Fig. 3.

Illustration of the proofs for Theorems 9–12. Consider the trace and determinant of $\mathbf{Q}+\mu \mathbf{R}$ as polynomials in $\mu$ with $\mathbf{A}$’s eigenvalues ${\mu }_1,\dots ,{\mu }_k$ being points in the domain. By stability assumptions, $p_2\left({\mu }_i\right)>0$ and $p_1\left({\mu }_i\right)<0$ for all $i$ when $\lambda <{\lambda }_0$. If $p_1$ or $p_2$ has a root that is some ${\mu }_i$ a bifurcation may occur. The trace is linear in $\mu$, and when $\text{det}\left(\mathbf{R}\right)=0$, the determinant is linear. Assuming that one of the graphs intersects $\left({\mu }_i,0\right)$ for some $i$, we can determine those $i$ satisfying $p_1\left({\mu }_i\right)=0$ or $p_2\left({\mu }_i\right)=0$ from the slope of each line