up with $N+1$ numbers $\phi_{N}(a),\ldots,
\phi_{1}(a),\phi_{0}(a)$. The last equation on the $\mbox{zero}^{th}$ shell yields then a solvability condition for the parameter $a$ $$F(a)=1-(\alpha \phi_{1}^{2}(a)+\beta \phi_{0}(a) \phi_{1}(a))=0
\label{pfsolva}$$ Figures \[shellsta\] and \[IRbound\] show how things work for $Q=2$, $R=10^{5}$ and $\beta =0.348$. At least in the range of Reynolds numbers investigated here, there exists only one solution and the resulting fixed point is drawn on Fig.\[ptfix\]. The emergence of oscillations of period 2 and growing amplitude on the infrared side of the spectrum, is a generic feature of the ON model. Similar phenomena involving period 3 are known to plague the GOY model. Here the explanation is the following : in the inertial range (i.e. $1\ll n\ll n_{d}$), the equation to be solved reduces to $$q_n^{-1}(1+Cq_n^{-1})-q_{n+1}(q_{n+1}+C)=0
\label{itermap}$$ where one has introduced the ratio $q_n={\phi_n\over \phi_{n-1}}$ and $C={\beta
\over \alpha}$. Equation (\[itermap\]) defines a map $q_{n}=g(q_{n+1})$, whose fixed point $q_{n}=1$ is easily seen to be unstable for $C\leq 1$. Only for discrete values of the Reynolds number (such that one of the crossing points in Fig.\[shellsta\] has ordinate 1), can one hope to get rid of these oscillations. We should say that this odd-even disymmetry, though a pathology of the ON model, 