### Bifurcations along a fold of cycles curve

In CL_MATCONT there are five generic codim 2 bifurcations that can be detected along a fold of cycles curve:

- Branch point. We will denote this bifurcation with `BPC`
- Strong 1:1 resonance. We will denote this bifurcation with `R1`
- Cusp point, denoted by `CPC`
- Fold - flip, Limit Point - Period Doublingor We will denote this bifurcation by `LPPD`
- Fold - Neimark-Sacker, denoted by `LPNS`

A Generalized Hopf (`GH`) marks the end (or start) of an LPC curve.
To detect the generic singularities, we first define test functions, where is the number of branch parameters:

- (normal form coefficient)

In the expressions is the vector computed in (57) and (branch parameter) is a component of .

In the second expression , we compute by solving

| (73)

By discretization we obtain

To normalize we require . Then is approximated by and if this quantity is nonzero, is rescaled so that .

So the third expression for the normal form coefficient becomes

In the fourth expression, is the monodromy matrix.

In the fifth expression, is the matrix that restricts the matrix to the space orthogonal to the two-dimensional left eigenspace of the two multipliers that are closest to .
The number of branch parameters is not fixed. If the number of branch parameters is then this matrix has three more rows and columns. This singularity matrix is automatically extended: