vanishing and infinite delay. However, also for moderate delays the approximation sufficiently matches the numerical results. \[pic:SLfirstHopf\]](SL_lattice_firstHopf){width="34.00000%"}

The illustration by the numerically computed eigenvalues is shown in Fig. \[pic: SL-eig-stst\] for the system of $3\times3$ coupled SL oscillators for three cases: stable, critical, and unstable. One can observe also how multiple Hopf bifurcation may emerge after the destabilization. In the following section we discuss the plane waves arising in these Hopf bifurcations.

![(a) Destabilization of the zero steady state in $3\times3$ lattice of delay coupled SL oscillators with $C=2$, $\beta=0.5$, and $\tau=20$. The plot shows numerically computed eigenvalues (solutions of Eq. (\[eq: SLststChEq-1\])) and the continuous large delay approximation (\[eq: SLststpcs\]) by the red line. At $\alpha=-2.5$ the stationary state is stable, $\alpha=-2$ is the critical case, and for $\alpha=-1.6$ an unstable steady state is shown. All eigenvalues accumulate along the curves $\gamma(\Omega,k_{-})$ given by Eq. (\[eq: SLststpcs\]) with maxima at $\Omega=\pm\beta$. (b) Destabilization of the stationary state in an infinitely large lattice, where $k_{-}$becomes a continuous parameter. The density plot shows the scaled real part of the eigenvalues $\operatorname{Re} \gamma = \lambda/\tau$. The parameters are the same as on the left. \[pic: SL-eig-stst\]](SL_2torus_stst_HDR+eig){width="\textwidth"}

Nonlinear plane waves and their stability
