Chaotic streamlines in the ABC flows

The particle paths of the Arnold-Beltrami-Childress (ABC) flows \ u = (A z+ C y, B x + A z, C y + B x). \ are investigated both analytically and numerically. This three-parameter family of spatially periodic flows provides a simple steady-state solution of Euler's equations. Nevertheless, the streamlines have a complicated Lagrangian structure which is studied here with dynamical systems tools. In general, there is a set of closed (on the torus, T 3) helical streamlines, each of which is surrounded by a finite region of KAM invariant surfaces. For certain values of the parameters strong resonances occur which disrupt the surfaces. The remaining space is occupied by chaotic particle paths: here stagnation points may occur and, when they do, they are connected by a web of heteroclinic streamlines. When one of the parameters A, B or C vanishes the flow is integrable. In the neighbourhood, perturbation techniques can be used to predict strong resonances. A systematic search for integrable cases is done using Painlevé tests, i. e. studying complex-time singularities of fluid-particle trajectories. When ABC ≠ 0 recursive clustering of complex time singularities occurs that seems characteristic of non-integrable behaviour.