Bifurcations along a flip curve

In MATCONT / CL_MATCONT there are four generic codim 2 bifurcations that can be detected along a flip curve: To detect these singularities, we define 4 test functions: where $c$ is the coefficient defined in (52), $M$ is the monodromy matrix and $\odot$ is the bialternate product.

The $v_1$'s and $\psi_1$'s are obtained as follows. For a given $\zeta \in {{\cal C}}^1([0,1],{\mathbb{R}^n})$ we consider three different discretizations:

Formally we further introduce $L_{C\times M}$ which is a structured sparse matrix that converts a vector $\zeta_M$ of values in the mesh points into a vector $\zeta_C$ of values in the collocation points by $\zeta_C=L_{C\times M}\zeta_M$. We compute $v_{1M}$ by solving

\begin{displaymath}
\left[\begin{array}{c}D-TA(t)\\
\delta_0+\delta_1
\end{array}\right]_{disc}v_{1M}=0.
\end{displaymath} (67)

The normalization of $v_{1M}$ is done by requiring $\sum_{i=1}^N\sum_{j=0}^m\sigma_j\langle(v_{1M})_{(i-1)m+j},(v_{1M})_{(i-1)m+j}\rangle\Delta_i=1$ where $\sigma_j$ is the Gauss-Lagrange quadrature coefficient and $\Delta_i$ is the length of the i-th interval. By discretization we obtain

\begin{displaymath}(v_{1W}^*)^T\left[\begin{array}{c}D-TA(t)\\ \delta_0+\delta_1\end{array}\right]_{disc}=0.\end{displaymath}

To normalize $(v_1^*)_{W_1}$ we require $\sum_{i=1}^N\sum_{j=1}^m\left\vert((v_1^*)_{W_1})_{ij}\right\vert _1=1$. Then $\int_{0}^{1} {\langle v_1^*(t),v_1(t)\rangle dt}$ is approximated by $(v_1^*)^T_{W_1}L_{C\times M}v_{1M}$ and if this quantity is nonzero, $v^*_{1W}$ is rescaled so that $\int_{0}^{1} {\langle v_1^*(t),v_1(t)\rangle dt}=\frac{1}{2}$. We compute $\psi_{1W}^*$ by solving

\begin{displaymath}(\psi_{1W}^*)^T\left[\begin{array}{c}D-TA(t)\\ \delta_0-\delta_1\end{array}\right]_{disc}=0\end{displaymath}

and normalize $\psi_{1W_1}^*$ by requiring $\sum_{i=1}^N\sum_{j=1}^m\left\vert((\psi_1^*)_{W_1})_{ij}\right\vert _1=1$. Then $\int_{0}^{1} {\langle \psi_1^*(t),F(u_{0,1}(t))\rangle dt}$ is approximated by $(\psi_1^*)^T_{W_1}(F(u_{0,1}(t)))_C$ and if this quantity is nonzero, $\psi^*_{1W}$ is rescaled so that $\int_{0}^{1} {\langle \psi_1^*(t),F(u_{0,1}(t))\rangle dt}=1$. $a_1$ can be computed as $(\psi^*_{W_1})^T(B(t,v_{1M},v_{1M}))_C$. The computation of $(h_{2,1})_M$ is done by solving

\begin{displaymath}
\left\{\begin{array}{rcl}
(D-TA(t))_{C\times M}(h_{2,1})_M...
...*_{W_1})^TL_{C\times M}(h_{2,1})_M&=&0.
\end{array}
\right.
\end{displaymath}

The expression for the normal form coefficient $c$ becomes

\begin{displaymath}
\begin{array}{c}
c=\frac{1}{3}((v^*_{1W_1})^T(C(t;v_{1M},\...
...
-6\frac{a_1}{T}(v^*_{1W_1})^T(A(t)v_{1}(t))_C).
\end{array}
\end{displaymath}

The singularity matrix is:
\begin{displaymath}
S = \left(\begin{array}{ccc}
0 & - & - \\
- & 0 & -\\
1 & 1 & 0
\end{array}\right).
\end{displaymath} (68)