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Mathematical Definition

A BPC can be characterized by adding two extra constraints G1=0 and G2=0 to (50) where G1 and G2 are the Branch Point test functions. The complete BVP defining a BPC point using the minimal extended system is

ì
ï
ï
ï
í
ï
ï
ï
î
dx

dt

- Tf(x,a)
= 0
x(0) - x(1)
= 0
ó
õ
1

0 
áx(t),
×
x
 

old 
(t) ñdt
= 0
G[x,T,a]
= 0
(86)

where

G= æ
ç
ç
è
G1
G2
ö
÷
÷
ø

is defined by requiring

N æ
ç
ç
è
v1
v2
G1
G2
ö
÷
÷
ø
= æ
ç
ç
ç
ç
ç
ç
è
0
0
0
0
0
0
1
0
0
1
ö
÷
÷
÷
÷
÷
÷
ø
.
(87)

Here v1 and v2 are functions, G1 and G2 are scalars and

N = é
ê
ê
ê
ê
ê
ê
ë
D-Tfx(x(·),a)
    -f(x(·),a)
    -Tfb(x(·),a)
   w01
d1-d0
    0
    0
    w02
Int[x\dot]old(·)
    0
    0
    w03
v11
    v12
    v13
    0
v21
    v22
    v23
    0
ù
ú
ú
ú
ú
ú
ú
û
(88)

where the bordering operators v11,v21, function w01, vector w02 and scalars v12,v22,v13,v23 and w03 are chosen so that N is nonsingular [7][8].