### Bifurcations along a fold curve

In continuous-time systems there are four generic codim 2 bifurcations that can be detected along a fold curve:

- Bogdanov - Takens. We will denote this bifurcation by `BT`
- Zero - Hopf point, denoted by `ZH`
- Cusp point, denoted by `CP`
- Branch point, denoted by `BP`

To detect these singularities, we first define test functions, where is the number of branch parameters:

- (cf. formula (39))

- | (58)

In these expressions are the vectors computed in (56) and(57) respectively, is the bialternate matrix product, , are vectors chosen so that the square matrix in (58) is non-singular, and (branch parameters) are components of . The singularity matrix for is:

| (59)

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