Next: Continuation of torus bifurcation Up: Continuation of fold bifurcation Previous: Mathematical definition   Contents

Bifurcations

In CL_MatCont there are five generic codim 2 bifurcations that can be detected along the fold curve:

A Generalized Hopf (GH) marks the end (or start) of an LPC curve.
To detect the generic singularities, we first define bp+5 test functions, where bp is the number of branch parameters:

In the yi expressions w is the vector computed in (60) and bi (branch parameter) is a component of a.

In the second expression ybp+1, we compute v1M by solving

é
ê
ê
ê
ê
ë
D-TA(t)
d0-d1
ó
õ
1

0 
F(u0,1(t))LC×M
ù
ú
ú
ú
ú
û






disc 
v1M = é
ê
ê
ê
ë
TF(u0,1(t))
0
0
ù
ú
ú
ú
û






disc 
.
(75)

By discretization we obtain

(j1W*)T é
ê
ê
ë
D-TA(t)
d0-d1
ù
ú
ú
û




disc 
=0.

To normalize (j1*)W1 we require åi=1Nåj=1m|((v1*)W1)(i-1)m+j|1=1. Then ò01 áj1*(t),v1(t)ñdt is approximated by (j1*)TW1LC×Mv1M and if this quantity is nonzero, j*1W is rescaled so that ò01 áj1*(t),v1(t)ñdt=1.

So the third expression for the normal form coefficient b becomes

b = 1

2

((j1W1*)T((B(t;v1M,v1M)C+2(A(t)v1(t))C)).

In the fourth expression, M is the monodromy matrix.

In the fifth expression, M2 = (M - In)2, restricted to the subspace without the two eigenvalues with smallest norm.

The number of branch parameters is not fixed. If the number of branch parameters is 3 then this matrix has three more rows and columns. This singularity matrix is automatically extended:

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