三次系统极限环及其稳定和分岔的一种算法

An Algorithm of Stability and Bifurcation of Limit Cycles for Cubic System

引进适当的参数,求出该参数近似为零时系统的解答;以此解答为初值,给参数以小增量(即参数摄动);将平面三次多项式微分系统极限环相图的x坐标假设为广义谐函数;将y坐标和频率作富氏展开;相应于参数的增量,得到极限环振幅、偏心距以及y坐标和频率的富氏系数的增量;用谐波平衡法得到以这些增量为独立变量的线性代数方程组;求解该方程组,得到各相关增量;以这些增量与初值的和为下一参数增量步骤相应的初值,重复上述过程,直至参数还原至原系统为止,从而得到极限环及其频率、周期、稳定性指标,以及极限环关于参数分岔曲线的近似解析表达式。文末给出算例。

With a suitable parameter, the solution of the system is solved as this parameter equaled zero. This solution is taken as the initial value, and the parameter is given a small increment. The x coordinate of limit cycle phase portraits for planar cubic polynomial differential systems are supposed as the generalized harmonic function. And the y coordinate and the frequency of limit cycle are expanded as Fourier series. Corresponding to the increments of the parameter, the increment of the amplitude, eccentricity and the Fourier coefficients ofy coordinate and the frequency of limit cycle are obtained. The linear algebra equations about these increments are got with harmonic balance. Solving these equations, these increments are obtained. The procedure is repeated with the initial value of the next step as the sum of the increments and the initial value, until the parameter is returned to original state. And then the approximate analytical expressions of frequency, periodic, stability index and bifurcation of limit cycles about the parameter are calculated. An example is shown at the end.