Understanding Lorenz Equations & Stability: A Homework Guide

[piareround]

Homework Statement

http://imageshack.us/photo/my-images/535/newpicture3.jpg/

Homework Equations

Honestly i used the equation in Strogatz 10.3.3 so I am not sure how to do it for others.

The Attempt at a Solution

http://imageshack.us/photo/my-images/406/worke.png/
http://imageshack.us/photo/my-images/406/worke.png/
As you can see I found the fixed points by treating the problem as a map instead of as somehow related to the Lorenz Equations. I am really not sure how they want us to use the Lorentz euqations to do the same thing.
Moving on though I found the stability like I normally do by taking a derivative and setting that equal to zero. However, in this case I am really not sure how to interperate that derivative. Does anyone know after find the derivative you supposed to show stability for Lorenz systems?
Finally I am not really sure what they mean by period doubling orbit. I know what period doubling is and I know an orbit is like a circular trajectory, but I am really confused at what the two together are supposed to mean. Could anyone help clarify what the last portion of the question is asking and how you are supposed to show it? Prehaps by showing a similar example?

 

[piareround]

Sorry if that last post is confusing, I guess the question I really need answer is: what is period-2 orbit?