the minimal $\alpha=\alpha_{0}$ with $$\alpha_{0}(k_{-},k_{\tau})=R\cos k_{\tau}\frac{1-2(\cos^{2}k_{-}-\sin^{2}k_{\tau})}{\cos^{2}k_{-}-\sin^{2}k_{\tau}},$$ where a plane wave with particular $k_{-}$ and $k_{\tau}$ stabilizes.

The bifurcation diagram in Fig. \[fig:SLBD\] summarizes and illustrates the obtained results, showing the regions where the plane waves are stable (light gray), weakly unstable (darker gray, labeled with $U$ and $M$), and strongly unstable (dark gray, labeled with $S$).

![Stability diagrams for periodic solutions of the SL-system (\[eq: SL\]) with different values of $k_{-}$: (a) $k_{-}=0$ and (b) $k_{-}=\pi/4$. The regime of stable periodic solutions is filled with a light gray and is located above the $M$-labeled curve in this projection. The curves denote the upper boundaries of the different stability regimes discussed in the text: $M$ modulational instability, $U$ uniform instability and $S$ strong instability. The size of the stable regime decreases with the increasing of $k_{-}$. \[fig:SLBD\]](SL_stability_map_lattice_paper){width="0.75\linewidth"}

### Qualitative summary for the plane waves in coupled Stuart-Landau model

The main qualitative conclusions of this relatively technical section are as follows: The family of plane wave solutions (\[eq:persol\]) is located on the circles (\[eq: SLpsolCir\]) (for a fixed $k_{-}$), and the number of plane waves grows accordingly to (\[eq:Npsol\]). The stability of a plane wave is governed by the characteristic equation (\[eq: 