$\phi={\phi}_{ij}$ is the phase difference between clock $i$ and clock $j$.]{}

The value of $\varepsilon$ is the above mentioned $\varepsilon=\frac
{8\mu\alpha}{h^{2}}$ from [[@OlMe] where $\mu$ is the dry friction coefficient, $\frac{h^{2}}{2}$ is the kinetic energy furnished by the internal escape mechanism of each clock once per cycle and $\alpha$ the interaction coefficient between the clocks. The greater the $\alpha$ is, the greater the mutual influence among the clocks. In this paper, we do not need to particularize ]{}$\varepsilon$, since we are not interested in doing experimental computations. Therefore, we are interested in the fundamental result of symmetry between three oscillators subject to very weak mutual symmetric interaction.

Most of the reasonings are independent on the form of the function $P\left(
\phi\right)  $, therefore we consider a general differentiable odd function of the real variable $\phi$, $P(\phi)$, for the development of the model, and consider it of the form (\[Perturb1\]) when we analyze the model in section 4.

Observe that $|\sin(x+\varepsilon\sin y)-\sin x|<{\varepsilon}$ when $\varepsilon$[ is assumed to be sufficiently small. Therefore, we restrict our model to first order. ]{}We consider all the values of variables and constants in IS units.

Construction of the model
-------------------------

We now construct a dynamical 