We study the dynamical system associated with fluid particle motions o
f the Arnold-Beltrami-Childress (ABC) flow, defined by (x) over dot =
A sin z + C cos y, (y) over dot = B sin x + A cos z, (z) over dot = C
sin y + B cos x, where A, B, C are real parameters and \C\ much less t
han 1. First, we reduce this system to action-angle-angle coordinates.
Then, by using the new-KAM-like theorems for perturbations of a three
-dimensional, volume-preserving map, we obtain the conditions of exist
ence of invariant tori in the ABC flow. In addition, by using a high-d
imensional generalization of the Melnikov method, we obtain the analyt
ical criterion for the existence of chaotic streamlines in the ABC flo
w. (C) 1998 Published by Elsevier Science B.V.