Hopf bifurcation for some analytic differential systems in$\R^3$ via averaging theory

We study the Hopf bifurcation from the singular point witheigenvalues $a$ε$ \ \pm\ bi$ and $c $ε located at the origenof an analytic differential system of the form $ \dot x= f(x)$, where $x \in \R^3$. Under convenient assumptions we provethat the Hopf bifurcation can produce $1$, $2$ or $3$ limit cycles.We also characterize the stability of these limit cycles. The maintool for proving these results is the averaging theory of first andsecond order.