Chaotic and Resonant Streamlines in the ABC Flow

Abstract

Using the method of Melnikov, the dynamical system associated with fluid particle motion of the Arnold-Beltrami-Childress (ABC) flow is studied in this paper. Let \[ ( 1 )\qquad \dot x = A\sin z + C\cos y,\qquad\dot y = B\sin x + A\cos z,\qquad\dot z = C\sin y + B\cos x \], where $A,B,C$ are real parameters and satisfy $A > B > C > 0$ and $C \ll 1$. The analytical criteria for the existence of chaotic streamlines and resonant streamlines in (1) have been obtained.