- If you add an $i$-independent constant to $\theta_i$ then the difference $\theta_k-\theta_i$ that appears in the amplitude equation is unchanged.
- Global phase symmetry means that if $\psi_n=a_ne^{i\theta_n}$ is a solution, then also $\psi_n e^{i\phi t}$ with a constant phase $\phi$ is a solution. A local phase symmetry would have a $\phi_n$ that depends on $n$, which does not appply here.
- The limit cycle is semi-stable, rather than stable, because only amplitude perturbations decay in time, phase perturbations do not.
Carlo Beenakker
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