Bifurcation control and universal unfolding for Hopf-zero singularities with leading solenoidal terms (Q2808179)

The authors consider systems with a Hopf-zero singularity of the form NEWLINE\[NEWLINE \dot{x}= f(x,y,z), \quad \dot{y}= z+g(x,y,z), \quad \dot{z}=-y+h(x,y,z), NEWLINE\]NEWLINE for \((x,y,z) \in \mathbb{R}^3\) and where \(f,g,h\) do not have linear and constant terms. The authors study the \(n\)-universal asymptotic normal form of this system because it facilitates the (finitely determined) local bifurcation analysis in terms of the unfolding parameters. They compute the relations between the unfolding parameters and the parameters of the original system. This reveals the impact of control parameters on the nonlinear control system. The authors take cylindrical coordinates and make some slight assumptions on the parameters so as to prove their results. Bifurcation of equilibria and limit cycles are also studied.