一个阿贝尔积分根的数目的下界

A Lower Bound for the Number of Zeroes of an Abelian Integral

研究了一个近哈密尔顿系统的阿贝尔积分孤立零点的最大个数的下界,由此给出了该系统最大数目极限环的下界.对于系统=H(x,y)/y(1+x)+εP(x,y),=H(x,y)x(1+x)+εQ(x,y),其中H(x,y)=y2/2+x2k/(2k),k≥1是一个整数,ε是一个小参数且P和Q是次数至多为n的关于x的多项式.利用霍尔普夫极限环分支理论,得到Z(1,2)=1,Z(1,3)=1,其中Z(n,k)为M(h)最大独立根的个数.

In this paper a polynomial near-Hamiltonian system is studied,where a lower bound for the maximum number of isolated zeroes of the corresponding Abelian integral is given,which gives a lower bound for the maximum number of limit cycles.a polynomial near-Hamiltonian system x=aH（x,y）/ay（1＋x）＋εP（x,y）,y=aH（x,y）ax（1＋x）＋εQ（x,y）,where H（x,y） = y^2/2 ＋ x^2k/（2k）;k was studied 1 is an integer number,ε is a small parameter and P and Q are polynomials in x of degree at most n.By using a general theorem on Hopf bifurcation of limit cycles,that Z（1,2） = 1,Z（1,3） = 1;where Z（n,k） denotes the maximum number of isolated zeroes of the integral M（h）.