几个多项式微分系统极限环的个数

研究了可积系统(称为未扰系统).{xx=-y(1+x4).y=x(1+x4).在几类多项式扰动之下极限环的个数.即当未扰系统加上低次扰动后,考虑扰动系统:.xx=-y(1+x4.)x=-y(1+x4),.y=x(1+x4)+εPn(x,y),+εQn(x,y),1≤n≤4,其中Pn,Qn是任意的n次多项式,讨论了它们从未扰系统的周期环处分支出极限环的个数.通过计算扰动系统的一阶M eln i-kov函数以及估计其根的个数得到从未扰系统的周期轨处分支出极限环的最大个数.证明了未扰系统加上1次或者2次扰动项时,扰动系统最多有1个极限环;加上3次或者4次扰动项时,扰动系统最多有4个极限环.

The number of limit cycles of some polynomial differential systems

We consider the number of limit cycles of the following integrable system, which is called the unperturbed system, with the perturbation of some polynomials:{(x)x =-y(1+x4) (y)=x(1+x4).That is, considering the perturbed system under the low degree perturbations:(x)x = -y(1 +x4)x =-y(1 +x4) ,(y) =x(1 +x4) +εPn(x,y), +εQn(x,y) with 1 ≤n≤4,where Pn,Qn are arbitrary polynomials of degree n, the author studies the number of limit cycles bifurcating from the period annulus of the unperturbed system. By calculating the first order Melnikov function of the system and estimating the number of zeros of the function, we obtain the maximal number of limit cycles which can bifurcate from the periodic orbits of the system. One can prove that with the perturbation of degree 1 or 2, the perturbed system has at most 1 limit cycle; and with the perturbation of degree 3 or 4, the perturbed system has at most 4 limit cycles.