Find the lyapunov function to prove the asymptotic stability

Question

Let's the following non linear system: $\begin{cases} \dot{x_1}=x_2&\\ \dot{x_2}=-x_1^3&\\ \end{cases}$ determine if the origin is asymptotically stable and in this case if it is globally asymptotically stable. I have tried to linearized the system but I have obtained two eigenvalues equal to $0$. I have tried to find out an opportune $V$ to prove that the origin is AS but I can't.
Anyone can help me?

Answer 1

There is no asymptotic stability. Taken the system

$$ \cases{ \dot x_1 = x_2\\ \dot x_2 = -x_1^3 } $$

and multiplying by $x_1^3, x_2$ as

$$ \cases{ x_1^3\dot x_1 = x_1^3x_2\\ x_2\dot x_2 = -x_1^3x_2 } $$

and adding we have

$$ \frac 12x_1^4+x_2^2 = C $$

so those orbits characterize a center around the origin.

Answer 2

Hint: Try something with $x_1^4$ instead of $x_1^2$.