Computation of the normal form as well as the unfolding of the vector field with zero-zero-Hopf bifurcation at the origin

Abstract

The normal forms of the vector fields with local bifurcations at the equilibrium points can be employed to describe the topological structure in the neighborhood of the critical points. Furthermore, the relationship between the coefficients of the normal form and the original system is very important to understand the behaviors of the practical dynamics. Though many results related to low co-dimensional local bifurcations are presented, the normal forms as well as the computation with high co-dimensional local bifurcations still remain an open problem to be investigated. The main purpose of this paper is to derive the normal form of a vector field with codimension-3 zero-zero-Hopf bifurcation at the origin and develop an uniform program to compute the coefficients of the normal form from a general system. By employing the central manifold theory and the normal form theory, all the expressions of the coefficients of the nonlinear transformations and the normal form up to any desired order related to the local bifurcation are presented, which can be computed via a software program based on the symbolic language Maple, attached in the appendix. Perturbation of the vector field at the bifurcation point can also be derived accordingly, which can be used to explore the topological property of the bifurcation.