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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 $0\le\alpha<\frac{1}{29}$, which is more precise than those given by {\it 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 34C28 Complex behavior, chaotic systems (ODE)
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##### References:
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