# Hopf bifurcation and limit cycles: problem in understanding

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

$$\lambda_{1/2} = \frac{1}{2}\left[\mu \pm \sqrt{\mu^2 - 4\beta} \right]$$

Assuming that $4\beta > \mu^2$ we have $$\begin{matrix} \mu < 0: &\text{ stable spiral}\\ \mu = 0: &\text{ limit cycle}\\ \mu > 0: &\text{ non-stable spiral} \end{matrix}$$

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?

• Sorry for my bad english. I unstable spiral has eigenvalues with non-zero real and imaginary parts. The real part of the eigenvalue of a limit cycle is zero. So why can spirals and limit cycles coexist? – Alexander Tille May 18 '17 at 7:32
• The type of limit cycle is determined by eigenvalues of Jacobi matrix that is computed at fixed point of Poincaré mapping. In your case this mapping is one-dimensional interval mapping and Jacobi matrix computed at fixed point has only one real eigenvalue. That's not in accordance with your undestanding of limit cycle. – Evgeny May 18 '17 at 7:55
• Oh, I get it now. You are confusing terminology and linear case with non-linear. Limit cycle is a closed trajectory which has no closed trajectories in some of its neighbourhoods. When you have linear system with purely imaginary eigenvalues, you don't have limit cycles: all trajectories are closed and in any neighbourhood of any trajectory there is another closed trajectory. Andronov-Hopf bifurcation shows what really happens in non-linear system: your intution and habits from linear case don't work here, in typical case only one closed trajectory appears after the bifurcation. – Evgeny May 19 '17 at 21:04
• Ok, thanks. I think I got your point – Alexander Tille May 22 '17 at 12:50