Center problem and \(\nu \)-cyclicity of polynomial zero-Hopf singularities with non-singular rotation axis
The family of systems \[ \dot{x}=-y +F_1(x,y,z;\lambda), \dot{y}=x +F_2(x,y,z;\lambda), \dot{z}=F_3(x,y,z;\lambda) \] is considered, where \(F_k\) are polynomials of \((x,y,z)\) without constant or linear terms and \(\lambda\in\mathbb{R}^p\) collects the set of their coefficients. It is also assumed \(F_1=F_2=0\), \(F_3\not\equiv 0\) at \(x=y=0\). The purpose of the paper is to obtain some three-dimensional extension (near the origin) to the classical planar Poincaré-Lyapunov center problem and Bautin theory. In particular, the cubic truncated normal form of the system is considered as an example.
scientific article; zbMATH DE number 7379529