## 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 $$M$$.

### MSC:

 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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### References:

 [1] Pecora L M, Carroll T L. Synchronization in chaotic systems[J]. Phys. Rev. Lett., 1990, 64: 821–824. · Zbl 0938.37019 [2] Boccaletti S, Kurths J, Osipov G, et al. The synchronization of chaotic systems [J]. Phys. Rep., 2002, 366:1–101. · Zbl 0995.37022 [3] Gonzdlez- Miranda J M. Chaotic systems with a null conditional Lyapunov exponent under nonlinear driving [J]. Phys. Rev. E, 1996, 53: R5–8. [4] Mainieri R, Rehacek J. Projective synchronization in three-dimensional chaotic systems [J]. Phys. Rev. Lett., 1999, 82:3042–3045. [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 [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 [7] Xu Dao-lin, La Zhi-gang. Controlled projective synchronization in nonpartially-linear chaotic systems [J]. Int. J. Bifur. Chaos, 2002, 12:1395–1402. [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. [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 [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 [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 [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. [14] Kocarev L, Parlitz U. Synchronizing spatiotemporal chaos in coupled nonlinear oscillators [J]. Phys. Rev. Lett., 1996, 77: 2206–2209. [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. [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 [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 [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 [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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