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Bifurcations

In continuous-time systems there are two generic codim 1 bifurcations that can be detected along the equilibrium curve (no derivations will be done here; for more detailed information see [10]):

The equilibrium curve can also have branching points. These are denoted with BP. To detect these singularities, we first define 3 test functions:

f1(u,a)
=
det
æ
ç
ç
è
Fx
vT
ö
÷
÷
ø
,
(35)
f2(u,a)
=
æ
ç
ç
ç
ç
ç
è
é
ê
ê
ë
(2fu(u,a) \odot In)
bw
bvT
d
ù
ú
ú
û
 \ æ
ç
ç
ç
ç
ç
è
0
...
0
1
ö
÷
÷
÷
÷
÷
ø
ö
÷
÷
÷
÷
÷
ø







n+1 
,
(36)
f3(u,a)
=
vn+1,
(37)

where \odot is the bialternate matrix product. Using these test functions we can define the singularities:

A proof that these test functions correctly detect the mentioned singularities can be found in [10]. Here we only notice that f2=0 not only at Hopf points but also at neutral saddles, i.e. points where fx has two real eigenvalues with sum zero. So, the singularity matrix is:

S = æ
ç
ç
ç
è
0
-
-
-
0
-
1
-
0
ö
÷
÷
÷
ø
(38)

For each detected limit point, the corresponding quadratic normal form coefficient is computed:

a= 1

2

pTfuu[q,q],
(39)

where fu q=fTu p = 0, pTq = 1. At a Hopf bifurcation point, the first Lyapunov coefficient is computed by the formula

l1= 1

2

Re ì
í
î
pT æ
è
fuuu[q,q,
-
q
 
]-2fuu[q,F-1ufuu[q,
-
q
 
]]+fuu[
-
q
 
,(2iwIn-fu)-1fuu[q,q]] ö
ø
ü
ý
þ
,
(40)

where fuq=iwq, fTu p=-iwp, [`p]Tq=1.