Change ODE system to Polar to apply Poincare-Bendixson

[wrolsr]

Question:
Show that the system

x'= x-y-x[x^2 + (3/2)y^2]
y'= x+y -y[x^2 + (1/2)y^2]

has at least one periodic orbit.


I know that I need to apply Poincare-Bendixson Theorem. I can prove the first three points of it easily, but to create a trapping region, I believe that I need to switch this system to polar. I know that I need to make the substitutions y=r*sin(θ), dy= sin(θ)*dr + r*cos(θ) θ', x= r*cos(θ), dx= cos(θ)*dr - r*sin(θ)θ'. But when I do make the substitution, it makes the equations worse than before. Is there some other way to come up with a r min and r max for the trapping region? I just assumed it was a polar type question because its phase plane has a limit cycle on it.

 

[pasmith]

If V(x,y) = \frac12(x^2 + y^2) = \frac12 r^2 then
\dot V = x\dot x + y \dot y<br /> = x^2 + y^2 - x^4 - \frac{5}{2} x^2y^2 - \frac12 y^4
which after some rearrangement yields
\dot V = x^2 + y^2 - \left(x^2 + \frac54y^2\right)^2 + \frac{17}{16}y^4
Looking at that, you can see that there exist 0 &lt; m &lt; M such that if 0 &lt; r &lt; m then \dot V &gt; 0 (because the origin is unstable, so trajectories are locally away from it) and if r &gt; M then \dot V &lt; 0 (because if |y| is fixed then \dot V &lt; 0 for |x| sufficiently large), so that the trapping region is m \leq r \leq M.

I don't think it's necessary to calculate m and M; it is enough to show that they exist.