Consider the system:
$\dot{x_{1}}=x_{2}+x_{1}(1-x_{1}^{2}-x_{2}^{2})$
$\dot{x_{2}}=-x_{1}+x_{2}(1-x_{1}^{2}-x_{2}^{2})$
Transform the system into polar coordinates and use this to show that the unit circle is an exponentially stable limit cycle on $D=\left\{ x:x_{1}^{2}+x_{2}^{2}\geq\epsilon>0\right\}$
I have done the transformation to polar coordinates with $x_{1}=rcos(\theta ) , x_{2}=rsin(\theta)$ and got the system:
$\dot{\theta}=-1$
$\dot{r}=r(1-r^{2})$
And now I am not sure how to proceed. Intuitively I would like to use a Poincare map and show that $\frac{\partial P}{\partial r}(r^{*})<1$ where $r^{*}=1$ but I don't know how to do it in practice or even if this is the correct approach.
edit: as suggested solving the ODE yields
$P(r(0),\theta(0))=(\frac{e^{t}}{\sqrt{−1+r(0)−2+e^{2t}}},t+\theta(0))$
and for the strip $H:=\left\{ (r,\theta):r>0,\theta=2k\pi,k\in Z\right\}$ with return time to $H$, $ \tau(r(0),2\pi)=2\pi$
I eventually get:
$P(r(0))=\frac{e^{2π}}{\sqrt{−1+r(0)−2+e^{4π}}},r(0)\neq0⇒P(1)=1$
and $|\frac{\partial P}{\partial r(0)}(1)|=|e^{-4\pi}|<1$ but this only shows local asymptotic stability. I don't know what needs to be modified to make it show exponential stability on the aforementioned $D$