& -1 & 0 & 0\\
\epsilon & 0 & 0 & 0\\
0 & 0 & \xi & -1\\
0 & 0 & \epsilon & 0\end{array} \right)
\delta{\bf x}(t) 
+ \frac{1}{\epsilon} \left( \begin{array}{cccc}
0 & 0 & C & 0\\
0 & 0 & 0 & 0\\
C & 0 & 0 & 0\\
0 & 0 & 0 & 0\end{array} \right)    
\delta{\bf x}(t-\tau)\end{aligned}$$ where $\xi=1-a^2-C$. The ansatz $$\delta\bf{x}(t)=e^{\lambda t} \bf{u}$$ where $\bf{u}$ is an eigenvector of the Jacobian matrix, leads to the characteristic equation for the eigenvalues $\lambda$: $$(1- \xi \lambda+\epsilon \lambda^2 )^2-(\lambda C e^{-\lambda \tau })^2=0,$$ which can be factorized giving $$1- \xi \lambda+\epsilon \lambda^2 \pm \lambda C e^{-\lambda \tau }=0.
\label{char_eq}$$

This transcendental equation has infinitely many complex solutions $\lambda$. Fig. \[fig:eigenmodes\] shows the real parts of $\lambda$ for various values of $C$.

![\[fig:eigenmodes\] Real parts Re($\lambda$) of the eigenvalues of the fixed point vs. time delay $\tau$ for $a\!=\!1.05$, $\epsilon=0.01$, and (a) C=0.1, (b) C=0.4, (c) C=1, (d) C=2.](fig07.eps){width="50.00000%"}

As can be seen in Fig. \[fig:eigenmodes\] the real parts of all eigenvalues are negative throughout, i.e., the fixed point of the coupled system remains stable for all $C$. This can be shown analytically for $a>1$ 