I have some issues to understand the Hopf bifurcation. To keep it simple I would like to discuss the 2 dimensional case.
In that case the eigenvalues of the Jacobian at a critical point look like
\begin{equation} \lambda_{1/2} = \frac{1}{2}\left[\mu \pm \sqrt{\mu^2 - 4\beta} \right] \end{equation}
Assuming that $4\beta > \mu^2$ we have \begin{equation} \begin{matrix} \mu < 0: &\text{ stable spiral}\\ \mu = 0: &\text{ limit cycle}\\ \mu > 0: &\text{ non-stable spiral} \end{matrix} \end{equation}
I can test if the limit cycle is stable by adding a small value $\epsilon$ to the critical point and calculate the time-derivative.
A supercritical Hopf-Bifurcation occurs when a stable spiral ($\mu < 0 $) changes into a unstable spiral ($\mu > 0 $) surrounded by a elliptical limit cycle. [Strogatz1994]
In this image I understand how the hopf-bifurcation works:
negative real part: stable spiral
no real part: limit cycle
positive real part: unstable spiral
But the next image confuses me.
Here exists a limit cycle for non-zero real parts. I cannot explain the existance of the limit cycle. Can someone help me?
