# zbMATH — the first resource for mathematics

Finite-time synchronization of uncertain unified chaotic systems based on CLF. (English) Zbl 1183.34072
Consider the master system
\begin{aligned} \dot x_1 & = (25\alpha+ 10)(x_2- x_1),\\ \dot x_2 & = (28-35\alpha) x_1- x_1 x_3+ (29\alpha- 1)x_2,\\ \dot x_3 & = x_1 x_2- {(8+ \alpha)\over 3} x_3.\end{aligned}\tag{1} For $$\alpha\in [0,1]$$ system (1) is chaotic, for certain $$\alpha$$-values it is related to the Lorenz, Lü and Chen system. Representing (1) in the form $$\dot x= f(x,\alpha)$$, the authors consider together with (1) the slave system $$\dot y= f(y,\alpha)+ u$$. The goal of the authors is to find a control $$u$$ such that the slave system synchronizes the master system in finite time, that is, the corresponding error system
$\dot e=\widetilde f(e,y,\alpha)+ u\text{ with }e= y- x$ has the property that their solutions tend to zero in a finite time. Of course, this requires that the error system is not Lipschitzian in $$e$$. The authors construct such a control by means of a control Lyapunov function. Moreover, they show that this control is robust against perturbations of some coefficients of (1).

##### MSC:
 34D06 Synchronization of solutions to ordinary differential equations 34C28 Complex behavior and chaotic systems of ordinary differential equations 34H05 Control problems involving ordinary differential equations
Full Text:
##### References:
 [1] Colet, P.; Roy, R., Digital communication with synchronizationed chaotic lasers, Opt. lett., 19, 2056-2058, (1994) [2] Sugawara, T.; Tachikawa, M.; Tsukamoto, T.; Shimizu, T., Observation of synchronization in laser chaos, Phys. rev. lett., 72, 3502-3505, (1994) [3] Lu, J.A.; Wu, X.Q.; Lü, J.H., Synchronization of a unified chaotic system and the application in secure communication, Phys. lett. A, 305, 365-370, (2002) · Zbl 1005.37012 [4] S. Bhat, D. Bernstein, Finite-time stability of homogeneous systems, in: Proceedings of ACC, Albuquerque, NM, 1997, pp. 2513-2514 [5] Haimo, V.T., Finite time controllers, SIAM J. control optim., 24, 760-770, (1986) · Zbl 0603.93005 [6] Lü, J.H.; Chen, G.R; Cheng, D.Z., S.celikovsky, bridge the gap between the Lorenz and the Chen system, Int. J. bifurcation chaos, 12, 12, 2917-2926, (2002) · Zbl 1043.37026 [7] Sontag, E.D., A ‘universal’ construction of artstein’s theorem on nonlinear stabilization, System control lett., 13, 117-123, (1989) · Zbl 0684.93063 [8] Artstein, Z., Stabilization with relaxed controls, Nonlinear anal. TMA, 7, 11, 1163-1173, (1983) · Zbl 0525.93053 [9] Khalil, H.K., Nonlinear systems, (2002), Prentice-Hall New Jersey, pp. 102-103 [10] Wang, F.Q.; Liu, C.X., Synchronization of unified chaotic system based on passive control, Physica D, 225, 55-60, (2007) · Zbl 1119.34332 [11] Park, J.H., On synchronization of unified chaotic systems via nonlinear control, Chaos solitons fractals, 25, 699-704, (2005) · Zbl 1125.93469 [12] Tao, C.H.; Xiong, H.X.; Hu, F., Two novel synchronization criterions for a unified chaotic system, Chaos solitons fractals, 27, 115-120, (2006) · Zbl 1083.37514 [13] Yan, J.P.; Li, C.P., Generalized projective synchronization of a unified chaotic system, Chaos solitons fractals, 26, 1119-1124, (2005) · Zbl 1073.65147
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.