Coarse Graining the State Space of a Turbulent Flow Using Periodic Orbits

We show that turbulent dynamics that arise in simulations of the three-dimensional Navier-Stokes equations in a triply periodic domain under sinusoidal forcing can be described as transient visits to the neighborhoods of unstable time-periodic solutions. Based on this description, we reduce the original system with more than 10^{5} degrees of freedom to a 17-node Markov chain where each node corresponds to the neighborhood of a periodic orbit. The model accurately reproduces long-term averages of the system's observables as weighted sums over the periodic orbits.