Normal Form and Unfolding of Vector Field with Codimension-3 Triple Hopf Bifurcation

Abstract

The universal unfolding of a normal form can be employed to reveal the general behaviors of a specific local bifurcation, while the computation of the normal form for high codimensional bifurcation still remains unsolved. This paper focuses on a vector field with codimension-3 triple Hopf bifurcation. Besides 1:1 internal resonance for two frequencies in semi-simple form, two cases are considered, corresponding to internal resonance and noninternal resonance between the first two frequencies and the third frequency, respectively. Based on a combination of center manifold and normal theory, all the coefficients in the normal form and the nonlinear transformation are derived explicitly in terms of the coefficients of the original vector field. Upon the recursive procedure established, a user friendly computer program can be easily developed using a symbolic computation language Maple to compute the coefficients up to an arbitrary order for a specific vector field with triple Hopf bifurcation. Furthermore, universal unfolding of the normal form is obtained, which can be used to display the topological structure in the neighborhood of bifurcation point. It is pointed out that different choices of the remaining terms in the nonlinear transformation may lead to different expressions of the normal form and the unfolding, which are qualitatively equivalent to each other.