Damped rational double-barrier phase portrait
= Damped rational double-barrier phase portrait
For
$$
\ddot x+k\dot x+
\frac{2x(1-x^2)}{(1+x^2)^3}=0,
$$
the mechanical energy is
$$
E(x,y)=\frac12y^2+\frac{x^2}{(1+x^2)^2},
\qquad
\dot E=-ky^2.
$$
The origin is a center for $k=0$ and an asymptotically stable equilibrium for $k>0$, while $(\pm1,0)$ are saddles. For positive damping, their stable manifolds form the boundary between the basin of the origin and the two escape regions.