DYNAMICS OF A 3-DIMENSIONAL INCOMPRESSIBLE-FLOW WITH STAGNATION POINTS

The nonlinear dynamics of the streamlines of a two parameter family of incompressible flows is investigated. In a range of parameters, the flow contains two fixed points, or stagnation points, which are created and destroyed by saddle-node and saddle-node-Hopf bifurcations. Between the parameter values where these local bifurcations occur, there is a global bifurcation that creates an unstable closed orbit. Further global bifurcations lead to chaotic scattering and horseshoes. The detailed nature of the scattering depends upon whether the stagnation points are spiralling, i.e., upon whether their eigenvalues have imaginary parts. In the vicinity of a curve in the parameter space, a reversed cascade of period doubling bifurcations occurs, after which KAM (Kolmogorov, Arnold and Moser) tori appear, giving rise to nonhyperbolic chaotic scattering. As the parameter value for the saddle-node-Hopf bifurcation is approached, the region of chaos shrinks rapidly, leaving a vortex structure of KAM tori of size square-root epsilon, where epsilon is the distance in parameter space to the saddle-node-Hopf bifurcation.