bifurcations, or periodic solutions respectively, can be estimated by $$\begin{aligned}
\mathcal{N}_{H} \approx \begin{cases}
\frac{2C\tau}{\pi} N \frac{1}{\sin{\left(\frac{\pi}{2 N}\right)}} & N\,\mathrm{even},\\
\frac{2C\tau}{\pi} N \cot{\left( \frac{\pi}{2N} \right)} & N\,\mathrm{odd}.
\end{cases}\end{aligned}$$ In the case of large $N$ the asymptotic number of solutions is given by $$\mathcal{N}_{H}\sim\frac{4C\tau}{\pi^{2}}N^{2}.\label{eq:Npsol}$$ Thus, in the case of large delay or lattice-size the number of solutions grows and any point on the circles (\[eq: SLpsolCir\]) refers to a periodic solution, i.e. the circle disk is densely filled with points $(a,\Omega)$ corresponding to the existing periodic solutions. Indeed, for an infinitely large lattice, the wavevector $\mathbf{k}$ admits any values from $\mathbb{R}^{2}$. Therefore one obtains continuous parameters $k_{-}\in\left(-\pi,\pi\right)$ and $k_{\tau}=(k_{+}-\Omega\tau)\in\left(0,2\pi\right)$, which parametrize all periodic solutions of Eq. (\[eq: SL\]).

![(a) Frequency $\Omega$ versus amplitude $a$ of periodic solution (\[eq:persol\]). The admissible parameters are located on the circles with radii $R$ [\[]{}Eq. \[pic: SLpsolEll\][\]]{}, except the parts with $a^{2}<0$. (b) Branches of periodic solutions emerge from the steady state within the interval $\alpha\in\left[-\left|R\right|,\left|R\right|\right]$. \[pic: SLpsolEll\]](sl_hopf+psol)

### Stability of plane wave solutions

The local asymptotic stability of plane wave solutions can be studied using the linearized equation for small perturbations $\delta z_{m,n}(t)$. In co-rotating coordinates $\xi_{m,n}$ determined by $$z_{m,n}(t)=e^{i(\Omega t-k_{1}m-k_{2}n)}u_{m,n}(t),\label{eq:COR}$$ the plane wave is just 