Next: Continuation of codim 2 Up: Continuation of torus bifurcation Previous: Mathematical definition   Contents

Bifurcations

In continuous-time systems there are eight generic codim 2 bifurcations that can be detected along the torus curve:

To detect these singularities, we first define 6 test functions:

where v1M is computed by solving

é
ê
ê
ë

D-TA(t)+iqI

d0-d1

ù
ú
ú
û





D 

v1M=0.

(80)

The normalization of v1M is done by requiring åi=1Nåj=0msjá(v1M)ij,(v1M)ijñti=1 where sj is the Gauss-Lagrange quadrature coefficient. By discretization we obtain

(v1W*)H

é
ê
ê
ë

D-TA(t)+iq

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. We compute j1W* by solving

(j1W*)T

é
ê
ê
ë

D-TA(t)

d0-d1

ù
ú
ú
û





disc 

=0

and normalize j1W1* by requiring åi=1Nåj=1m|((j1*)W1)ij|1=1. Then ò01 áj1*(t),F(u0,1(t))ñdt is approximated by (j1*)TW1(F(u0,1(t)))C and if this quantity is nonzero, j*1W is rescaled so that ò01 áj1*(t),F(u0,1(t))ñdt=1. We compute h20,1M by solving

é
ê
ê
ë

D-TA(t)+2iqI

d0-d1

ù
ú
ú
û





disc 

h20,1M =

é
ê
ê
ë

B(t,v1M(t),v1M(t))

0

ù
ú
ú
û

.

a1 can be computed as (j*W1)T(B(t,v1M,[`v]1M))C.

The computation of (h11,1)M is done by solving

é
ê
ê
ê
ë

(D-TA(t))C×M

d(0)-d(1)

(j*W1)TLC×M

ù
ú
ú
ú
û

(h11,1)M =

é
ê
ê
ê
ê
ë

B(t;v1M,


v

 


1M 

))C - a1(F(u0,1(t)))C

0

0

ù
ú
ú
ú
ú
û

The expression for the normal form coefficient d becomes

d =

1


2

((v1W1*)T,(B(t;h11,1M,v1M))C+(B(t;h20,1M,


v

 


1M 

))C+

1


T

(C(t;v1M,v1M,


v

 


1M 

))C)

-

a1


T

(v1W1*)T(A(t)v1(t))C+

ia1q


T2

.

In the 7th test function, M is the monodromy matrix.

In the 8th test function, M2 = (M - In)2, restricted to the subspace without the two eigenvalues with smallest norm.


The singularity matrix is:

S =

æ
ç
ç
ç
ç
ç
ç
ç
è

0

-

-

-

-

-

-

0

-

-

-

-

-

-

0

-

-

-

-

-

-

0

-

-

-

-

-

-

0

-

-

-

-

-

1

0

ö
÷
÷
÷
÷
÷
÷
÷
ø

.

(81)