On the uniqueness of limit cycles for generalized Liénard systems

Abstract In this article, the general Liénard system d x d t = ϕ ( y ) − F ( x ) , d y d t = − g ( x ) \left\{\begin{array}{l}\frac{{\rm{d}}x}{{\rm{d}}t}=\phi (y)-F\left(x),\\ \frac{{\rm{d}}y}{{\rm{d}}t}=-g\left(x)\end{array}\right. is studied. By using the Filippov transformation, combined with the careful estimation of divergence along the closed orbit, we prove the sufficient conditions for the uniqueness of limit cycles in this system. Our results extend almost all the related existing studies on the Liénard system.