Spatially localized unstable periodic orbits of a high-dimensional chaotic system

Abstract

Using an innovative damped-Newton method, we report the calculation and analysis of many distinct unstable periodic orbits (UPOs) for a high-fractal-dimension $(D=8.8)$ extensively chaotic solution of a partial differential equation. A majority of the UPOs turn out to be spatially localized in that time dependence occurs only on portions of the spatial domain. With a escape-time weighting of 127 UPOs, the attractor's fractal dimension can be estimated with a relative error of 2%. Statistical errors are found to decrease as $1/\sqrt{N}$ as the number $N$ of known UPOs increases.