\section{The Double Toroid}

This dynamical system has an invariant manifold which is a genus
two compact manifold.  The system is a singular perturbation and is created
by considering relaxation oscillators, see Deng \cite{Deng:constructing}.
The equations are
\begin{eqnarray*}
x' & = & a(1.5-z)x(x^2-8)/16+b z(\alpha (\alpha x + \beta y) (8-(\alpha x + \beta y)^2) /16
	- \beta^2 x + \alpha y) \\
y' & = & a(1.5-z)y+b z(\beta (\alpha x + \beta y )(8-(\alpha x + \beta y)^2)/116 + \alpha
	(\beta x + \alpha y)) \\
z' & = & z(1.5-z)(z-1+H(x,y))/\epsilon+(.75-z)
\end{eqnarray*}
where
\begin{eqnarray*}
H(x,y) & = & (x-y)^4/64 -(x^2-y^2)/2+(x+y)^2/4 \\
\alpha & = & cos(\theta) \\
\beta & = & sin(\theta)
\end{eqnarray*}
This is a three dimensional system with parameters $a,b,\theta$ and
singular perturbation parameter $\epsilon$.

The file {\tt double\_toroid1.dat} contains a set of parameter values where
the invariant two-manifold contains two attracting periodic orbits.



\section{The Perturbed Double Toroid}

This dynamical system has an invariant manifold which is a genus
two compact manifold.  The system is a singular perturbation and is created
by considering relaxation oscillators, see Deng \cite{Deng:constructing}.
The equations are
\begin{eqnarray*}
x' & = & a(1.5-z)(-\frac{dH}{dy})/16+b z(\alpha (\alpha x + \beta y) (8-(\alpha x + \beta y)^2) /16
	- \beta^2 x + \alpha y) \\
y' & = & a(1.5-z)(\frac{dH}{dx}+cy)+b z(\beta (\alpha x + \beta y )(8-(\alpha x + \beta y)^2)/116 + \alpha
	(\beta x + \alpha y)) \\
z' & = & z(1.5-z)(z-1+H(x,y))/\epsilon+(.75-z)
\end{eqnarray*}
where
\begin{eqnarray*}
H(x,y) & = & (x-y)^4/64 -(x^2-y^2)/2+(x+y)^2/4 \\
\alpha & = & cos(\theta) \\
\beta & = & sin(\theta)
\end{eqnarray*}
This is a three dimensional system with parameters $a,b,c,\theta$ and
singular perturbation parameter $\epsilon$.

The file {\tt perturbed\_dtoroid1.dat} contains a set of parameter values where
a chaotic attractor exists.
%The file {\tt perturbed\_dtoroid2} contains a set of parameter values where
%a knotted attracting periodic orbit exists.
