ansatz $\lambda=\frac{1}{\tau}\gamma(\Omega)+i\Omega$ in the characteristic equation (\[eq: SLststChEq-1\]). One obtains $$i\Omega=\alpha+i\beta+C\cos\left(k_{-}\right)e^{ik_{+}}e^{-\gamma-i\Omega\tau},$$ where the small term $\gamma/\tau$ has been neglected. It can be solved as $$Y:=e^{-\gamma}e^{i(k_{+}-\Omega\tau)}=-\frac{\alpha+i(\beta-\Omega)}{C\cos{\left(k_{-}\right)} }$$ and finally for the curve $\gamma(\Omega)$ we obtain $$\gamma(\Omega)=-\frac{1}{2}\log|Y|^{2}=-\frac{1}{2}\log\left(\frac{\alpha^{2}+(\beta-\Omega)^{2}}{C^{2}\cos^2{\left(k_{-}\right)} }\right).\label{eq: SLststpcs}$$ Note that the spectrum is invariant with respect to complex conjugation. It is easy to see that the spatial mode $k_{-}=0$ corresponds to the maximal values of $\gamma(\Omega)$. Thus, the spatial modes with $k_{-}=0$ are most unstable. Moreover, it is easy to check that the leading value of $\gamma$ is negative if $\left| \alpha \right|>C$ and positive otherwise. This implies that the homogeneous steady state is asymptotically stable for $\alpha<-C$ and unstable for $\alpha>-C$ (we take also into account that there is a strongly unstable spectrum for $\alpha>0$). Hence, the destabilization takes place at $\alpha=-C$ via Hopf bifurcation at the frequency $\Omega\approx\beta$. Here we should remark that our conclusions are rigorous for long enough delay $\tau$. In our case, however, the destabilization threshold is well approximated by this value already for moderate delays, see Fig. \[pic:SLfirstHopf\].

![Destabilization threshold of the homogeneous stationary state in lattices of coupled Stuart-Landau oscillators as a function of the coupling delay. The asymptotic result $\alpha_{H}=-C$ is exact for 