A BPC can be characterized by adding two extra constraints
and
to (47) where
and
are the Branch Point test functions. The complete BVP defining a BPC point using the minimal extended system is
![$\displaystyle \left\{ \begin{array}{ll}
\frac{dx}{dt} - Tf(x,\alpha) & = 0 \\
...
...t),\dot x_{old}(t) \rangle dt & = 0 \\
G[x,T,\alpha] & = 0
\end{array} \right.$](img585.png) |
|
|
(84) |
where
is defined by requiring
 |
(85) |
Here
and
are functions,
and
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}](img588.png) |
(86) |
where the bordering operators
, function
, vector
and scalars
and
are chosen so that
is nonsingular [15][16].