INVARIANT TORI AND CHAOTIC STREAMLINES IN THE ABC FLOW

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.