What is the difference between a half-stable and a saddle node in two and three dimensions?

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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.

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+1 for my friend here. – Babak S. Mar 22 '13 at 19:58
@BabakS.: Thank you! – Amzoti Mar 22 '13 at 20:01
Thank you ... so is the idea roughly that half-stable limit cycles at the critical bifurcation value are analogous to half-stable points on the line? What about situations in which a fixed point in $\mathbb{R}^2$ is truly "half-stable," like where the left half-plane is unstable and the right stable? – tacos_tacos_tacos Mar 31 '13 at 9:55
@tacos_tacos_tacos Yes on the your first statement (it is useful to use information from thing we know and that is how these things were defined and built - up). The latter can happen and the above is an example. It is stable on one side of the plane, goes through a transition and then is stable on the other side of the plane. WHy would this be an issue? – Amzoti Mar 31 '13 at 12:46
+1 dear friend! – amWhy Apr 17 '13 at 0:51