The problem of the existence of a third isolating integral of motion in an axisymmetric potential is investigated by numerical experiments. It is found that the third integral exists for only a limited range of initial conditions.

Lorenz (1963) has investigated a system of three first-order differential equations, whose solutions tend toward a "strange attractor". We show that the same properties can be observed in a simple mapping of the plane defined by: x i+1 = y i -\-l -axj, y i+1 = bx t . Numerical experiments are carried out for a = lΛ, b = 03. Depending on the initial point (x Oi y o ), the sequence of points obtained by iteration of the mapping either diverges to infinity or tends to a strange attractor, which appears to be the product of a onedimensional manifold by a Cantor set.

The particle paths of the Arnold-Beltrami-Childress (ABC) flows \[ u = (A \sin z+ C \cos y, B \sin x + A \cos z, C \sin y + B \cos x). \] 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, T3) 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.

Lorenz (1963) has investigated a system of three first-order differential equations, whose solutions tend toward a "strange attractor". We show that the same properties can be observed in a simple mapping of the plane defined by: x i+1 = y i -\-l -axj, y i+1 = bx t . Numerical experiments are carried out for a = lΛ, b = 03. Depending on the initial point (x Oi y o ), the sequence of points obtained by iteration of the mapping either diverges to infinity or tends to a strange attractor, which appears to be the product of a onedimensional manifold by a Cantor set.