动力系统极限环计算方法的改进

A Modification of the Perturbation-Incremental Method for Limit Cycle Analysis of Dynamic Systems

对计算极限环的摄动一增量法作了改进，解的表达形式更一般化，更适合一般平面动力系统极限环及其分叉的计算．该方法的特点是用有限Ｆｏｕｒｉｅｒ级数给出极限环的近似表达式，把微分方程化为线性增量方程，应用谐波平衡法建立Ｆｏｕｒｉｅｒ系数的线性代数方程组，再用迭代法求解，计算方法直观、简单，求出了以前原方法难于计算的二次系统的４个极限环，也求出了其具有争议的算例的极限环的个数．算例表明该方法是有效的．求出了改进前的摄动－增量法所不能求出的极限环．

The perturbation-incremental method is modified for limit cycle analysis of dynamic systems. The solu- tion of the limit cycles is assumed to he a more general form such that the method is suitable for the computation of limit cycle and its bifurcation of the general planar dynamic systems. The essence of this method is to construct an approximate analytical expression in terms of finite Fourier series for the limit cycle. Firstly, the governing differen tial equations are transformed into linearized incremental equations. Then. a set of linearied algebraic equations in terms of Fourier coefficients is set up by using the harmonic balance method. This method is visual and simple. The numerical examples show that the present method is very efficient and accurate. Some limit cycles, which could not be found by using the former method, have been calculated.