The number of near-small-amplitude limit cycles in arbitrary polynomial systems

讲座时间 Datetime: 2022年5月21日，星期六，14:30-15:30

地点 Venue: 腾讯线上会议：370-526-311

 主持人 Host：刘长剑

报告人 Speaker:赵丽琴

单位 Affiliation:北京师范大学

报告摘要 Abstract:

In this paper, we study the number of near small amplitude limit cycles in arbitrary polynomial systems. It is found that almost all the results for the number of small amplitude limit cycles are obtained by calculating Lyapunov constants and determining the order of the corresponding Hopf bifurcation. It is well known that the difficulty in calculating the Lyapunov constants increases with the increasing of the degree of polynomial systems. By applying the method developed by C.Christopher &N.Lloyd, M.Han & J. Li,  we first obtained the lower bound of the number of small amplitude limit cycles for m=6,……14 , where m is the degree of the planar polynomial systems,  and then we proved that M(m) ≥ m^2 if m ≥ 23. Finally, we obtained  that M(m) grows as least as rapidly as 18/25 ·1/2 ln 2(m + 2)2 ln(m + 2) for all large m (it is proved by M.Han & J. Li in “J. Differential Equations, 252 (2012), 3278-3304” that the number of all limit cycles in arbitrary polynomial systems with degree m grows as least as rapidly as 2 ln 2 1 (m + 2)2 ln(m + 2)), where we denote by M(m) the number of near-small-amplitude limit cycles of arbitrary polynomial systems with degree m.