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Estimating the bounds for the Lorenz family of chaotic systems. (English) Zbl 1061.93506
Summary: We derive a sharper upper bound for the Lorenz system, for all the positive values of its parameters $a$, $b$ and $c$. Comparing with the best result existing in the current literature, we fill the gap of the estimate for $0<b\leqslant 1$ and get rid of the singularity problem as $b\to 1^+$. Furthermore, for $a>1$, $1\leqslant b<2$, we obtain a more precise estimate. Along the same line, we also provide estimates of bounds for a unified chaotic system for $0\leqslant \alpha <\frac {1} {29}$. When $\alpha =0$, the estimate agrees precisely with the known result. Finally, the two-dimensional bounds with respect to $x-z$ for the Chen system, Lü system and the unified system are established.

93C10Nonlinear control systems
37D45Strange attractors, chaotic dynamics
Full Text: DOI
[1] Lorenz, E. N.: Deterministic non-periods flows. J. atmos. Sci. 20, 130-141 (1963)
[2] Chen, G. R.; Ueta, T.: Yet another chaotic attractor. Int. J. Bifurcat. chaos 9, 1465-1466 (1999) · Zbl 0962.37013
[3] Vaněcěk, A.; Cělikovsky, S.: Control systems: from linear analysis to synthesis of chaos. (1996)
[4] Lü, J.; Chen, G.: A new chaotic attractor coined. Int. J. Bifurcat. chaos 12, No. 3, 659-661 (2002) · Zbl 1063.34510
[5] Lü, J.; Chen, G.; Cheng, D.; Čelikovsky\breve{}, S.: Bridge the gap between the Lorenz system and the Chen system. Int. J. Bifurcat. chaos 12, 2917-2926 (2002) · Zbl 1043.37026
[6] Chen, G.; Lü, J.: Dynamical analysis, control and synchronization of the Lorenz systems family. (2003)
[7] Lu, J.; Wu, X.; Lü, J.: Synchronization of a unified chaotic system and the application in secure communication. Phys. lett. A 305, 365-370 (2002) · Zbl 1005.37012
[8] Zhou, T.; Chen, G.; Tang, Y.: A universal unfolding of the Lorenz system. Chaos, solitons & fractals 20, 979-993 (2004) · Zbl 1048.37032
[9] Wu, X.; Lu, J.: Parameter identification and backstepping control of uncertain Lü system. Chaos, solitons & fractals 18, 721-729 (2003) · Zbl 1068.93019
[10] Lü, J.; Lu, J.: Controlling uncertain Lü system using linear feedback. Chaos, solitons & fractals 17, 127-133 (2002)
[11] Leonov, G.; Bunin, A.; Koksch, N.: Attractor localization of the Lorenz system. Zamm 67, 649-656 (1987) · Zbl 0653.34040
[12] Pogromsky, A. Yu.; Santoboni, G.; Nijmeijer, H.: An ultimate bound on the trajectories of the Lorenz systems and its applications. Nonlinearity 16, 1597-1605 (2003) · Zbl 1050.34078
[13] Zhou, T.; Tang, Y.; Chen, G.: Complex dynamical behaviors of the chaotic Chen’s system. Int. J. Bifurcat. chaos 9, 2561-2574 (2003) · Zbl 1046.37018