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I need to check the stability of the equilibrium point of the following system, $n \in \Bbb N$: $$ \left\{ \begin{array} \dot \dot x_1=x_2 \\ \dot x_2=-x_1^n \end{array} \right. $$

I tried using linearization, but the eigenvalues are zero, which means it's not the way to go. I also searched for a Lyapunov function, but couldn't find one. Any ideas?

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Hints:

If $n$ is even it is not hard to directly show that $(0,0)$ is unstable. Consider a starting point $(-\epsilon, -\epsilon)$ for small $\epsilon >0$. Note that $x_2$ is non increasing, hence $x_2(t) \le -\epsilon$ for all $t$. What does that say about $x_1$?

If $n$ is odd, look at the function $V(x) = {1 \over n+1} x_1^{n+1} + {1 \over 2} x_2^2$.

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  • $\begingroup$ Thank you. For even $n$, I'm not sure I understand how to show that the point is unstable. For odd $n$, how did you come up with that funcion? $\endgroup$
    – user401516
    Jul 3, 2018 at 17:05
  • $\begingroup$ By analogy with the $n=1$ case. Take $ax_1^b+c x_2^d$ and pick $a,b,c,d$ accordingly. $\endgroup$
    – copper.hat
    Jul 3, 2018 at 17:09
  • $\begingroup$ I added an additional hint. $\endgroup$
    – copper.hat
    Jul 3, 2018 at 17:13
  • $\begingroup$ @user401516 Notice that $V(x)$ is related to the first integral of the system written as a single equation: differentiate the first equation and substitute the second to get $\ddot{x}_1 = -x_1^n$, multiply by $\dot{x}_1$ and integrate to get $\frac{1}{2}\dot{x}_1^2 = -\frac{1}{n+1}x_1^{n+1} + C$, then rearrange and substitute to get $C = \frac{1}{n+1}x_1^{n+1} + \frac{1}{2}x_2^2 = V(x)$, which is the first integral. $\endgroup$
    – Alex Jones
    Jul 3, 2018 at 19:07

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