高维Hopf分岔系统的最简规范形

Simplest Normal Form of High-Dimensional Hopf Bifurcation System

针对高维Hopf动态分岔问题,研究了不经计算其传统规范形,直接计算高维任意阶数的Hopf分岔系统的最简规范形.利用中心流形定理,将原n维动力系统降为二维的中心流形,根据规范形理论,对中心流形上流的方程进一步化简,在不经过计算传统规范形的情况下,直接计算出其最简规范形中只包含的三阶和五阶项.编写了Mathematica程序,利用该程序,可直接由原n维动力系统计算出其最简规范形.通过3个算例验证了该方法的正确性和计算程序的高效性.

Aiming at the problem of Hopf dynamic bifurcation, a simplest normal form of any high-dimensional Hopf bifurcation system was obtained without calculating its traditional normal form. With the center manifold theory, the n-dimensional system was reduced to a two-dimensional center manifold. On the basis of normal form theory, the central manifold equations were further reduced to the simplest normal form which was only composed of the 3rd-order and 5th-order terms. A program in mathematica language was designed for immediate extraction of the simplest normal form from n-dimensional system. Three examples were given in order to verify the method and to show the efficiency of the program.