Degenerate Fold-Hopf Bifurcations in a Rössler-Type System

We study the Hopf and the fold--Hopf bifurcations of the R\"ossler--type differential system * =-y-z, =x ay, =-cz byz, * with b 0. We show that the classical Hopf bifurcation cannot be applied to this system for detecting the fold--Hopf bifurcation, which here is studied using the averaging theory. Our results show that a Hopf bifurcation takes place at the equilibrium (-ac/b,c/b,-c/b) when c=a<0. This Hopf bifurcation becomes a fold--Hopf bifurcation when c=a=0