which is quite efficient and easily extended to any shell model. It should however be said from the beginning that our results about the ON model are in complete agreement with the conclusions reached in [@N88].\
By plugging the Ansatz (\[selfsim\_1\]) into (\[nlb\_n\]) and introducing the logarithmic variable $\xi =n+{1\over z \log Q}\log (t^{*}-t)$, one arrives at the following equation for $f$ ($f$ is actually divided by $z \log Q$ to make the result a bit simpler) $$f'(\xi )-z\log Q f(\xi )=(f(\xi -1)f(\xi )+{\beta\over \alpha}
Q^{2\over 3} f(\xi-1)^{2}) - {1\over Q^{2}}(f(\xi+1)^{2}+{\beta\over \alpha}
Q^{2\over 3}f(\xi )f(\xi+1))
\label{selfsimequ}$$ If square integrability of $f$ is required, Eq. (\[selfsimequ\]) is nothing but a non-linear eigenvalue problem for the unknown $z$, which is very difficult to solve directly, either analytically or numerically. To make progress, we can try to approach $f$ dynamically. Rather than coming back to the original equations of the model, let us introduce a fictitious dynamics leaving the norm of the $(N+1)$-dimensional vector $\vec b$ invariant $$\frac{d}{d\tau} \vec{b}= \vec{N}[\vec{b}]-\frac{<\vec{N}[\vec{b}],\vec{b}>}
{<\vec{b},\vec{b}>}\, \vec{b}
\label{projdyn}$$ In the equation above, $\vec{N}[\vec{b}]$ is the vector of components $N_{n}[\vec{b}]$, whose expression was given in Eq. (\[nlb\_n\]) and $\langle \vec{A},\vec{B}\rangle =\sum_{n=0}^{N} A_{n}B_{n}$ is the usual euclidean scalar product. The 