Generalized projective synchronization for the chaotic Lorenz system and the chaotic Chen system. (English) Zbl 1118.34050

Two systems are said to have the “projective synchronization” if their state variables \(x\in \mathbb{R}^n\) and \(y \in \mathbb{R}^n\) satisfy asymptotically \(\| x(t)-\alpha y(t)\| \to 0\) for \(t\to \infty\) and all initial conditions with some scalar \(\alpha\). The authors consider the following control setup in order to achieve the projective synchronization: \[ x'=f(x),\quad y'=f(y) + u, \] where the control \(u(x,y)\) is chosen in such a way that the error system admits the linear form \((\alpha y-x)'=M(\alpha y-x)\) with a stable matrix \(M\).


34H05 Control problems involving ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34D35 Stability of manifolds of solutions to ordinary differential equations
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[1] Pecora L M, Carroll T L. Synchronization in chaotic systems[J]. Phys. Rev. Lett., 1990, 64: 821–824. · Zbl 0938.37019 · doi:10.1103/PhysRevLett.64.821
[2] Boccaletti S, Kurths J, Osipov G, et al. The synchronization of chaotic systems [J]. Phys. Rep., 2002, 366:1–101. · Zbl 0995.37022 · doi:10.1016/S0370-1573(02)00137-0
[3] Gonzdlez- Miranda J M. Chaotic systems with a null conditional Lyapunov exponent under nonlinear driving [J]. Phys. Rev. E, 1996, 53: R5–8. · doi:10.1103/PhysRevE.53.R5
[4] Mainieri R, Rehacek J. Projective synchronization in three-dimensional chaotic systems [J]. Phys. Rev. Lett., 1999, 82:3042–3045. · doi:10.1103/PhysRevLett.82.3042
[5] Li Zhi-gang, Xu Dao-lin. Stability criterion for projective synchronization in three-dimensional chaotic systems [J]. Phys. Lett. A, 2001, 282:175–179. · Zbl 0983.37036 · doi:10.1016/S0375-9601(01)00185-2
[6] Xu Dao-lin, Li Zhi-gang, Bishop S R. Manipulating the scaling factor of projective synchronization in three-dimensional chaotic systems [J]. Chaos, 2001, 11: 439–442. · Zbl 0996.37075 · doi:10.1063/1.1380370
[7] Xu Dao-lin, La Zhi-gang. Controlled projective synchronization in nonpartially-linear chaotic systems [J]. Int. J. Bifur. Chaos, 2002, 12:1395–1402. · doi:10.1142/S0218127402005170
[8] Xu Dao-lin, Chee Chin-yi. Controlling the ultimate state of projective synchronization in chaotic systems of arbitrary dimension [J]. Phys. Rev. E, 2002, 66:046218. · doi:10.1103/PhysRevE.66.046218
[9] Xu Dao-lin, Ong Wee-leng, Li Zhi-gang. Criteria of the occurrence of projective synchronization in chaotic systems of arbitrary dimension [J]. Phys. Lett. A, 2002, 305:167–172. · Zbl 1001.37026 · doi:10.1016/S0375-9601(02)01445-7
[10] Chee Chin-yi, Xu Dao-lin. Control of the formation of projective synchronization in lower-dimensional discrete-time systems [J]. Phys. Lett. A, 2003, 318:112–118. · Zbl 1098.37512 · doi:10.1016/j.physleta.2003.09.024
[11] Xu Dao-lin, Chee Chin-yi, Li Chang-pin. A necessary condition of projective synchronization in discrete-time systems of arbitrary dimensions [J]. Chaos, Solitons and Fractals, 2004, 22: 175–180. · Zbl 1060.93535 · doi:10.1016/j.chaos.2004.01.012
[12] Afraimovich V S, Verichev N N, Rabinovich M I. Izvestiya Vysshikh Uchebnykh Zavedenii Radiofizika [J]. 1986, 29(9): 1050–1060.
[13] Rulkov N F, Sushchik M M, Tsimring L S, etal. Generalized synchronization of chaos in directionally coupled chaotic systems [J]. Phys. Rev. E, 1995, 51: 980–994. · doi:10.1103/PhysRevE.51.980
[14] Kocarev L, Parlitz U. Synchronizing spatiotemporal chaos in coupled nonlinear oscillators [J]. Phys. Rev. Lett., 1996, 77: 2206–2209. · doi:10.1103/PhysRevLett.77.2206
[15] Abarbanel H D I, Rulkov N F, Sushchik M M. Generalized synchronization of chaos: The auxiliary system approach [J]. Phys. Rev. E, 1996, 53: 4528–4535. · doi:10.1103/PhysRevE.53.4528
[16] Sparrow C. The Lorenz Equations, Bifurcation, Chaos, and Strange Attractors [M]. Springer-Verlag, New York, 1982. · Zbl 0504.58001
[17] Li Li-kang, Yu Chong-hua, Zhu Zheng-hua. Numerical Methods for Differential Equations [M]. Fudan University Press, Shanghai, 1999.
[18] Chen Guan-Rong, Ueta T. Yet another chaotic attractor [J]. Int. J. Bifur. Chaos, 1999, 9: 1465–1466. · Zbl 0962.37013 · doi:10.1142/S0218127499001024
[19] Li Chang-Pin, Chen Guan-Rong. A note on Hopf bifurcation in Chen’s system [J]. Int. J. Bifur. Chaos, 2003, 13: 1609–1615. · Zbl 1074.34045 · doi:10.1142/S0218127403007394
[20] Li Chang-Pin, Peng Guo-jun. Chaos in the Chen’s system with a fractional order [J]. Chaos, Solitons and Fractals, 2004, 22: 443–450. · Zbl 1060.37026 · doi:10.1016/j.chaos.2004.02.013
[21] Yan Jian-ping, Li Chang-pin, On synchronization of three chaotic systems [J]. Chaos, Solitons and Fractals, 2005, 23: 1683–1688. · Zbl 1068.94535
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