Proving periodic orbits using Poincare Bendixson's Theorem and a trapping region

[Rubik]

Homework Statement

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?

Homework Equations

By using Poincare Bendixson's Theorem

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?

 

[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.

 

[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

 

[Dick]

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

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?

 

[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?

 

[Dick]

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?

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?

 

[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?

 

[Dick]

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?

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?

 

[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..

 

[Dick]

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..

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?

 

[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?

 

[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?

 

[Rubik]

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

 

[Dick]

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

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

 

[Rubik]

No problem thanks so much for your help!