Understanding a Chaotic Saddle with Focus on a 9-Variable Model of Planar Couette Flow

Abstract

We study the transition between laminar and turbulent states in a Galerkin representation of a parallel shear flow, where a stable laminar flow and a transient turbulent flow state coexist. We find trajectories on the chaotic saddle in a model of plane Couette flow, and then use those trajectories to calculate the spectrum of Lyapunov exponents and the dimension of the system. We are able to estimate the fractal dimension of the both the saddle set and its stable manifolds. At moderate values of Reynolds number, these dimension estimates indicate that the stable set is nearly dense in many regions of phase. We find that the regions of initial conditions where the transient lifetimes show strong heterogeneity and appear sensitively dependendent on the initial conditions are separated from the regions with a smooth variation of lifetimes by an previously undescribed invariant structure, which we call the edge of chaos. We describe a technique to identify and follow the edge of chaos and provide evidence that it is a smooth manifold. For some values of Reynolds numbers we find that the edge of chaos coincides with the stable manifold of a periodic orbit, whereas in other ranges of the parameter, the edge is the stable set of a higherdimensional chaotic object. We provide evidence that this invariant edge structure may be a typical attribute of high dimensional transient chaos.