$3$ - dimensional Hopf bifurcation via averaging theory of second order

Abstract

We study the Hopf bifurcation occurring in polynomial quadraticvector fields in $\R^3$. By applying the averaging theory ofsecond order to these systems we show that at most $3$ limit cycles canbifurcate from a singular point having eigenvalues of the form$\pm bi$ and $0$. We provide an example of a quadratic polynomial differential system for which exactly $3$ limit cycles bifurcate from a such singularpoint.