Bifurcations along a fold curve

In continuous-time systems there are four generic codim 2 bifurcations that can be detected along a fold curve: To detect these singularities, we first define $bp+3$ test functions, where $bp$ is the number of branch parameters: In these expressions $v, w$ are the vectors computed in (56) and(57) respectively, $\odot$ is the bialternate matrix product, $v_1$, $w_1$ are $\frac{n(n-1)}{2}$ vectors chosen so that the square matrix in (58) is non-singular, and $\beta_i$ (branch parameters) are components of $\alpha$. The singularity matrix for $bp=0$ is:
\begin{displaymath}
S = \left(\begin{array}{ccc}
0 & 0 & -\\
1 & 0 & -\\
- & - & 0
\end{array}\right)
\end{displaymath} (59)

The number of branch parameters is not fixed. If the number of branch parameters is $2$ then this matrix has two more rows and columns. This singularity matrix is automatically extended:

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