Time averaged properties along unstable periodic orbits of the Kuramoto-Sivashinsky equation

It has been reported in some dynamical systems in fluid dynamics that only a few UPOs (unstable periodic orbits) with low periods can give good approximations to mean properties of turbulent (chaotic) solutions. By employing the Kuramoto-Sivashinsky equation we compare time averaged properties of a set of UPOs embedded in a chaotic attractor and those of a set of segments of chaotic orbits, and report that the distributions of the time average of a dynamical variable along UPOs with lower and higher periods are similar to each other, and the variance of the distribution is small, in contrast with that along chaotic segments. The result is similar to those for low dimensional ordinary differential equations including Lorenz system, Rössler system and economic system.