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⩽1 and get rid of the singularity problem as b→1+. Furthermore, for a>1, 1⩽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⩽α< 1 29 . When α=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.