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Mathematical definition

In the MatCont / CL_MatCont toolbox Hopf curves are computed by minimally extended defining systems, cf. [9] § 4.3.4. The Hopf curve is defined by the following system

ì
ï
í
ï
î
f(u,a)
=
0,
gi1j1(u,a,k)
=
0
gi2j2(u,a,k)
=
0
(63)

with the unknowns u,a,k,(i1,j1,i2,j2) Î {1,2} and where g=(

g11
g12
g21
g22

) is obtained by solving

æ
ç
ç
è
fu2+kIn
Wbor
VborT
O
ö
÷
÷
ø
æ
ç
ç
è
V
G
ö
÷
÷
ø
= æ
ç
ç
è
0n,2
I2
ö
÷
÷
ø
,
(64)

where fu has eigenvalues ±iw,w > 0, k=w2 and Vbor,Wbor Î Rn×2 are chosen such that the matrix in (64) is nonsingular. i1,j1,i2,j2,Vbor and Wbor are auxiliary variables that can be adapted. This method is implemented in the curve definition file hopf.m.