Abstract. Shell models of turbulence have been employed as toy models which, in their chaotic states, show statistical properties similar to real fluid turbulence, including Kolmogorov energy spectrum and intermittency. These models are interesting because, at the present stage, it is still quite difficult or almost impossible to study relations between those traditional statistical properties and the structure of the chaos underlying the real fluid turbulence because of huge dimension of the chaotic attractor. In this paper we will give a brief review on the chaotic properties of a shell model (GOY model), with emphasis on its Lyapunov spectrum and unstable periodic orbits, in relation to the Kolmogorov scaling law of the turbulence.
Statistical properties of nonlinear shell models of turbulence from linear advection models: rigorous results