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Abstract
In this paper, we employ generalized Lyapunov functions to derive new estimations of the ultimate boundary for the trajectories of two types of Lorenz systems, one with parameters in finite intervals and the other in infinite intervals. The new estimations improve the results reported so far in the literature. In particular, for the singular cases: b → 1+ and a → 0+, we have obtained the estimations independent of a. Moreover, our method using elementary algebra greatly simplifies the proofs in the literature. This is an interesting attempt in obtaining information of the attractors which is difficult when merely based on differential equations. It indicates that Lyapunov function is still a powerful tool in the study of qualitative behavior of chaotic systems.
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