Unstable Periodic Orbits in Turbulence

Recently both shear turbulence and isotropic turbulence have been investigated by means of unstable periodic orbits. These orbits are embedded in a high dimensional chaotic attractor that represents the turbulent flow. In both cases, the periodic motion can be shown to bear a strong similarity to the turbulent motion. Therefore we can learn about the nature of turbulence by studying the periodic motion. This gives us a number of tools that can not directly be applied to complex, non periodic flow, such as parameter continuation, computation of the Lyapunov spectrum and analysis of stable and unstable manifolds. In this paper we review recent work on plane Couette flow and introduce a low-order model to illustrate the structure of phase space as revealed by the study of unstable periodic orbits. We also present new work on isotropic turbulence and speculate about future applications and theory for systems with many degrees of freedom.