Outward escape from the rational potential barrier
= Outward escape from the rational potential barrier
For $k>0$, the trajectory beginning at $(x,y)=(1,y_0)$ with $y_0>0$ remains in $x>1$, $y>0$ and obeys
$$
0<y(t)<\sqrt{y_0^2+\frac12}.
$$
It enters every strip $0<y<\varepsilon$: otherwise $\dot E=-ky^2\leq-k\varepsilon^2$ would eventually make the nonnegative energy negative.