Orbital Normal Forms of Hopf-zero Singularity

Consider a Dynamical system x'=F(x,µ) such that its linear part has a pair of imaginary eigenvalues and one zero eigenvalue (Hopf Zero singularity). Recently, the simplest normal form for this singular system has been obtained by sl(2) Lie algebra theory and the decomposition of space into variant and invariant subspaces. The normal form of this singular system is divided into three general cases. In this paper, the obtained results will be extended to orbital normal form and one of the three aforementioned cases will be discussed. The orbital obtained normal form will be simpler than the previous simplest normal form.