Hopf bifurcation theorem

Consider a planar system of ordinary differential equations, written in such a form as to make explicit the dependence on a parameter $\mu$:

$x^{\prime }$$=$$f_1(x,y,\mu )$

$y^{\prime }$$=$$f_2(x,y,\mu )$

Assume that this system has the origin as an equilibrium for all $\mu$. Suppose that the linearization $Df$ at zero has the two purely imaginary eigenvalues ${\lambda }_1(\mu )$ and ${\lambda }_2(\mu )$ when $\mu ={\mu }_c$. If the real part of the eigenvalues verify

$$\frac{d}{d\mu }{\left(\mathrm{\Re }\left({\lambda }_{1,2}(\mu )\right)\right)}_{|\mu ={\mu }_c}>0$$

and the origin is asymptotically stable at $\mu ={\mu }_c$, then