Given the following autonomous differential equations, which illustrates an elliptic limit cycle,
$r' = \alpha -\left(\frac{r}{a}\right)^2 \cos^2\theta - \left(\frac{r}{b}\right)^2 \sin^2\theta \\ \theta ' = -1$
Where, $r= r(t),\\ \theta = \theta(t),\\ a,b > 0 \\\alpha \quad \text{is a scaling constant, known from the equation of an ellipse as 1}$
Analyzing the system for possible bifurcation, where i startet with finding the Jacobian matrix, which gave the following eigenvalues,
$\lambda_{0} = 0 \\ \lambda_{2}=-\frac{2r\left(\cos^2\theta \cdot b^2+\sin^2\theta\cdot a^2\right)}{a^2b^2}$
I found the fix points for the system, but the size made it difficult to use them in any way.
Where did it go wrong? And are there something that might make this bifurcation analysis ane easier ?