Computation of Normal Form and Unfolding of Codimension-3 Zero-Hopf–Hopf Bifurcation

Abstract

The computation of the normal form as well as its unfolding is a key step to understand the topological structure of a bifurcation. Though a lot of results have been obtained, it still remains unsolved for higher co-dimensional bifurcations. The main purpose of this paper is devoted to the computation of a codimension-3 zero-Hopf–Hopf bifurcation, at which a zero as well as two pairs of pure imaginary eigenvalues can be found from the matrix evaluated at the equilibrium point. Different distributions of eigenvalues are considered, which may behave in a non-semisimple form for 1:1 internal resonance. Based on the combination of center manifold and normal form theory, all the coefficients of normal forms and nonlinear transformations are derived explicitly in terms of parameters of the original vector field, which are obtained via a recursive procedure. Accordingly, a user friendly computer program using a symbolic computation language Maple is developed to compute the coefficients up to an arbitrary order for a specific vector field with zero-Hopf–Hopf bifurcation. Furthermore, universal unfolding parameters are derived in terms of the perturbation of physical parameters, which can be employed to investigate the local behaviors in the neighborhood of the bifurcation point. Here, we emphasize that though different norm forms based on different choices may exist, their topological structures are the same, corresponding to qualitatively equivalent dynamics.