Given the following system
$\dot{x} = y$
$\dot{y} = y(9-x^2-2y^2) – x$
verify whether it has periodic solutions and if so are they attracting or repelling.
I thought:
The critical points or fixed point is (0,0) but is this correct and if the answer is yes then is that a periodic solution? And in general, how does one find out the periodic solution?
Finding fixed points is easy, you set $\dot{x}$ and $\dot{y}$ equal to zero and to verify the stability you find the derivatives of the fixed points. But I don't know how to find the periodic solutions…
Maybe rewriting the system in polar coordinates helps somehow?
Best Answer
In polar form we have $$r\dot{r} = x\dot{x} +y\dot{y} = r^2\sin^2\theta(9-r^2(1+\sin^2\theta)).$$ Now $$\dot{r} <0 \implies r^2 \ge\frac{9}{1+\sin^2\theta}\ge9$$ and $$\dot{r}>0 \implies r^2\le\frac{9}{1+\sin^2\theta} \le \frac{9}{2}$$ This tells us that orbits cannot leave the annulus $3/\sqrt{2}\le r\le3$.
By Poincare-Bendixson theorem, the limit set of the of orbits entering the annulus must be either a limit cycle, a fixed point, or some sort of homoclinic or heteroclinic connection. It is thus sufficient to show that there are no fixed points in this region.
Returning to the original equations; we find that the fixed point is at $(x,y)=(0,0)$. Which is not inside the annulus, hence the limit set must be a limit cycle, since the orbits can't leave the annulus, we know that the limit cycle must be stable. Hence, there is at least one stable limit cycle and it is inside the annulus.
Edit 1: Some comments
I don't have my Guckenheimer and Holmes text with me at the moment; but according to Scholarpedia, Poincare-Bendixson is sometimes stated as follows:
So the points $\theta=n\pi$ are no problem as the trajectories remain in the closed annulus.