Question about graph Ginzburg-Landau equation

Question

I am reading the following paper: http://ieeexplore.ieee.org/document/7798534/

For the equation (5):
$$\dot{a}=(1-a_i^2)a_i-\beta\sum_{k-i}(a_k\cos(\theta_k-\theta_i)-a_i)$$ $$\dot{\theta}_i=\beta\sum_{k-i}\frac{a_k}{a_i}\sin(\theta_i-\theta_k)$$

1. Both are dynamics for amplitude $a$ and phase $\theta$.
2. $k-i$ means there is an edge linking the node $k$ and $i$.
3. $\beta$ is just a coefficient

My question is the following:

1. Why the amplitude dynamics are invariant?

and the following:

2. Why it is "global phase symmetry"
3. Why the equilibria are semi-stable with respect to rotation?

Thanks!

Answer 1

1. If you add an $i$-independent time-dependent term $\delta\theta=\omega t+\gamma$ to $\theta_i$ then the difference $\theta_k-\theta_i$ that appears in the amplitude equation is unchanged. The phase equation is changed by a term $d\delta\theta/dt=\omega$.
2. 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.
3. The limit cycle is semi-stable, rather than stable, because only amplitude perturbations decay in time, phase perturbations do not.