×

Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system. (English) Zbl 1104.37024

From the introduction: We investigate the ultimate bound and positively invariant set for the Lorenz system and the unified chaotic system using a technique combining the generalized Lyapunov function theory and optimization. For the Lorenz system, we derive an ellipsoidal bound and positively invariant set for all the positive values of its parameters \(a\) and \(b\), and also give the minimum point and minimum value for the volume of the ellipsoid. Comparing with the best results existing in the current literature, our new results fill up the gap of the estimate for the case of \(0<a<1\) and \(0<b<2\). Along the same line, we also obtain estimates of ellipsoidal and cylindrical bounds for the unified chaotic system for its parameter range \(0\leq\alpha<\frac{1}{29}\), which is more precise than those given by M. V. Klibanov [Diff. Equations 22, 1232–1240 (1986; Zbl 0618.42012)]. Furthermore, we derive the minimum point and minimum value for the ellipsoid. These theoretical results are important and useful in chaos control, chaos synchronization and their applications.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34C28 Complex behavior and chaotic systems of ordinary differential equations

Citations:

Zbl 0618.42012
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Lorenz, E. N., Deterministic non-periods flows, J. Atoms Sci., 20, 130-141 (1963) · Zbl 1417.37129
[2] Chen, G.; Ueta, T., Yet another chaotic attractor, Int. J. Bifur. Chaos, 9, 1465-1466 (1999) · Zbl 0962.37013
[3] Vaněcěk, A.; Cělikovesky, S., Control Systems: From Linear Analysis to Synthesis of Chaos (1996), Prentice Hall: Prentice Hall London
[4] Lü, J.; Chen, G., A new chaotic attractor coined, Int. J. Bifur. Chaos, 12, 659-661 (2002) · Zbl 1063.34510
[5] Lü, J.; Chen, G.; Cheng, D.; Čelikovsky˘, S., Bridge the gap between the Lorenz system and the Chen system, Int. J. Bifur. Chaos, 12, 2917-2926 (2002) · Zbl 1043.37026
[6] Chen, G.; Lü, J., Dynamical Analysis, Control and Synchronization of the Lorenz Systems Family (2003), Science Press: Science Press China
[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] Leonov, G., Bound for attractors and the existence of homoclinic orbit in the Lorenz system, J. Appl. Math. Mech., 65, 19-32 (2001)
[9] Zhou, T.; Tang, Y.; Chen, G., Complex dynamical behaviors of the chaotic Chen’s system, Int. J. Bifur. Chaos, 9, 2561-2574 (2003) · Zbl 1046.37018
[10] Pogromsky, A.; 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
[11] Leonov, G.; Bunin, A.; Koksch, N., Attractor localization of the Lorenz system, Z. Angew. Math. Mech., 67, 649-656 (1987) · Zbl 0653.34040
[12] Li, D.; Lu, J.; Wu, X.; Chen, G., Estimating the bounds for the Lorenz family of chaotic systems, Chaos Solitons Fractals, 23, 529-534 (2005) · Zbl 1061.93506
[13] Liao, X., On the global basin of attraction and positively invariant set for the Lorenz chaotic system and its application in chaos control and synchronization, Sci. China Ser. E, 34, 1404-1419 (2004)
[14] Lefchetz, S., Differential Equations: Geometric Theory (1963), Interscience: Interscience New York
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.