High-order Normal Forms of Nilpotent Vector Fields with Symmetry

Normal forms, which bear all the dynamics properties of the original system at the area neighbouring the equilibrium point, are powerful analytical tools for studying problems concerning dynamic bifurcation of nonlinear vector fields. On the basis of Ushiki normal form theory, the nonlinear vector fields, if their Jacobian matrices of the linear parts are nilpoten matrices with γ-symmetry, their third- or fifth-order normal forms with γ-symmetry in one-dimensional and two-dimension nilpotent vetor fiels can be worked out by aid of infinitesimal deformation method and further deduced and proved is the k-order normal form of the vector fields with a retrogressed 1-section and γ-symmetry.