Half-stable vs saddle node What is the difference between a half-stable and a saddle node in two and three dimensions?
 A: For the 1-D case, we can look at the fixed points for:
$$x' = x^2$$
This is considered a hybrid case of the stable and unstable case, so is called half-stable, since the fixed point is attracting from the left and repelling from the right.
This simple example sets the stage for what happens in higher dimensions.
In the 2-D case, we can look at Saddle-node bifurcation of cycles.
When two limit cycles coalesce and annihilate, this is called a fold or saddle-node bifurcation of cycles.
For example:
$$r' = \mu r + r^3-r^5$$
A saddle-node bifurcation occurs when $\mu_C = -\frac{1}{4}$. If you look at this in a 2-D space, these fixed points look like circular limit cycles. You would consider $\mu \lt \mu_C$, $\mu = \mu_C$ and $0 \gt \mu \gt \mu_C$.
At $\mu \lt \mu_C$, a stable single cycle exists, at $\mu_C$, a half-stable cycle is magically born. As $\mu$ increases, this splits off into a pair of limit cycles where one is stable and other is unstable. 
That should provide enough to do the 3-D case also - or you can search for it on the web.
