\section{Cubic Vector Field}

\noindent This model file contains the defining equations for a four parameter
planar vector field studied by Danglemayr and Guckenheimer \cite{dangle1}:
\begin{eqnarray*}
  \dot{x} & = & y \\
  \dot{y} & = & -(x^3 + r x^2 + n x + m ) + y(b-x^2) \; \; .
\end{eqnarray*}
Study of these equations are relevant to problems in fluid convection  and
the dynamics of simple models for chemical reactors. 
In \cite{dangle1} the unfolding for the full
system is described and, based on analytical and numerical evidence, it
is conjectured that Equation (1) is, in fact, a versal unfolding of the
system,
\begin{eqnarray*}
  \dot{x} & = & y \\ 
  \dot{y} & = & -x^3-x^2y  
\end{eqnarray*}
in the appropriate space of vector fields.
We note that for the special parameter values $r=m=0$ the equations above
are the normal form for a codimension-2 bifurcation equivarient with respect
to rotation by $\pi$ described by Takens \cite{takens1}.
\medskip

\noindent The data files {\bf cubic.dat} shows the loci of Hopf and saddle node
bifurcations in the $(n,b)$ parameter plane for a particular choice of $r$ and $m$.
Also shown is the phase portrait for a selected point in this parameter plane 
showing the two limit cycles which arise through Hopf Bifurcation.  These data
correspond to an instance of stability diagram (B) shown in Figure 2 of
Danglemayr and Guckenheimer \cite{dangle1}. 
