Limit cycles of a class of cubic Liénard equations (Q1936522)

In the focus of this paper there is the following class of polynomial Liénard systems of the form \[ \dot x=y-(a_1x+a_2x^2+a_3x^3),\;\dot y=-(b_1x+b_2x^2+b_3x^3) \] with real coefficients. By using Filippov transformations and Zhang's theorem, the authors derive some conditions for the existence, non-existence and uniqueness of a limit cycle. It is proved that the mentioned system has at most one limit cycle surrounding the origin if \(a_1 a_3<0\) or \(b_2=0\). The paper contains an example illustrating that the considered system can have three limit cycles.