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A new fractional-order chaotic complex system and its antisynchronization. (English) Zbl 1472.37045

Summary: We propose a new fractional-order chaotic complex system and study its dynamical properties including symmetry, equilibria and their stability, and chaotic attractors. Chaotic behavior is verified with phase portraits, bifurcation diagrams, the histories, and the largest Lyapunov exponents. And we find that chaos exists in this system with orders less than 5 by numerical simulation. Additionally, antisynchronization of different fractional-order chaotic complex systems is considered based on the stability theory of fractional-order systems. This new system and the fractional-order complex Lorenz system can achieve antisynchronization. Corresponding numerical simulations show the effectiveness and feasibility of the scheme.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
26A33 Fractional derivatives and integrals
34A08 Fractional ordinary differential equations
34D06 Synchronization of solutions to ordinary differential equations
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[1] Hartley, T. T.; Lorenzo, C. F.; Qammer, H. K., Chaos in a fractional order Chua’s system, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 42, 8, 485-490 (1995) · doi:10.1109/81.404062
[2] Li, C.; Chen, G., Chaos and hyperchaos in the fractional-order Rössler equations, Physica A: Statistical Mechanics and its Applications, 341, 1-4, 55-61 (2004) · doi:10.1016/j.physa.2004.04.113
[3] Grigorenko, I.; Grigorenko, E., Chaotic dynamics of the fractional Lorenz system, Physical Review Letters, 91, 3 (2003)
[4] Li, C.; Chen, G., Chaos in the fractional order Chen system and its control, Chaos, Solitons and Fractals, 22, 3, 549-554 (2004) · Zbl 1069.37025 · doi:10.1016/j.chaos.2004.02.035
[5] Lu, J. G., Chaotic dynamics of the fractional-order Lü system and its synchronization, Physics Letters A, 354, 4, 305-311 (2006) · doi:10.1016/j.physleta.2006.01.068
[6] Pan, L.; Zhou, W.; Fang, J.; Li, D., Synchronization and anti-synchronization of new uncertain fractional-order modified unified chaotic systems via novel active pinning control, Communications in Nonlinear Science and Numerical Simulation, 15, 12, 3754-3762 (2010) · Zbl 1222.34063 · doi:10.1016/j.cnsns.2010.01.025
[7] Si, G.; Sun, Z.; Zhang, Y.; Chen, W., Projective synchronization of different fractional-order chaotic systems with non-identical orders, Nonlinear Analysis: Real World Applications, 13, 4, 1761-1771 (2012) · Zbl 1257.34040 · doi:10.1016/j.nonrwa.2011.12.006
[8] Chen, L.; Chai, Y.; Wu, R., Lag projective synchronization in fractional-order chaotic (hyperchaotic) systems, Physics Letters A: General, Atomic and Solid State Physics, 375, 21, 2099-2110 (2011) · Zbl 1242.34094 · doi:10.1016/j.physleta.2011.04.015
[9] Roldán, E.; de Valcárcel, G. J.; Vilaseca, R.; Mandel, P., Single-mode-laser phase dynamics, Physical Review A, 48, 1, 591-598 (1993) · doi:10.1103/PhysRevA.48.591
[10] Ning, C.-Z.; Haken, H., Detuned lasers and the complex Lorenz equations: subcritical and supercritical Hopf bifurcations, Physical Review A, 41, 7, 3826-3837 (1990) · doi:10.1103/PhysRevA.41.3826
[11] Toronov, V. Y.; Derbov, V. L., Boundedness of attractors in the complex Lorenz model, Physical Review E, 55, 3, 3689-3692 (1997) · doi:10.1103/PhysRevE.55.3689
[12] Mahmoud, G. M.; Bountis, T., The dynamics of systems of complex nonlinear oscillators: a review, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 14, 11, 3821-3846 (2004) · Zbl 1091.34524 · doi:10.1142/S0218127404011624
