Liénard equations with bounded trajectories in the Liénard plane (Q1108437)

The simple Liénard equation \(\ddot x+f(x)\dot x+g(x)=0\) is studied in the Liénard plane \((\dot x=y-F(x)\), \(\dot y=-g(x))\). It is assumed that \(F'(x)=f(x)\) and g(x) are continuous functions for all real x. It is shown that there exists some Jordan curve K in the Liénard plane surrounding the point (0,0) such that any trajectory starting in the exterior of K reaches K and eventually stays in the interior of K - provided some extra conditions are fulfilled. The result contains a classical theorem of Dragilev as a special case. If in particular g(x) is odd some further results are derived by separating F(x) in two parts one of which is even.