The number of small amplitude limit cycles in arbitrary polynomial systems

主讲人：赵丽琴  北京师范大学教授

时间：2021年11月26日9：00

地点：腾讯会议 725 730 864    密码 123456

举办单位：数理学院

主讲人介绍：赵丽琴，北京师范大学教授，博士研究生导师，研究方向：向量场的分支理论. 在极限环的分支理论方面做了一些工作. 多次主持国家自然科学基金。

内容介绍：In this paper, we study the number of 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. So, it is necessary and valuable  for us to achieve some general results about the number of small amplitude limit  cycles in arbitrary polynomial systems with degree m, which is denoted by M(m).  In this paper, by applying the method developed by C. Christopher and N. Lloyd  in 1995, and M. Han and J. Li in 2012, we first obtain the lower bounds for  M(6)-M(14), and then prove that M(m)≥m^2 if m≥23. Finally, we obtain that M(m)  grows as least as rapidly as 18/25 1/2ln2(m+2)^2ln(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, denoted by H(m), grows as least as rapidly as 1/2ln2(m+2)^2ln(m+2).