Normal Forms Of Nilpotent Vector Fields And Codimension Two Bifurcation

Normal forms are powerful analytical tools for studying the qualitative behavior of nonlinear vector fields. Many authors have discussed the further reduction of normal forms and some of them have discussed uniqueness of normal forms.Ushiki introduced a systematic method by which nonlinear parts are also used to simplify higher order terms. In contrast to the classical method of normal form theory where only one Lie bracket is used to simplify the higher order terms while Ushiki's method allows more Lie brackets about the simplification.In this paper our formulation is based on the method of infinitesimal deformation and the theory of Ushiki's normal forms. We can derive various degenerate or non-degenerate 2-order and 3-order normal forms of nonlinear nilpotent vector fields. Some vector fields are imbued with symmetry. We can also obtain their normal forms by the method of infinitesimal deformation.Universal unfolding contains all perturbations of vector fields in equilibrium. So we can use universal unfolding to analyze the dynamical character of vector fields. We study the universal unfolding of 2-order normal forms.