Intersection with the vertical isocline in the generalized Liénard equations (Q996862)

The paper deals with the generalized Liénard system \[ \dot{x}=\frac{h(y)-F(x)}{a(x)},\quad \dot{y}=-a(x)g(x), \] where \(a(x)>0\) for all real \(x\). Under the assumption that the origin is a unique equilibrium, the authors consider the problem whether all trajectories of this system intersect the vertical isocline \(h(y)=F(x).\) The given sufficient and necessary conditions extend and improve recent results in this direction.