# 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 $\mu$:

 $\displaystyle x^{\prime}$ $\displaystyle=$ $\displaystyle f_{1}(x,y,\mu)$ $\displaystyle y^{\prime}$ $\displaystyle=$ $\displaystyle f_{2}(x,y,\mu)$

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

 $\frac{d}{d\mu}\left(\Re\left(\lambda_{1,2}(\mu)\right)\right)_{|\mu=\mu_{c}}>0$

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

 $roman]{enumerate}\item\mu_{c}isabifurcationpoint;\item forsome\mu_{1}\in\bar{% \mathbb{R}}suchthat\mu_{1}<\mu<\mu_{c},theoriginisastablefocus;\item forsome% \mu_{2}\in\bar{\mathbb{R}}suchthat\mu_{c}<\mu<\mu_{2},theoriginisunstable,% surroundedbyastablelimitcyclewhosesizeincreaseswith\mu.\par Thisisasimplifiedversionofthetheorem% ,correspondingtoasupercriticalHopfbifurcation.\end{enumerate}\par SometimestheHopftheoremiscalled% \emph{Poincar\'{e}-Andronov-Hopf theorem}% sinceitwasindependentlydiscoveredbyAndronovin1929andHopfin1943andPoincar\'{e}% haddiscussionofsuchresultin1892.\@@cite[cite]{[\@@bibref{Refnum}{HK}{}{}]}% \thebibliography\bibitem[HK]{HK}Hale,JackH.\&Ko\,cak,H\"{u}seyin:% DynamicsandBifurcations.Springer-Verlag,NewYork,1991.\endthebibliography% \begin{flushright}\begin{tabular}[]{|ll|}\hline Title&Hopf bifurcation theorem% \\ Canonical name&HopfBifurcationTheorem\\ Date of creation&2013-03-22 13:18:45\\ Last modified on&2013-03-22 13:18:45\\ Owner&Daume (40)\\ Last modified by&Daume (40)\\ Numerical id&8\\ Author&Daume (40)\\ Entry type&Theorem\\ Classification&msc 34C05\\ Synonym&Poincar\'{e}-Andronov-Hopf\\ \hline\end{tabular}\end{flushright}\end{document}$