一类快慢耦合Duffing-van der Pol系统的平衡点分析

Equilibrium Analysis of a Class of Fast-Slow Coupled Duffing-van der Pol System

本文主要研究了一类5次项和3次项共存的快慢耦合Duffing-van der Pol系统的平衡点问题。对系统求平衡点,得到了一个一元五次方程。使用一元次方程的降次解法,将平衡点方程的五次降到四次。当平衡点方程降到四次时,计算得到常数项的系数为零,所以可以直接降次为三次方程。采用盛金公式来对三次方程进行分类求解。综合这四类情形,得到了平衡点方程的解在F-u1平面上的个数分布。

This paper studies the equilibrium point problem of a class of fast-slow coupled Duffing-van der Pol system with the coexistence of quintic term and cubic term. Quintic equation of one variable is obtained by solving the equilibrium points of the system. By using the method of reducing the order of n degree equation of one variable, the order of the equilibrium equation is reduced from five to four. When the equilibrium equation is reduced to the quartic, the coefficient of the constant term is calculated to be zero, so it can be directly reduced to the cubic equation. Morigane formula is used to solve the cubic equation. The distribution of the solution of the equilibrium equation on the F-u1 plane is obtained by synthesizing these four cases.