三类Liénard系统极限环的下界  

The lower bounds of limit cycles for three classes of Liénard systems

作　　者：熊峰 黄文韬[2] XIONG Feng;HUANG Wentao(School of Information Management and Engineering,Shanghai University of Finance and Economics,Shanghai 200433,China;Department of Science,Guilin University of Aerospace Technology,Guilin 541004,China)

机构地区：[1]上海财经大学信息管理与工程学院,上海200433 [2]桂林航天工业学院理学部,广西桂林541004

出　　处：《中山大学学报（自然科学版）》2018年第6期154-161,共8页Acta Scientiarum Naturalium Universitatis Sunyatseni

基　　金：国家自然科学基金(11361017);广西自然科学基金重点项目(2016GXNSFDA380031)

摘　　要：分别研究了(m,n)=(9,7)、(m,n)=(8,7)和(m,n)=(7,8)三类Liénard系统在原点邻域内的极限环数目问题。首先,应用计算机代数软件Mathematica和奇点量方法计算其伴随复系统的前10个,9个和9个奇点量,最后,通过雅克比行列式方法证明了这三类Liénard系统在原点充分小邻域内能够产生10个,9个和9个极限环。首次给出了H(9,7),H(8,7),H(7,8)的一个下界估计,即H(9,7)≥10,H(8,7)≥9,H(7,8)≥9。The problem of the number of limit cycles for three classes of Liénard systems,that is,(m,n)=(9,7),(m,n)=(8,7)and(m,n)=(7,8),are studied in the neighbourhood of the origin.The first ten,nine and nine singular point values for the corresponding accompany complex systems are calculated by using the computer algebra system,Mathematica,and the method of singular point values.Then,three kinds of Liénard system can generate at least ten,nine and nine limit cycles at the sufficiently small neighborhood of the origin respectively.And the exitence of limit cycle is proved by applying the Jacobian determinant method.It is the first time for giving the lower bound estimates for such systems,that is,H^(9,7)≥10,H^(8,7)≥9,H^(7,8)≥9.

关 键 词：LIÉNARD系统 奇点量 细焦点 极限环 

分 类 号：O175.12[理学—数学]