Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
add comment

1 Answer

up vote 4 down vote accepted

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.

share|improve this answer
    
+1 for my friend here. –  B. 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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.