is less visible in the chaotic state to be described below. Furthermore, it does not affect other physical quantities like the energy transfer $\epsilon_{n}$ defined in (\[flux\_n\]).\
We turn now to a qualitative discussion of the phase diagram of the model, which is observed as $\beta$ varies (for the particular values $Q=2$ and $R=10^{5}$). Although the static solution exists for every strictly positive $\beta$, it becomes unstable for $\beta \leq 0.355 \pm 10^{-3}$ and evolves towards a periodic limit cycle through a first Hopf bifurcation. A scenario à la Ruelle Takens, similar to the one already discovered in the GOY model [@BLLP], leads then to chaos for $\beta \leq \beta^{*} = 0.349 \pm
 10^{-3}$. We did not try to get very precise estimates of these two thresholds, whose position is expected to vary with $R$. From a slightly unrelated analysis of movable singularities of the ON model, presented in Section 5, we shall speculate later in the paper that the asymptotic value of $\beta^{*}$ in the limit of infinite Reynolds number is of order 0.394. In the quasiperiodic regime, the shells oscillate in a coherent way around the static fixed point discussed above. Their oscillations remain of moderate amplitude even 