; Yang, Y.; Xu, Z., Impulsive control of projective synchronization in chaotic systems, Physics Letters A, 372, 18, 3228-3233 (2008) · Zbl 1220.34079 · doi:10.1016/j.physleta.2008.01.054
[39] Hamiche, H.; Kemih, K.; Ghanes, M.; Zhang, G.; Djennoune, S., Passive and impulsive synchronization of a new four-dimensional chaotic system, Nonlinear Analysis: Theory, Methods and Applications, 74, 4, 1146-1154 (2011) · Zbl 1216.34048 · doi:10.1016/j.na.2010.09.051
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