Mathematical definition

In the MATCONT / CL_MATCONT toolbox Hopf curves are computed by minimally extended defining systems, cf. [22] §4.3.4. The Hopf curve is defined by the following system
\begin{displaymath}
\left\{
\begin{array}{ccl}
f(u,\alpha) & = & 0, \\
g_{i...
...0\\
g_{i_2j_2}(u,\alpha,k) & = & 0
\end{array}
\right.
\end{displaymath} (61)

with the unknowns $u,\alpha,k,(i_1,j_1,i_2,j_2)\in\{1,2\}$ and where $g=\left(
\begin{array}{cc}
g_{11} & g_{12} \\
g_{21} & g_{22}
\end{array}
\right)$ is obtained by solving
\begin{displaymath}
\left(
\begin{array}{cc}
f_u^2+kI_n & W_{bor} \\
V_{bor...
...ft(
\begin{array}{c}
0_{n,2}\\ I_2
\end{array}
\right),
\end{displaymath} (62)

where $f_u$ has eigenvalues $\pm i\omega,\omega>0$, $k=\omega^2$ and $V_{bor},W_{bor} \in \ensuremath{\mathbf{R}}^{n\times 2}$ are chosen such that the matrix in (62) is nonsingular. $i_1,j_1,i_2,j_2,V_{bor}$ and $W_{bor}$ are auxiliary variables that can be adapted. This method is implemented in the curve definition file hopf.m.