一类平面系统极限环存在性及个数的新判别法

One new method of determing the existence and numbers of limit cycle for one kind of the planar systems

利用向量场通量方法给出平面系统极限环存在性与个数以及局部Hopf分支问题的判定定理,并将其物理意义加以说明,得到只含有1个奇点的平面N(N2)次多项式系统最多有N个极限环的结论,改善了Hilbert第16个问题的研究现状.同时揭示出该方法的局限性.最后给出几个特例并结合数值模拟对判定定理的可行性加以验证.

The judgement theorems are given by the method of flux quantity of vector field for the existence ofp lanar systems' limit cycles and its numbers,then its physical meaning can be explained and it is shown that N-order planar polynomial systems of only one singularity have at most N limit cycles for Hilbert 16th current research problems improved. Meanwhile,the limitation of the method has been announced. Finally,some spacial examples are given by combining numercial stimulation for checking the theorems' feasibility.