1
$\begingroup$

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?

$\endgroup$
2
$\begingroup$

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 $V(x) = {1 \over n+1} x_1^{n+1} + {1 \over 2} x_2^2$.

$\endgroup$
  • $\begingroup$ 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 '18 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 '18 at 17:09
  • $\begingroup$ I added an additional hint. $\endgroup$ – copper.hat Jul 3 '18 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. $\endgroup$ – AlexanderJ93 Jul 3 '18 at 19:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.