shown to increase with the delay. Further in Sec. \[sec:FHN-hom\] similar results are obtained for the FHN systems. Section \[sec:non\] considers the case when the delays are not identical. In this case it is shown how a variety of spatiotemporal patterns can be created by varying the coupling delays. Finally, additional illustrative examples are presented in Sec. \[sub:Ex\].

Stuart-Landau oscillators with homogeneous coupling delays\[sec:SLhom\]
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In this section we start with a lattice of SL oscillators with homogeneous delays $\tau_{m,n}^{\rightarrow}=\tau_{m,n}^{\downarrow}=\tau$ as a model for the local dynamics: $$\begin{aligned}
\begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}t}z_{m,n}(t) &= (\alpha+i\beta)z_{m,n}(t) - z_{m,n}(t)|z_{m,n}(t)|^{2}+ \\
&\quad + \frac{C}{2}\left(z_{m-1,n}(t-\tau)+z_{m,n-1}(t-\tau)\right).\label{eq: SL}
\end{aligned}\end{aligned}$$ The variables $z_{m,n}$ are complex-valued. The parameter $\alpha$ controls the local dynamics without coupling, i.e. a stable steady state exists for $\alpha<0$ and a stable limit cycle for $\alpha>0$; $\beta$ is the frequency of this limit cycle. The coupling strength is determined by $C>0$.

It this section we study the bifurcation scenario, which is associated with the destabilization of the homogeneous steady state $z=0$ and the appearance of various plane waves. Many aspects of this scenario can be studied analytically due to the $S^{1}$ equivariance of the system: $\mathbf{F}(e^{i\nu}\cdot,e^{i\nu}\cdot)=e^{i\nu}\mathbf{F}(\cdot,\cdot)$ for any real $\nu$. At some places we will assume additionally 