How do you determine if a function is Lyapunov or asymptotically stable? The definitions do not seem to tell us how to prove whether a solution is stable or unstable.

For example, I am trying to determine the stability of


I have found the equilibrium solutions $x=0,1,-1$ but do not know where to go from here.


1 Answer 1


Use Lyapunov's linearisation method.

Let $f(x) = x-x^3$. We have $f^{-1}(\{0\}) = \{-1,0,1\}$.

$f'(-1) < 0$, $f'(0) >0$, $f'(1) < 0$.

Hence $-1,1$ are exponentially stable equilibria, and $0$ is unstable.

  • $\begingroup$ @Did: Thanks for pointing that out, I have fixed the typo. $\endgroup$
    – copper.hat
    Jul 2, 2016 at 16:31

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .