# Poincare Bendixson's Theorem

1. Apr 9, 2012

### Rubik

1. The problem statement, all variables and given/known data
System in polar coordinates
$\dot{r}$ = 2r - r3(2 + sin($\theta$)),
$\dot{\theta}$ = 3 - r2

Use a trapping region to show there is at least one periodic orbit?

2. Relevant equations

By using Poincare Bendixson's Theorem

3. The attempt at a solution

I am struggling to know where to start.. Do I start by considering
g($\theta$) = 2 + sin($\theta$). Any help would be greatly appreciated?

2. Apr 9, 2012

### Dick

Your g(θ) is always between 3 and 1. Try and find a small value r1 such that r' on the circle r=r1 is always positive. Now find a larger value of r2 such that r' on the circle r=r2 is always negative. Doesn't that make the region between the two circles a trapping region? Now think about fixed points.

3. Apr 10, 2012

### Rubik

Okay so does this mean

1≤g(θ)≤3 So you get

$\dot{r}$ = 2r - r3g(θ)
$\dot{r}$ > 2r - 3r3 > 0 and so 2/3 > r2

$\dot{r}$ < 2r - r3 < 0 and so r2 > 2

and √(2/3) < r < √2

4. Apr 10, 2012

### Dick

No, you don't really have to solve for anything. Pick a small value of r like r=1/2. Can you show r'>0 if r=1/2?

5. Apr 10, 2012

### Rubik

So if I have to give bounds for the periodic orbit I can pick any value for r(small) provided r'>0 and then choose another value (this time larger) and show r'<0? How would I then determine its stability characteristics?

6. Apr 10, 2012

### Dick

If you can find such values then you have a trapped region between the two circles. Can you have a fixed point in between? Try and figure out why not. Where can a fixed point be? Then use Poincare Bendixson clearly, yes?

Last edited: Apr 10, 2012
7. Apr 10, 2012

### Rubik

There can't be any fixed points in trapping region. There can be a fixed point at the origin? So to determing the stability I determine the fixed point at the origin?

8. Apr 10, 2012

### Dick

Sure there is a fixed point at the origin. But that doesn't matter since it's not in your trapping region. Then what does your theorem tell you?

9. Apr 10, 2012

### Rubik

So the flow enters the trapping region about the origin which has no critical points which means it is a stable periodic orbit in the trapping region..

10. Apr 10, 2012

### Dick

Not quite. It tells you there IS a periodic orbit in the trapping region. What's the exact statement of the theorem you are using?

11. Apr 10, 2012

### Rubik

Suppose $\alpha$(x0) enters and does not leave some closed and bounded domain D that contains no critical points. This means that $\phi$(x0, t) $\in$ D for all t≥$\tau$, for some $\tau$≥0. Then there is at least one periodic orbit in D and this orbit is in the $\omega$-limit set of x0.

What does this mean in terms of stability? I have no idea how to determine the stability?

12. Apr 10, 2012

### Dick

I'm maybe overextending myself here, but didn't the original problem ask you to show that there is a periodic orbit? I don't think it said anything about stability?

13. Apr 10, 2012

### Rubik

Yes it did, but I didn't realise there was a second part that asks to determine its stability characteristics..

14. Apr 10, 2012

### Dick

Ok, dynamical systems aren't really something I know well, so I might have to recuse myself here. Sorry.

15. Apr 10, 2012

### Rubik

No problem thanks so much for your help!