Hopf bifurcations of ODE systems along the singular direction in the parameter plane (Q2566922)

The paper considers Hopf bifurcations of ordinary differential systems with two parameters along the singular direction (when this direction is nondegenerate) in the parameter plane. Hopf bifurcations of the same system along nonsingular directions have been partly studied recently by \textit{R. Zhang, J. Wei} and \textit{Q. Huang} [Int. J. Bifurcation Chaos Appl. Sci. Eng. 13, 1545--1560 (2003; Zbl 1061.34032)]. As an application of the main theoretical results, the existence and nonexistence for a degenerate Hopf bifurcation type (i.e., the type without transversality) of one-parameter ordinary differential systems are studied. But the stability of the bifurcated periodic solutions is not addressed.