Characterizing the edge of chaos for a shear flow model

Abstract

We characterize the edge of chaos, the boundary which separates initial conditions which lead to chaotic behavior from those which directly decay to the laminar state, for a nine-dimensional shear flow model. This boundary is the eight-dimensional stable manifold of an unstable periodic orbit, whose properties are characterized. Furthermore, we compute the probability that perturbations of a given energy will lead to transient chaos before decaying to the laminar state, or to a nontrivial attractor for a range of Reynolds numbers. Finally, we consider the relationship between the edge of chaos and linear transient growth, a mechanism which may trigger nonlinear effects that lead to turbulence in shear flows.