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Mathematical definition

A Fold bifurcation of limit cycles (Limit Point of Cycles, LPC) generically corresponds to a turning point of a curve of limit cycles. It can be characterized by adding an extra constraint G=0 to (50) where G is the Fold test function. The complete BVP defining a LPC point using the minimal extended system is

ì
ï
ï
ï
í
ï
ï
ï
î
du

dt

- Tf(u,a)
= 0
u(0) - u(1)
= 0
ó
õ
1

0 
áu(t),
×
u
 

old 
(t) ñdt
= 0
G[u,T,a]
= 0
(72)

where G is defined by requiring

N1 æ
ç
ç
ç
è
v
S
G
ö
÷
÷
÷
ø
= æ
ç
ç
ç
ç
ç
è
0
0
0
1
ö
÷
÷
÷
÷
÷
ø
.
(73)

Here v is a function, S and G are scalars and

N1 = é
ê
ê
ê
ê
ê
ë
D-Tfu(u(t),a)
    -f(u(t),a)
   w01
d1-d0
    0
    w02
Intf(u(·),a)
    0
    w03
Intv01
    v02
    0
ù
ú
ú
ú
ú
ú
û
(74)

where the bordering functions v01,w01, vector w02 and scalars v02 and w03 are chosen so that N1 is nonsingular []. This method (using system (73) and (74)) is implemented in the curve definition file limitpointcycle. The discretization is done using orthogonal collocation in the same way as it was done for limit cycles.