Invariant tori and chaotic streamlines in the ABC flow

Abstract We study the dynamical system associated with fluid particle motions of the Arnold-Beltrami-Childress (ABC) flow, defined by x = A sin z + C cos y , y = B sin x + A cos z , z = C sin y + B cos x , where A , B , C are real parameters and | C | ⪡ 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 existence of invariant tori in the ABC flow. In addition, by using a high-dimensional generalization of the Melnikov method, we obtain the analytical criterion for the existence of chaotic streamlines in the ABC flow.