TITLE: Find the Lyapunov function for the nonlinear second order system.
QUESTION [6 upvotes]: Consider the system
$$
\begin{aligned}
\dot{x}_1 &= -x_1+x_2,\\
\dot{x}_2 &= -x_2^3.
\end{aligned}
$$
The origin is obviously globally asymptotically stable. Indeed, $x_2\to 0$ and the dynamics of $x_1$ is linear and thus converges to zero for the converging input $x_2$.
However, I got stuck trying to find a Lyapunov function for this system. Could you please help?

REPLY [4 votes]: 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 $(x_1,x_2)$.