Find the Lyapunov function for the nonlinear second order system.

Define $v=x_1-x_2$. Then $\dot{v} = -x_1+x_2+x_2^3 = -v+x_2^3$. Consider the system $$ \begin{aligned} \dot{v} &= -v + x_2^3, \\ \dot{x}_2 &= -x_2^3. \end{aligned} $$ Choose $V = v^2+\frac{1}{2}x_2^4$. Then $$\dot{V} = -2v^2 + 2vx_2^3-2x_2^6.$$ Since $2vx_2^3 \le v^2+x_2^6$ we have $$\dot{V}\le -v^2-x_2^6.$$ Thus the origin $(v,x_2)=(0,0)$ is globally asymptotically stable, and the same holds for $(x_1,x_2)$.