Estimating the bounds for the Lorenz family of chaotic systems

In this paper, 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 less than or equal to 1 and get rid of the singularity problem as b --> 1(+). Furthermore, for a > 1, 1 less than or equal to 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 less than or equal to alpha < (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, Lu system and the unified system are established. (C) 2004 Elsevier Ltd. All rights reserved.