Unstable periodic solutions embedded in a shell model turbulence

Abstract

An approach to intermittency of a shell model turbulence is proposed from the viewpoint of dynamical systems. We detected unstable solutions of the Gledzer-Ohkitani-Yamada shell model and studied their relation to turbulence statistics. One of the solutions has an unstable periodic orbit (UPO), which shows an intermittency where the scaling exponents of the structure function have a nonlinear dependence on its order, quite similar to that of turbulence solution at the same parameter values. The attractor in the phase space is found to be well approximated by a continuous set of solutions generated from the UPO through a one-parameter phase transformation, which implies that the intermittency of the shell model turbulence is described by this UPO. © 2003 The American Physical Society.