Find the lyapunov function to prove the asymptotic stability

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?

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

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

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

so those orbits characterize a center around the origin. • So when I can't to find out a certain $V$ s.t the derivative is negative definite ,can I try to prove that around the origin the orbits of the non linear system are closed?
– fill
Feb 15 '21 at 17:17
• You can try to obtain a movement integral like the one found. Feb 15 '21 at 17:25

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

• Nothing changes... $\dot{V}=x_1^3(x_2)+x_2(-x_1^3)=0$
– fill
Feb 15 '21 at 16:15
• So you found a constant of motion, that's good news! But how did you get $\dot V=0$ with your original $V$, by the way? That must have been some mistake in the calculation. Feb 15 '21 at 18:40