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Bifurcations

In MatCont / CL_MatCont there are four generic codim 2 bifurcations that can be detected along the flip curve:

To detect these singularities, we define 4 test functions:

where c is the coefficient defined in (55), M is the monodromy matrix and \odot is the bialternate product.

The v1's and y1's are obtained as follows. For a given z Î C1([0,1],\mathbb Rn) we consider three different discretizations:

Formally we further introduce LC×M which is a structured sparse matrix that converts a vector zM of values in the mesh points into a vector zC of values in the collocation points by zC=LC×MzM. We compute v1M by solving

é
ê
ê
ë
D-TA(t)
d0+d1
ù
ú
ú
û




disc 
v1M=0.
(69)

The normalization of v1M is done by requiring åi=1Nåj=0msjá(v1M)(i-1)m+j,(v1M)(i-1)m+jñDi=1 where sj is the Gauss-Lagrange quadrature coefficient and Di is the length of the i-th interval. By discretization we obtain

(v1W*)T é
ê
ê
ë
D-TA(t)
d0+d1
ù
ú
ú
û




disc 
=0.

To normalize (v1*)W1 we require åi=1Nåj=1m|((v1*)W1)ij|1=1. Then ò01 áv1*(t),v1(t)ñdt is approximated by (v1*)TW1LC×Mv1M and if this quantity is nonzero, v*1W is rescaled so that ò01 áv1*(t),v1(t)ñdt=[1/2]. We compute y1W* by solving

(y1W*)T é
ê
ê
ë
D-TA(t)
d0-d1
ù
ú
ú
û




disc 
=0

and normalize y1W1* by requiring åi=1Nåj=1m|((y1*)W1)ij|1=1. Then ò01 áy1*(t),F(u0,1(t))ñdt is approximated by (y1*)TW1(F(u0,1(t)))C and if this quantity is nonzero, y*1W is rescaled so that ò01 áy1*(t),F(u0,1(t))ñdt=1. a1 can be computed as (y*W1)T(B(t,v1M,v1M))C. The computation of (h2,1)M is done by solving

ì
ï
í
ï
î
(D-TA(t))C×M(h2,1)M
=
(B(t;v1M,v1M))C + 2a1(F(u0,1(t)))C, t Î [0,1]
(d(0)-d(1))(h2,1)M
=
0,
(y*W1)TLC×M(h2,1)M
=
0.

The expression for the normal form coefficient c becomes

c= 1

3

((v*1W1)T(C(t;v1M, 1

T

v1M,v1M)C+3(B(t;v1M,(h2,1)M)C
-6 a1

T

(v*1W1)T(A(t)v1(t))C).

The singularity matrix is:

S = æ
ç
ç
ç
è
0
-
-
-
0
-
1
1
0
ö
÷
÷
÷
ø
.
(70)