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Mathematical definition

In the toolbox fold curves are computed by minimally extended defining systems cf. [9], § 4.1.2. The fold curve is defined by the following system

ì
ï
í
ï
î
f(u,a)
=
0,
g(u,a)
=
0,
(58)

where (u,a) Î Rn+2, while g is obtained by solving

æ
ç
ç
è
fu(u,a)
wbor
vborT
0
ö
÷
÷
ø
æ
ç
ç
è
v
g
ö
÷
÷
ø
= æ
ç
ç
è
0n
1
ö
÷
÷
ø
,
(59)

and wbor,vbor Î Rn are chosen such that the matrix in (59) is nonsingular. An advantage of this method is that the derivatives of g can be obtained easily from the derivatives of fu(u,a):

gz = -wT(fu)z v

where z is a state variable or an active parameter and w is obtained by solving

æ
ç
ç
è
fuT(u,a)
vbor
wborT
0
ö
÷
÷
ø
æ
ç
ç
è
w
g
ö
÷
÷
ø
= æ
ç
ç
è
0n
1
ö
÷
÷
ø
,
(60)

This method is implemented in the curve definition file limitpoint.m.