Hyperchaos behaviors and chaos synchronization of two unidirectional coupled simplified Lorenz systems

Abstract

To design a hyperchaotic generator and apply chaos into secure communication, a linear unidirectional coupling control is applied to two identical simplified Lorenz systems. The dynamical evolution process of the coupled system is investigated with variations of the system parameter and coupling coefficients. Particularly, the influence of coupling strength on dynamics of the coupled system is analyzed in detail. The range of the coupling strength in which the coupled system can generate hyperchaos or realize synchronization is determined, including phase portraits, Lyapunov exponents, and Poincaré section. And the critical value of the system parameter between hyperchaos and synchronization is also found with fixed coupled strength. In addition, abundant dynamical behaviors such as four-wing hyperchaotic, two-wing chaotic, single-wing coexisting attractors and periodic orbits are observed and chaos synchronization error curves are also drawn by varying system parameter c. Numerical simulations are implemented to verify the results of these investigations.

Keywords

hyperchaos / chaos synchronization / coupling control / simplified Lorenz system

References

[1] PecoraM, CarrollT L. Synchronization in chaotic systems [J]. Phys Rev Lett, 1990, 64(8): 821-824

[2] ShenL-q, WangM, LiuW-y, SunG-hui. Prediction based chaos control via a new neural network [J]. Phys Lett A, 2008, 372(46): 6916-6921

[3] VincentU E, GuoR-wei. Finite-time synchronization for a class of chaotic and hyperchaotic systems via adaptive feedback controller [J]. Phys Lett A, 2011, 375(24): 2322-2326

[4] YeG-dong. Image scrambling encryption algorithm of pixel bit based on chaos map [J]. Pattern Recognition Letters, 2010, 31(5): 347-354

[5] GrzybowskiJ M V, RafikovM, BalthazarJ M. Synchronization of the unified chaotic system and application in secure communication [J]. Communication Nonlinear Science Numerical Simulation, 2009, 14(6): 2793-2806

[6] RösslerO E. An equation for hyperchaos [J]. Phys Lett A, 1979, 71(2/3): 155-157

[7] WangF-z, ChenZ-q, WuW-j, YuanZ-zhi. A novel hyperchaos evolved from three dimensional modified Lorenz chaotic system [J]. Chin Phys B, 2007, 16(11): 3238-3243

[8] ChenZ-q, YangY, QiG-y, YuanZ-zhi. A novel hyperchaos system only with one equilibrium [J]. Phys Lett A, 2007, 360(5): 696-701

[9] GaoT-g, ChenG-r, ChenZ-q, CangS-jian. The generation and circuit implementation of a new hyperchaos based upon Lorenz system [J]. Phys Lett A, 2007, 361(1/2): 78-86

[10] PangS-q, LiuY-jian. A new hyperchaotic system from the Lü system and its control [J]. Journal of Computational and Applied Mathematics, 2011, 235(8): 2775-15

[11] NiuY-j, WangX-y, WangM-j, ZhangH-guang. A new hyperchaotic system and its circuit implementation [J]. Communication Nonlinear Science Numerical Simulation, 2010, 15(11): 3518-3524

[12] LiY-x, ChenG-r, TangW K S. Controlling a unified chaotic system to hyperchaotic [J]. Circuits and Systems II: Express Briefs, IEEE Transactions on, 2005, 52(4): 204-207

[13] TamL M, ChenH J, ChenH K, Si TouW-meng. Generation of hyperchaos from the Chen-Lee system via sinusoidal perturbation [J]. Chaos Solitons & Fractals, 2008, 38(3): 826-839

[14] WuX-q, LuJ-a, IuH H C, WongS-chuag. Suppression and generation of chaos for a three-dimensional autonomous system using parametric perturbations [J]. Chaos Solitons & Fractals, 2007, 31(4): 499-503

[15] RechP C. Chaos control in a discrete time system through asymmetric coupling [J]. Phys Lett A, 2008, 372(24): 4434-4440

[16] OlusolaO I, VincentU E, NjahA N. Synchronization, multistability and basin crisis in coupled pendula [J]. Journal of Sound and Vibration, 2010, 329(14): 443-456

[17] ChenZ-y, BiQ-sheng. Bifurcations and chaos of coupled Jerk systems [J]. Acta Phys Sin, 2010, 59(11): 7669-7678

[18] GrassiG, SeveranceF L, MillerD A. Multi-wing hyperchaotic attractors from coupled Lorenz systems [J]. Chaos Solitons & Fractals, 2009, 41(1): 284-291

[19] MahmoudE E. Dynamics and synchronization of new hyperchaotic complex Lorenz system [J]. Mathematical and Computer Modelling, 2012, 55(7/8): 1951-1962

[20] ZhaoJ-kun. Adaptive Q-S synchronization between coupled chaotic systems with stochastic perturbation and delay [J]. Applied Mathematical Modelling, 2012, 36(7): 3306-3313

[21] YuanZ-l, XuZ-y, GuoL-xiao. Generalized synchronization of two bidirectionally coupled discrete dynamical systems [J]. Communication Nonlinear Science Numerical Simulation, 2012, 17(2): 992-1002

[22] DuanZ-s, ChenG-rong. Global robust stability and synchronization of networks with Lorenz-type nodes [J]. IEEE Transactions on Circuits and Systems-II, 2009, 56(8): 569-573

[23] DuanZ-s, ChenG-r, HuangLin. Disconnected synchronized regions of complex dynamical networks [J]. IEEE Transactions on Automatic Control, 2009, 54(4): 845-849

[24] SunK-h, SprottJ C. Dynamics of a simplified Lorenz system [J]. International Journal of Bifurcation and Chaos, 2009, 19(4): 1357-1366