一类具有对称性的幂零向量场的高阶规范形

High-order Normal Forms of Nilpotent Vector Fields with Symmetry

规范形是研究非线性向量场的动分岔问题强有力的工具,它包含了原系统在平衡点附近的所有动力学特性.对于一类具有γ对称的线性部分Jacobian矩阵为幂零矩阵的非线性向量场,在Ushiki规范形理论的基础上,利用无穷小形变的方法,得到了一维和二维的幂零向量场的具有γ对称的三阶、五阶规范形,并推导和证明了具有γ对称的且1-节退化的向量场的k阶规范形.

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.