On the uniqueness of limit cycles for generalized Liénard systems

Abstract

In this article, the general Liénard system dxdt=ϕ(y)−F(x),dydt=−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.

Keywords

• generalized liénard system
• limit cycles
• uniqueness
• 34a25
• 34c07
• 34c05
• 34c25