一类动力系统的极限环及其数目和分布

用摄动增量法求解一类平面二次动力系统,指出系统在有限域内只有环绕原点的四个环,幅值较小的三个是极限环(分别是稳定、不稳定和稳定),较大的是同宿环;标出无切曲线,以及两条渐近曲线的近似位置;计算结果表明,摄动增量法的近似极限环与数值积分法吻合良好。由三个极限环的速率曲线无公共交点这一事实,进一步具体说明平面多项式微分系统极限环的数目(即Hilbter第16问题第二部分)不能简单地由代数方法解决。

The Limit Cycles and Their Number and Distribution of a Dynamical System

A plane quadratic dynamical system is solved by using the perturbation incremental method. It is shown that there are only four cycles in the finite field of this system. Three of smaller amplitudes are the limit cycles (stability, instability and stability respectively) and the larger one is the homoclinic cycle. The un tangent curve and the two gradual curves are plotted. The computation result showed that the perturbation incremental method is in good agreement compared with the numerical integral method. It is shown further that the number of the limit cycles of the planar polynomial differential systems (the second part of Hilbters 16 problems) cant be simply solved with using the algebra from the fact that there isnt the common point of intersection of the rate curves of the three limit cycles.