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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?

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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.

enter image description here

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  • $\begingroup$ 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? $\endgroup$
    – fill
    Feb 15 at 17:17
  • $\begingroup$ You can try to obtain a movement integral like the one found. $\endgroup$
    – Cesareo
    Feb 15 at 17:25
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Hint: Try something with $x_1^4$ instead of $x_1^2$.

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  • $\begingroup$ Nothing changes... $\dot{V}=x_1^3(x_2)+x_2(-x_1^3)=0$ $\endgroup$
    – fill
    Feb 15 at 16:15
  • $\begingroup$ 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. $\endgroup$ Feb 15 at 18:40

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