The structure of the transition boundary for shear flows

The shape and properties of the basin of attraction of the stable laminar point is investigated for finite-dimensional models of shear flows. In some of these models, the basin boundary is the stable manifold of an equilibrium point Xlb, the lower-branch point. As parameters change, the boundary undergoes a topological change at which a periodic orbit P emerges via a homoclinic bifurcation, and thereafter the major part of the basin boundary coincides with the stable manifold of P. The stable manifold of Xlb is then detectable only as an ``edge,'' i.e., the boundary between sets having different relaminarization properties. Implications for the nature of the edge are discussed.