# 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

$$ |