3}[(1-\epsilon)\phi^{*}_{n-1}\phi^{*}_{n-2}
+ \epsilon \phi^{*}_{n-1}\phi^{*}_{n+1} - \phi^{*}_{n+1}\phi^{*}_{n+2}]
 -{Q^{2n}\over R}\phi_{n}\quad 2\leq n \leq N\\
\phi_{N+1} & = & \phi_{N+2}=0\end{aligned}$$

\[GOYmod\]

Here again the structure of non-linearities is fixed by a single parameter $\epsilon$, varying as $\beta$ between 0 and 1. We put for simplicity the forcing on the $\mbox{zero}^{th}$ shell, as first tried in [@BLLP]. The factor $-i$, which is usually kept in front of the non-linear terms, as a remnant of the Navier-Stokes equation, has been absorbed in an innocuous redefinition of variables ($u_{n} \rightarrow
iu_{n}$). The forcing also can be assumed to be real (here $F_{n}=(2-\epsilon) \delta
_{n,0}$), without loss of generality, thanks to the invariance of equations under the following phase transformation (see for instance [@BBP])

\[GOYsym\] $$\begin{aligned}
\phi_{3n} & \rightarrow & e^{i\theta_{0}} \phi_{3n}\\
\phi_{3n+1} & \rightarrow & e^{i\theta_{1}} \phi_{3n+1}\\
\phi_{3n+2} & \rightarrow & e^{-i(\theta_{0}+\theta_{1})} \phi_{3n+2}\end{aligned}$$

where $\theta_{0}$ and $\theta_{1}$ are arbitrary angles. There are obvious differences between the real ON model and the complex GOY model, which make their comparison interesting in its own right. The phase space which may be explored by the GOY model is [*a priori*]{} larger, at least for initial conditions not purely real. Also the range of non-linear interactions is wider than in 