一个阿贝尔积分根的数目的下界(英文)

A lower bound for the number of zeroes about an Abelian integral

霍尔普夫分支是动力系统分支理论中一个重要的部分,几乎所有的问题都和非退化中心附近的极限环的数目以及扰动相关.本文研究了一个近哈密尔顿系统x=H(x,y)y(1+x)+εP(x,y),y=-H(x,y)x(1+x)+εQ(x,y),其中H(x,y)=y2/2+x2k/(2k),k≥1.通过利用霍尔普夫极限环分支理论,得到相应的阿贝尔积分孤立零点的最大个数的下界,由此给出了最大数目极限环的下界.

Hopf bifurcation is an important part of bifurcation theory of dynamical systems. Almost all known works are concerned with the bifurcation and number of limit cycles near a nondegenerate focus or center. In this paper, we study a polynomial near-Hamiltonian system where H（x,y）=yZ/2%-xZk/（2k）,k≥1. By using a general theorem on Hopf bifurcation of limit cycles, a lower bound for the maximum number of isolated zeroes of the corresponding Abelian integral is gived, which give a lower bound for the maximum number of limit cycles.