The estimate of the amplitude of limit cycles of symmetric Liénard systems

详细信息    查看全文

• 作者：Yuli Cao ^{20144207005@stu.suda.edu.cn} ; Changjian Liu ; ^{liucj@suda.edu.cn}
• 关键词：34C07 ; 34C05 ; 34C14
• 刊名：Journal of Differential Equations
• 出版年：2017
• 出版时间：5 February 2017
• 年：2017
• 卷：262
• 期：3
• 页码：2025-2038
• 全文大小：264 K
• 卷排序：262

文摘

Symmetric Liénard system x˙=y&minus;F(x),y˙=&minus;g(x) (i.e. F(x)F(x) and g(x)g(x) are odd functions) is studied. It is well known that under some hypotheses, this system has a unique limit cycle. We develop a method to give both the upper bound and lower bound of the amplitude, which is the maximal value of the x  -coordinate, of the unique limit cycle. As an application, we consider van der Pol equation x˙=y&minus;μ(x3/3&minus;x),y˙=&minus;x, where μ>0μ>0. Denote by A(μ)A(μ) the amplitude of its unique limit cycle, then for any μ  , we show that A(μ)<2.0976A(μ)<2.0976 and for μ=1,2μ=1,2, we show that A(μ)>2A(μ)>2. Both the upper bound and the lower bound improve the existing ones.