Edge states in the transition to turbulence in pipe and other shear flows

We study the boundary of the laminar region in pipe and other shear flows near the onset of turbulence. Approaching the boundary from the laminar side, the lifetime of perturbations increases, and it diverges when the boundary is reached. Once this critical amplitude is exceeded the trajectory swings up to the turbulent regime, but its lifetime varies sensitively with amplitude, consistent with the strange saddle picture of the turbulence proposed earlier. The edge trajectory is asymptotic to a single well defined state, independent of the type of perturbation. The edge then becomes the stable manifold of this structure. In the case of a model shear flow, the edge states are simple or period doubled or chaotic trajectories. The case of pipe flow shows less variability and the edge state seems to remain close to a state with simple vortex structure.