Nonlocal deformations of autonomous invariant curves for Liénard equations with quadratic damping (Q2169722)

Using the transformation \(w=F(x)\) and the time scaling \(d \tau =G(x) dt\), the equation \[ x_{tt}+f(x)x_t+g(x)=0 \] turns into \[ w_{\tau\tau}+\tilde{f}(w)w_{\tau}+\tilde{g}(w)=0 \] by the restriction \(F_{xx}G-F_xG_x=0\). So it is possible to establish the connections of invariant curves and cofactors between two systems. As the applications, these can provide explicit solutions from the Painleve-Gamier classification of type Ince II and VII, which are valid to find traveling waves.