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Bifurcations

In continuous-time systems there are four generic codim 2 bifurcations that can be detected along the 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 (59) and(60) respectively, and bi (branch parameters) are components of a. The singularity matrix for bp=0 is:

S = æ
ç
ç
ç
è
0
0
-
1
0
-
-
-
0
ö
÷
÷
÷
ø
(61)

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:

S = æ
ç
ç
ç
ç
ç
ç
è
0
0
-
-
-
1
0
-
-
-
-
-
0
-
-
-
-
-
0
-
-
-
-
-
0
ö
÷
÷
÷
÷
÷
÷
ø