A proof of Wang-Kooij's conjectures for a cubic Liénard system with a cusp

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• 作者：Hebai Chen^a ; Xingwu Chen^b ; ^{xingwu.chen@hotmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Lié ; nard system ; Cuspidal loop ; Limit cycle ; Rotational vector field
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：445
• 期：1
• 页码：884-897
• 全文大小：803 K

文摘

In this paper the global dynamics of a cubic Liénard system with a cusp is studied to follow Wang and Kooij (1992) [13], who proved that the maximum number of limit cycles is 2 and stated two conjectures about the curves of the cuspidal loop bifurcation and the double limit cycle bifurcation. We give positive answers to those two conjectures and further properties of those bifurcation curves such as monotonicity and smoothness. Finally, associated with previous results we obtain the complete bifurcation diagram and all phase portraits, and demonstrate some numerical examples.