Consider the nonlinear model of a mass spring-damper: $$ \begin{align} \ddot{y} &= -2 \dot{y}\mid\dot{y}\mid - 5y -2y^3 \end{align} $$ where y is the vertical displacement and mass is assumed unitary for simplicity.
By using the states: $$ \begin{align} x_1 = y\\x_2 =\dot {y} \end{align} $$ and equilibrium positions: $$ \begin{align} x_1 = 0 \\ x_2=0 \end{align} $$ The matrix A of the linearised system around the equilibrium is found: $$ \begin{align} A= \begin{pmatrix} 0&1\\ -5 &0\end{pmatrix} \end{align} $$
A) How can I use the sum of the kinetic energy and potential energy stored in the spring tern as a Lyapunov function: $$ \begin{align} V(x_1,x_2) = \frac{1}{2}x_2^2 + \int_0^{x_1}({5z +2z^3})\,dz \end{align} $$ and prove global stability of the system?
B) Why can't this Lyapunov function be used to show global asymptotic stability of the system?