Mathematical Definition

A BPC can be characterized by adding two extra constraints $G_1=0$ and $G_2=0$ to (47) where $G_1$ and $G_2$ are the Branch Point test functions. The complete BVP defining a BPC point using the minimal extended system is
$\displaystyle \left\{ \begin{array}{ll}
\frac{dx}{dt} - Tf(x,\alpha) & = 0 \\
...
...t),\dot x_{old}(t) \rangle dt & = 0 \\
G[x,T,\alpha] & = 0
\end{array} \right.$     (84)

where

\begin{displaymath}
G=\left(\begin{array}{c}
G_{1}\\
G_{2}
\end{array}\right)\end{displaymath}

is defined by requiring
\begin{displaymath}
N\left(
\begin{array}{cc}
v_1&v_2\\
G_1&G_2
\end{arr...
...
0&0 \\
0&0\\
0&0\\
1&0\\
0&1
\end{array}\right).
\end{displaymath} (85)

Here $v_1$ and $v_2$ are functions, $G_1$ and $G_2$ are scalars and
\begin{displaymath}
N =
\left[
\begin{array}{cccc}
D-Tf_x(x(\cdot),\alpha) ...
...
v_{21} &~~~ v_{22}&~~~ v_{23} &~~~ 0
\end{array}
\right]
\end{displaymath} (86)

where the bordering operators $v_{11},v_{21}$, function $w_{01}$, vector $w_{02}$ and scalars $v_{12},v_{22},v_{13},v_{23}$ and $w_{03}$ are chosen so that $N$ is nonsingular [15][16].