between an unstable focus-node and a saddle-focus.

### Pseudo-continuous spectrum {#pseudo-continuous-spectrum .unnumbered}

The primary bifurcations of the steady state are captured by the pseudo-continuous spectrum. Just as in the previous section in the case of Stuart-Landau oscillators, this can be found by applying the ansatz $\lambda=\gamma/\tau+i\Omega$. Neglecting terms of order $\mathcal{O}(1/\tau)$ and introducing the variable $Y=e^{-\gamma}e^{i(k_{+}-\Omega\tau)}$ yields the modified characteristic equation $$\begin{aligned}
0 & =\det{\left|-i\Omega\operatorname{Id}+A+2B\cos k_{-}Y\right|}=\det{\left|\begin{array}{ccc}
a_{11}-i\Omega & a_{12} & 2b_{13}\cos{k_{-}}Y\\
a_{21} & a_{22}-i\Omega & 0\\
a_{31} & 0 & a_{33}-i\Omega
\end{array}\right|} = \\
 & =(a_{11}-i\Omega)(a_{22}-i\text{\ensuremath{\Omega}})(a_{33}-i\Omega)-2(a_{22}-i\Omega)a_{31}b_{13}\cos{k_{-}}Y-(a_{33}-i\Omega)a_{12}a_{21}.\end{aligned}$$ Due to the simple linear coupling-structure, this is a linear equation in $Y$, which can be solved as $$Y=\frac{a_{33}-i\Omega}{2a_{31}b_{13}\cos{k_{-}}}\left(a_{11}-i\Omega-\frac{a_{12}a_{21}}{a_{22}-i\Omega}\right),$$ leading to the asymptotic continuous spectrum $$\label{eq: FHNststhybrid}
\gamma(\Omega,k_{-}) = -\log{|Y(\Omega,k_{-})|}$$

This is a function of two parameters, determining the spectrum and stability of the steady state with respect to the perturbations with the spatial mode $k_{-}$ (and independent on $k_{+})$ and the delay-induced temporal modes $\Omega$. Some plots of this surface are illustrated in Fig. \[fig:FHMststhyb\]. Apparently the destabilization is similar to the case of Stuart-Landau oscillators described in Sec. \[sub:ststLong\]. The asymptotic weak spectrum is invariant with respect to $\Omega\to-\Omega$, $k_{-}\to-k_{-}$ and $k_{-}\to k_{-}+n\pi$, $n\in\mathbb{Z}$. The bifurcations of (\[eq: FHNststhybrid\]) lead to the 