An upper bound for the amplitude of limit cycles of Liénard-type differential systems

• Main Authors: Fangfang Jiang, Zhi Ji, Yan Wang
• Format: Article
• Language: English
• Published: University of Szeged 2017-05-01
• Series: Electronic Journal of Qualitative Theory of Differential Equations
• Subjects: liénard-type system; limit cycle; amplitude; upper bound
• Online Access: http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=5416

In this paper, we investigate the position problem of limit cycles for a class of Liénard-type differential systems. By considering the upper bound of the amplitude of limit cycles on $\{(x,y)\in\mathbb{R}^2: x<0\}$ and $\{(x,y)\in\mathbb{R}^2: x>0\}$ respectively, we provide a criterion concerning an explicit upper bound for the amplitude of the unique limit cycle of the Liénard-type system on the plane. Here the amplitude of a limit cycle on $\{(x,y)\in\mathbb{R}^2: x<0\}$ (resp. $\{(x,y)\in\mathbb{R}^2: x>0\}$) is defined as the minimum (resp. maximum) value of the $x$-coordinate on such a limit cycle. Finally, we give two examples including an application to predator-prey system model to illustrate the obtained theoretical result, and Matlab simulations are presented to show the agreement between our theoretical result with the simulation analysis.