[13] Mahmoud, E. E., Complex complete synchronization of two nonidentical hyperchaotic complex nonlinear systems, Mathematical Methods in the Applied Sciences, 37, 3, 321-328 (2014) · Zbl 1288.34045 · doi:10.1002/mma.2793
[14] Wu, Z.; Duan, J.; Fu, X., Complex projective synchronization in coupled chaotic complex dynamical systems, Nonlinear Dynamics, 69, 3, 771-779 (2012) · Zbl 1253.93060 · doi:10.1007/s11071-011-0303-0
[15] Zhang, F.; Liu, S., Full state hybrid projective synchronization and parameters identification for uncertain chaotic (hyperchaotic) complex systems, Journal of Computational and Nonlinear Dynamics, 9, 2 (2014) · doi:10.1115/1.4025475
[16] Mahmoud, G. M.; Mahmoud, E. E., Complex modified projective synchronization of two chaotic complex nonlinear systems, Nonlinear Dynamics, 73, 4, 2231-2240 (2013) · Zbl 1281.34074 · doi:10.1007/s11071-013-0937-1
[17] Liu, S.; Zhang, F., Complex function projective synchronization of complex chaotic system and its applications in secure communication, Nonlinear Dynamics, 76, 2, 1087-1097 (2014) · Zbl 1306.94072 · doi:10.1007/s11071-013-1192-1
[18] Luo, C.; Wang, X., Chaos in the fractional-order complex Lorenz system and its synchronization, Nonlinear Dynamics, 71, 1-2, 241-257 (2013) · Zbl 1268.34022 · doi:10.1007/s11071-012-0656-z
[19] Luo, C.; Wang, X., Chaos generated from the fractional-order complex Chen system and its application to digital secure communication, International Journal of Modern Physics C, 24 (2013) · doi:10.1142/S0129183113500253
[20] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0428.26004
[21] Podlubny, I., Fractional Differential Equations (1999), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0918.34010
[22] Mahmoud, G. M.; Bountis, T.; Mahmoud, E. E., Active control and global synchronization of the complex Chen and Lü systems, International Journal of Bifurcation and Chaos, 17, 12, 4295-4308 (2007) · Zbl 1146.93372 · doi:10.1142/S0218127407019962
[23] Diethelm, K.; Ford, N. J.; Freed, A. D., A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynamics, 29, 1-4, 3-22 (2002) · Zbl 1009.65049 · doi:10.1023/A:1016592219341
[24] Wang, X.; He, Y.; Wang, M., Chaos control of a fractional order modified coupled dynamos system, Nonlinear Analysis: Theory, Methods &; Applications, 71, 12, 6126-6134 (2009) · Zbl 1187.34080 · doi:10.1016/j.na.2009.06.065
[25] Wolf, A.; Swift, J. B.; Swinney, H. L.; Vastano, J. A., Determining Lyapunov exponents from a time series, Physica D: Nonlinear Phenomena, 16, 3, 285-317 (1985) · Zbl 0585.58037 · doi:10.1016/0167-2789(85)90011-9
[26] Pardalos, P. M.; Yatsenko, V. A., Optimization approach to the estimation and control of Lyapunov exponents, Journal of Optimization Theory and Its Applications, 128, 1, 29-48 (2006) · Zbl 1121.93061 · doi:10.1007/s10957-005-7554-1
[27] Nair, S. P.; Shiau, D.-S.; Principe, J. C.; Iasemidis, L. D.; Pardalos, P. M.; Norman, W. M.; Carney, P. R.; Kelly, K. M.; Sackellares, J. C., An investigation of EEG dynamics in an animal model of temporal lobe epilepsy using the maximum Lyapunov exponent, Experimental Neurology, 216, 1, 115-121 (2009) · doi:10.1016/j.expneurol.2008.11.009
[28] Matignon, D., Stability results for fractional differential equations with applications to control processing, Proceedings of the International IMACS IEEE-SMC Multiconference on Computational Engineering in Systems Applications, Lille, France
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