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### Branches of plane wave solutions

Because of the $S^{1}$-equivariance of the Stuart-Landau system (\[eq: SL\]), periodic solutions emerging from the homogeneous steady state via Hopf bifurcations have the following explicit form $$z(t)=ae^{i\Omega t-ik_{1}m-ik_{2}n}.\label{eq:persol}$$ The amplitude $a$, frequency $\Omega$, and the wavevector $\mathbf{k}=(k_{1},k_{2})^{T}$ of the periodic solutions satisfy the following relation $$i\Omega=\alpha+i\beta-a^{2}+e^{i(k_{+}-\Omega\tau)}C\cos{\left(k_{-}\right)},$$ which is obtained by substituting (\[eq:persol\]) into (\[eq: SL\]). Here we recall that $k_{\pm}$ is the wavevector in rotated coordinates (\[eq:Q\]). Taking real and imaginary parts, we obtain

$$\begin{aligned}
a^{2} & =\alpha+R\cos k_{\tau},\label{eq: SLpsolRe}\\
\Omega & =\beta+R\sin k_{\tau},\label{eq: SLpsolIm}\end{aligned}$$

where we denote $R:=C\cos k_{-}$ and $k_{\tau}:=k_{+}-\Omega\tau$. By excluding $k_{\tau}$ we obtain $$(a^{2}-\alpha)^{2}+(\Omega-\beta)^{2}=R^{2}.\label{eq: SLpsolCir}$$ Therefore all periodic solutions can be found on circles in the $(a^{2},\Omega)$-parameter space as shown in Fig. \[pic: SLpsolEll\]. Equation (\[eq: SLpsolIm\]) is known as *Kepler’s equation* and can be solved numerically with respect to $\Omega$. The number of solutions of (\[eq: SLpsolIm\]) matches the number of Hopf bifurcations and periodic solutions. All possible frequencies are confined to the interval $-\left|R\right|+\beta\leq\Omega\leq\left|R\right|+\beta$. The corresponding value for the parameter $\alpha$, at which Hopf bifurcation occurs, is then given by Eq. (\[eq: SLpsolRe\]) with $a^{2}=0$. For a system of the size $N\times N$, the number of Hopf 