Abstract
In this paper, the global analysis of a Liénard equation with quadratic damping is studied. There are 22 different global phase portraits in the Poincaré disc. Every global phase portrait is given as well as the complete global bifurcation diagram. Firstly, the equilibria at finite and infinite of the Liénard system are discussed. The properties of the equilibria are studied. Then, the sufficient and necessary conditions of the system with closed orbits are obtained. The degenerate Bogdanov-Takens bifurcation is studied and the bifurcation diagrams of the system are given.
Highlights
Introduction and Main ResultsLienard equations have a very wide application in many areas, such as mechanics, electronic technology, and modern biology; see [1,2,3,4]
People are strongly interested in the solution existence, vibration, and periodic solutions of Lienard equations, which promote the research of Lienard equations more and more deeply, as shown in [5,6,7,8,9]
In 2016, Llibre [10] studied the centers of the analytic differential systems and analyzed the focus-center problem
Summary
Lienard equations have a very wide application in many areas, such as mechanics, electronic technology, and modern biology; see [1,2,3,4]. Chen [11,12,13] investigated the dynamical behaviour of a cubic Lienard system with global parameters, analyzing the qualitative properties of all the equilibria and judging the existence of limit cycles and homoclinic loops for the whole parameter plane. They gave positive answers to Wang Kooij’s [14] two conjectures and further properties of those bifurcation curves such as monotonicity and smoothness.
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