Any suggestions on analyzing the long-term behavior of this system? I've computed the Jacobian, but I'm not sure how to gain much insight about the behavior from this. The eigenvalues appear to be messy, but perhaps that's the right approach? Here's the system:
$\dot{x}=rxz$
$\dot{y}=ryz$
$\dot{z}=-rxz-ryz$
where $r$ is a non-zero constant.
The system $$\begin {cases}0=rxz\\ 0=ryz\\ 0=-rxz-ryz=%-\dot{x}-\dot{y} \end{cases}$$ has solutions $z=0$ and $x=y=0$ which are equilibrium points of the system.
The matrix $$\begin{bmatrix} rz & 0&rx\\ 0&rz&ry\\ -rz&-rz&-rx-ry \end{bmatrix}$$ is sinugular and has an eigenvalues $0$, $r z$, $r (-x - y + z)$.
Since you have a zero eigenvalue you have an unstable saddles at the equilibrium points, that is they are sort of "saddle-sinks" and "saddle-sources".
At $z=0$ the only non zero eigenvalue is $-r (x+ y)$, so whenever $x+y>0$ you have "saddle-sink" if $r>0$ or "saddle-source" if $r<0$. Similarly for $x+y<0$ you get "saddle-source" if $r>0$ or "saddle-sink" if $r<0$.
For $x=y=0$ we have multiple eigenvalue $rz$ which which is "saddle-sink" if $rz<0$ and source otherwise.
You could try to visualize it using Mathematica, but it is not that clear, so you may need to play with options of their plotting function.
Also note $$\frac{\dot y}{\dot x}=\frac{dy}{dx}=\frac{ryz}{rxz}=\frac{y}{x}$$ therefore $y=c x$ and $$\dot{z}=-rxz-ryz=-(1+c)rxz=-(1+c)\dot{x}$$ Thus, $$\frac{\dot z}{\dot x}=\frac{dz}{dx}=-(1+c)$$ which gives $z=-(1+c)x$