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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.

MSC:
93C10Nonlinear control systems
37D45Strange attractors, chaotic dynamics
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References:
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