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:
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: