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### 系列学术活动之(34)An improvement on the number of limit cycles bifurcating from a non-degenerate center of homogeneous polynomial systems

 系列学术活动之（34） 题    目： An improvement on the number of limit cycles bifurcating from a non-degenerate center of homogeneous polynomial systems 摘    要： MIn the two articles [Appl. Math. Comput., 2012a, 2012b], J. Gine studied the number of small limit cycles bifurcating from the origin of the system:$\dot{x} = -y + P_n(x,y), \dot{y} = x + Q_n(x,y)$, where $P_n$ and $Q_n$ are homogeneous polynomials of degree $n$. It is shown that the maximal number of the small limit cycles, denoted by $M_h(n)$, satisfies $M_h(n) \ge 2n - 1$ for n = 4,5,6,7; and $M_h(8) \ge 13$, $M_h(9) \ge 16$. It seems that the correct answer for their case n = 9 should be $M_h(9)\ge 15$. In this paper, we apply Hopf bifurcation theory and normal form computation,and perturb the isolated, non-degenerate center (the origin) to prove that $M_h(n) \ge 2n$ for n = 4,5,6,7$; and$M_h(n) \ge 2(n-1)\$ for n =8,9, which improve Gine's results with one more limit cycle for each case. 报 告 人： 郁培，教授，加拿大西安大略大学 时    间： 2018年6月15日 15:00-16:00 地    点： 明义③-204室 报告人概况：郁培，加拿大西安大略大学(western university, canada)教授，博士生导师，常微分方程与动力系统领域著名专家，在非线性和混沌动力系统、稳定性理论和分支理论、微分方程的计算问题、生物数学以及物理和工程系统的应用问题等做出了杰出的工作。曾获安大略省长杰出研究奖，和合编辑在《siam review》等杂志发表近200篇论文以及springer等出版社出版了多部专著，其中《normal forms, melnikov functions and bifurcation of limit cycles》被列为applied mathematical sciences第181册，2012年在springer出版。郁培教授任《journal of applied analysis and computation》、《international journal of bifurcation and chaos》与《communications in nonlinear science and numerical simulation》等国际知名杂志编委。