Branch Point Locator

The location of BPC points in the non-generic situation (i.e. where some symmetry is present) as zeros of the test functions is numerically suspect because no local quadratic convergence can be guaranteed. This difficulty can be avoided by introducing an additional unknown $\beta\in \ensuremath{\mathbf{R}}$ and considering the minimally extended system:

$\displaystyle \left\{ \begin{array}{ll}
\frac{dx}{dt} - Tf(x,\alpha) +\beta p_1...
...{old}(t) \rangle dt+\beta p_3 & = 0 \\
G[x,T,\alpha] & = 0
\end{array} \right.$     (50)

where $G$ is defined as in (84) and $[p_1^T p_2^T p_3]^T$ is the bordering vector $[w_{01};w_{02};w_{03}]^T$ in (86). We solve this system with respect to $x,T,\alpha$ and $\beta$ by Newton's method with initial $\beta=0$. A branch point $(x,T,\alpha)$ corresponds to a regular solution $(x,T,\alpha,0)$ of system (50) (see [4],p. 165). We note, however that the second order partial derivatives (Hessian) of $f$ with respect to $x$ and $\alpha$ are required. The tangent vector $v_{1st}$ at the BPC singularity is approximated as $v_{1st}=\frac{v_1+v_2}{2}$ where $v_1$ is the tangent vector in the continuation point previous to the BPC and $v_2$ is the one in the next point.