Bifurcations along a Hopf curve

In continuous-time systems there are four generic codim 2 bifurcations that can be detected along a Hopf curve: To detect these singularities, we first define 4 test functions: where $v_1$, $w_1$ are carefully constructed and updated $\frac{n(n-1)}{2} \times 2$ matrices.

In this case the singularity matrix is:

\begin{displaymath}
S = \left(\begin{array}{cccc}
0 & 0 & - & -\\
1 & 0 & - & -\\
- & - & 0 & -\\
- & - & - & 0
\end{array}\right)
\end{displaymath} (63)