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An improvement on the number of limit cycles bifurcating from a non-degenerate center of homogeneous polynomial systems

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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:

201861515:00-16:00

地    点:

明义-204

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 sciences181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。







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