differentiable and invertible in $S=\left[  0,2\pi\right]  \times\left[
0,2\pi\right]  $ when $\varepsilon>0$ is small. The perturbation map $\Omega\left(  x,y\right)  $ is periodic in $%
\mathbb{R}
^{2}$. This implies that the solution of the problem in the set $S$ is a dynamical system and not the usual semi-dynamical system associated with discrete time. That will provide a reasonable simple structure to the problem of the stability of fixed points and will enable to derive global properties. Moreover, we prove that for small $\varepsilon$ the set $S$ is invariant for the dynamics of $F$, meaning that the two phase differences of oscillators $O_{2}$ and $O_{3}$ relative to oscillator $O_{1}$ stay in the interval $\left[  0,2\pi\right[  $.

In particular, the map $\Omega$ has the zeros $\left(  \pi,\pi\right)  $, $\left(  \frac{2}{3} \pi,\frac{4}{3} \pi\right)  $ and $\left(  \frac{4}{3}
\pi,\frac{2}{3} \pi\right)  $ in the interior of the set $S=\left[
0,2\pi\right]  \times\left[  0,2\pi\right]  $, which are fixed points of the model $F$. There are also four trivial fixed points, $\left(  0,0\right)  $, $\left(  0,2\pi\right)  $, $\left(  2\pi,0\right)  $ and $\left(  2\pi
,2\pi\right)  $ at the corners of $S$, and the four fixed points $\left(
0,\pi\right)  $, $\left(  \pi,2\pi\right)  $, $\left(  2\pi,\pi\right)  $ and $\left(  \pi,0\right)  $ on the 