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Synchronization of two rank-one chaotic systems without and with delay via linear delayed feedback control. (English) Zbl 1243.93035

Summary: This paper illustrates the presence of chaos in rank-one chaotic systems with delay via a binary test (called 0-1 test) for chaos. Chaotic synchronization between two rank-one chaotic systems without and with delay is achieved by means of Lyapunov functional and linear delayed feedback control method. Numerical simulations are implemented to verify the effectiveness of the proposed chaos synchronization scheme.

MSC:

93B52 Feedback control
34H10 Chaos control for problems involving ordinary differential equations
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[1] 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 · doi:10.1103/PhysRevLett.64.821
[2] R. Mainieri and J. Rehacek, “Projective synchronization in three-dimensional chaotic systems,” Physical Review Letters, vol. 82, no. 15, pp. 3042-3045, 1999. · doi:10.1103/PhysRevLett.82.3042
[3] J. Yan and C. Li, “Generalized projective synchronization of a unified chaotic system,” Chaos, Solitons and Fractals, vol. 26, no. 4, pp. 1119-1124, 2005. · Zbl 1073.65147 · doi:10.1016/j.chaos.2005.02.034
[4] G. Wen and D. Xu, “Nonlinear observer control for full-state projective synchronization in chaotic continuous-time systems,” Chaos, Solitons and Fractals, vol. 26, no. 1, pp. 71-77, 2005. · Zbl 1122.93311 · doi:10.1016/j.chaos.2004.09.117
[5] Y. W. Wang and Z. H. Guan, “Generalized synchronization of continuous chaotic system,” Chaos, Solitons and Fractals, vol. 27, no. 1, pp. 97-101, 2006. · Zbl 1083.37515 · doi:10.1016/j.chaos.2004.12.038
[6] 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. · doi:10.1103/PhysRevE.66.046218
[7] D. Xu, “Control of projective synchronization in chaotic systems,” Physical Review E, vol. 63, no. 2, Article ID 027201, 4 pages, 2001. · doi:10.1103/PhysRevE.63.027201
[8] Z. Li and D. Xu, “Stability criterion for projective synchronization in three-dimensional chaotic systems,” Physics Letters, Section A: General, Atomic and Solid State Physics, vol. 282, no. 3, pp. 175-179, 2001. · Zbl 0983.37036 · doi:10.1016/S0375-9601(01)00185-2
[9] C. Y. Chee and D. Xu, “Secure digital communication using controlled projective synchronisation of chaos,” Chaos, Solitons and Fractals, vol. 23, no. 3, pp. 1063-1070, 2005. · Zbl 1068.94010 · doi:10.1016/j.chaos.2004.06.017
[10] G. Álvarez, S. Li, F. Montoya, G. Pastor, and M. Romera, “Breaking projective chaos synchronization secure communication using filtering and generalized synchronization,” Chaos, Solitons and Fractals, vol. 24, no. 3, pp. 775-783, 2005. · Zbl 1068.94002 · doi:10.1016/j.chaos.2004.09.038
[11] J. H. Park, “Adaptive controller design for modified projective synchronization of Genesio-Tesi chaotic system with uncertain parameters,” Chaos, Solitons and Fractals, vol. 34, no. 4, pp. 1154-1159, 2007. · Zbl 1142.93428 · doi:10.1016/j.chaos.2006.04.053
[12] G. H. Li, “Generalized projective synchronization of two chaotic systems by using active control,” Chaos, Solitons and Fractals, vol. 30, no. 1, pp. 77-82, 2006. · Zbl 1144.37372 · doi:10.1016/j.chaos.2005.08.130
[13] W. He and J. Cao, “Generalized synchronization of chaotic systems: An auxiliary system approach via matrix measure,” Chaos, vol. 19, no. 1, Article ID 013118, 2009. · Zbl 1311.34113 · doi:10.1063/1.3076397
[14] J. Cao, D. W. C. Ho, and Y. Yang, “Projective synchronization of a class of delayed chaotic systems via impulsive control,” Physics Letters, Section A: General, Atomic and Solid State Physics, vol. 373, no. 35, pp. 3128-3133, 2009. · Zbl 1233.34017 · doi:10.1016/j.physleta.2009.06.056
[15] W. Xia and J. Cao, “Pinning synchronization of delayed dynamical networks via periodically intermittent control,” Chaos, vol. 19, no. 1, Article ID 013120, 2009. · Zbl 1311.93061 · doi:10.1063/1.3071933
[16] J. Cao, Z. Wang, and Y. Sun, “Synchronization in an array of linearly stochastically coupled networks with time delays,” Physica A: Statistical Mechanics and its Applications, vol. 385, no. 2, pp. 718-728, 2007. · doi:10.1016/j.physa.2007.06.043
[17] Q. Wang and L. S. Young, “Strange attractors with one direction of instability,” Communications in Mathematical Physics, vol. 218, no. 1, pp. 1-97, 2001. · Zbl 0996.37040 · doi:10.1007/s002200100379
[18] Q. Wang and L. S. Young, “From invariant curves to strange attractors,” Communications in Mathematical Physics, vol. 225, no. 2, pp. 275-304, 2002. · Zbl 1080.37550 · doi:10.1007/s002200100582
[19] Q. Wang and L. S. Young, “Strange Attractors in Periodically-Kicked Limit Cycles and Hopf Bifurcations,” Communications in Mathematical Physics, vol. 240, no. 3, pp. 509-529, 2003. · Zbl 1078.37027 · doi:10.1007/s00220-003-0902-9
[20] Q. Wang and A. Oksasoglu, “Strange attractors in periodically kicked Chua’s circuit,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 15, no. 1, pp. 83-98, 2005. · Zbl 1073.37039 · doi:10.1142/S0218127405012028
[21] F. Chen and M. Han, “Rank one chaos in a class of planar systems with heteroclinic cycle,” Chaos, vol. 19, no. 4, Article ID 043122, 2009. · Zbl 1311.37044 · doi:10.1063/1.3263945
[22] G. A. Gottwald and I. Melbourne, “A new test for chaos in deterministic systems,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 460, no. 2042, pp. 603-611, 2004. · Zbl 1042.37060 · doi:10.1098/rspa.2003.1183
[23] G. A. Gottwald and I. Melbourne, “Testing for chaos in deterministic systems with noise,” Physica D: Nonlinear Phenomena, vol. 212, no. 1-2, pp. 100-110, 2005. · Zbl 1097.37024 · doi:10.1016/j.physd.2005.09.011
[24] G. A. Gottwald and I. Melbourne, “On the implementation of the 0 - 1 test for chaos,” SIAM Journal on Applied Dynamical Systems, vol. 8, no. 1, pp. 129-145, 2009. · Zbl 1161.37054 · doi:10.1137/080718851
[25] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 1992. · Zbl 0752.34039
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