# Showing asymptotic orbital stability

1. Apr 6, 2017

### Mattbringssoda

1. The problem statement, all variables and given/known data

Show that every trajectory, except r = 0, is asymptotically orbitally stable

2. Relevant equations

I have a hint that: I should think of two initially close trajectories (say a(0) = |x(0) - y(0)| sufficiently small and see how the difference evolves over time (the rate a'(t).) I need to make |x(t)-y(t)| bounded.

3. The attempt at a solution

I think I'm getting confused by the symbology and I'm not sure how to operate the equations.

First I tried to calculate the fixed points. The only fixed point I get is the origin, r = 0. (If I tried r = 1, then I cant get theta ' = 0, so r = 0 must be the fixed point to look at, I'm guessing.)

So, next, I can see that r' = r(1-r) will go to zero as r increases in value. I think that's something I'm looking for.

Next, I think I need theta' to be bounded. Well, I know the first part, 1 + sin^2, will always be less than 2.....but there's nothing to bound (1-r)^2. As r increases, the term will just keep increasing without bound....

These are my thoughts on the problem so far and I don't know where to go with it from here......

Thanks for any help!

2. Apr 11, 2017

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.