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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 $x\in \bbfR^n$ and $y \in \bbfR^n$ satisfy asymptotically $\Vert x(t)-\alpha y(t)\Vert \to 0$ 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 $M$.

34H05ODE in connection with control problems
34C15Nonlinear oscillations, coupled oscillators (ODE)
34D35Stability of manifolds of solutions of ODE
Full Text: DOI
[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