%0 Journal Article
%T Transition to Chaos in a Shell Model of Turbulence
%A L. Biferale
%A A. Lambert
%A R. Lima
%A G. Paladin
%J Physics
%D 1994
%I arXiv
%R 10.1016/0167-2789(95)90065-9
%X We study a shell model for the energy cascade in three dimensional turbulence at varying the coefficients of the non-linear terms in such a way that the fundamental symmetries of Navier-Stokes are conserved. When a control parameter $\epsilon$ related to the strength of backward energy transfer is enough small, the dynamical system has a stable fixed point corresponding to the Kolmogorov scaling. This point becomes unstable at $\epsilon=0.3843...$ where a stable limit cycle appears via a Hopf bifurcation. By using the bi-orthogonal decomposition, the transition to chaos is shown to follow the Ruelle-Takens scenario. For $\epsilon > 0.3953..$ the dynamical evolution is intermittent with a positive Lyapunov exponent. In this regime, there exists a strange attractor which remains close to the Kolmogorov (now unstable) fixed point, and a local scaling invariance which can be described via a intermittent one-dimensional map.
%U http://arxiv.org/abs/chao-dyn/9402005v1