between $O_{1}$ and $O_{3}$ is symmetric, in the sense that $$(AC)_{0}={\psi}_{1}^{0}-{\psi}_{3}^{0}=-{\psi}_{3}^{0}=-(CA)_{0}.$$ The phase difference between $O_{2}$ and $O_{1}$ is $$(BA)_{0}={\psi}_{2}^{0}-{\psi}_{1}^{0}={\psi}_{2}^{0}$$ and the phase difference between $O_{1}$ and $O_{2}$ is $$(AB)_{0}={\psi}_{1}^{0}-{\psi}_{2}^{0}=-{\psi}_{2}^{0}=-(BA)_{0}.$$ The phase difference between $O_{3}$ and $O_{2}$ is $$(CB)_{0}={\psi}_{3}^{0}-{\psi}_{2}^{0}$$ and the phase difference between $O_{2}$ and $O_{3}$ is $$(BC)_{0}={\psi}_{2}^{0}-{\psi}_{3}^{0}=-(CB)_{0}.$$

**STEPS LEADING TO THE CONSTRUCTION OF THE MODEL**

**STEP 1:** first impact. Interactions of $O_{1}$ on $O_{2}$ and of $O_{1}$ on $O_{3}$, at $t=0$.

When the system in position A attains phase $0$ $(\operatorname{mod}2\pi)$ it receives a sudden supply of energy, for short a kick, from its escape mechanism, this kick propagates in the common support of the three clocks and reaches the other two clocks.

Now, the phase difference between $O_{3}$ and $O_{1}$ is corrected by the perturbative value $P$: $$\left(  CA\right)  _{I}=\left(  CA\right)  _{0}+P\left(  \left(  CA\right)
_{0}\right)  ={\psi}_{3}^{0}+P\left(  {\psi}_{3}^{0}\right)  .$$ The phase difference between $O_{1}$ and $O_{3}$ is $$\left(  AC\right)  _{I}=\left(  AC\right)  _{0}+P\left(  \left(  AC\right)
_{0}\right)  =-{\psi}_{3}^{0}+P\left(  -{\psi}_{3}^{0}\right)  =-(CA)_{I},$$ since $P$ must be an odd function of the mutual phase difference.

The phase difference between $O_{2}$ and $O_{1}$ is $$\left(  BA\right)  _{I}=\left(  BA\right)  _{0}+P\left(  \left(  BA\right)
_{0}\right)  ={\psi}_{2}^{0}+P\left(  {\psi}_{2}^{0}\right)  ,$$ and the symmetric phase difference between $O_{1}$ and $O_{2}$ 