Damped mechanical energy as a Lyapunov function

For $\ddot{x}=-U^{\prime }(x)-\mu \dot{x}$ with $\mu >0$,If $U$ is radially unbounded, every trajectory is bounded, and LaSalle's principle reduces its omega-limit set to invariant points with zero velocity.

Forthe mechanical energy isThe origin is a center for $k=0$ and an asymptotically stable equilibrium for $k>0$, while $(\pm 1,0)$ are saddles. For positive damping, their stable manifolds form the boundary between the basin of the origin and the two escape regions.

For $k>0$, the trajectory beginning at $(x,y)=(1,y_0)$ with $y_0>0$ remains in $x>1$, $y>0$ and obeysIt enters every strip $0<y<\epsilon$: otherwise $\dot{E}=-k{y}^2\le -k{\epsilon }^2$ would eventually make the nonnegative energy negative.