# Prove there are at least two periodic solutions

Could anyone comment on the following ODE problem? Thank you.

Given a 2-d system in polar coordinates: $$\dot{r}=r+r^{5}-r^{3}(1+\sin^{2}\theta)$$ $$\dot{\theta}=1$$

Prove that there are at least two nonconstant periodic solutions to this system.

It's easy to prove that there is a noncostant periodic solution using Poincare-Bendixson theorem, but I don't know how to prove the existantce of two nonconstant periodic solutions.

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My guess (not sure if correct): if a spiral one way is a solution, then maybe the same spiral in the opposite direction will also be a solution... – Adam Rubinson May 8 '12 at 3:03
Yeah, if $(r(t), \theta(t))$ is nonconstant, periodic, and solves the system, then $(r(t), \theta(t) + \pi)$ (aka $(-r(t), \theta(t))$, if you allow negative "$r$"'s in your polar coordinates) also has these properties. – leslie townes May 8 '12 at 3:12
@leslietownes Yes this is the answer. Thank you very much! – user7762 May 8 '12 at 16:09
Nice question. But it appears $\dot r \ge r + r^5 - 2 r^3 = r (1 - r^2)^2 \ge 0$. So $r$ should go straight to infinity, shouldn't it? Then how come there is any periodic solution? – hbp Oct 28 '15 at 22:19