Invariant tori and chaotic streamlines in the ABC flow

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Original languageEnglish
Pages (from-to)136-140
Journal / PublicationPhysics Letters, Section A: General, Atomic and Solid State Physics
Issue number3
Publication statusPublished - 5 Jan 1998


We study the dynamical system associated with fluid particle motions of the Arnold-Beltrami-Childress (ABC) flow, defined by ẋ = A sin z + C cos y, ẏ = B sin x + A cos z, ż = 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. © 1998 Published by Elsevier Science B.V.

Research Area(s)

  • ABC flow, Action-angle-angle, Chaos, Invariant tori, Melnikov method