### Bifurcations along a Hopf curve

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

- Bogdanov - Takens. We will denote this bifurcation by `BT`
- Zero - Hopf point, denoted by `ZH`
- Double - Hopf point, denoted by `HH`
- Generalized Hopf point, denoted by `GH`

To detect these singularities, we first define 4 test functions:

- (first Lyapunov coefficient, see (40)),

where , are carefully constructed and updated matrices.

In this case the singularity matrix is:

| (63)