Liénard boundedness function ($W=V(x)+z^2/2-xz$)

For the Liénard system $\dot{x}=z-F(x)$, $\dot{z}=-g(x)$ with $F=x^3/3-2x$, $g=x^3-x$, and $V^{\prime }=g$, the displayed function is coercive up to a constant. Differentiation gives $\dot{W}=-gF-z^2+Fz+xg\le -z^2/2-5x^6/18+8x^4/3-x^2$. Its derivative is negative outside a compact set, providing bounded forward dynamics even though the original oscillator has negative damping near the origin. The coordinate $z=y+F(x)$ exposes this restoring and damping balance.