一个阿贝尔积分根的数目的下界

摘要

Abstract: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 x=(δ)H(x,y)/（δ）y (1+x)+εp(x,y),y=-（δ）H(x,y)/（δ)x (1+x)+εQ(x,y),where H(x,y)=y2/2+x2k/(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.%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=(δ)H(x,y)/(δ)y (1+x)+εp(x,y),y=-(δ)H(x,y)/(δ)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.