Lyapunov and Asymptotic stability

Question

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

$x'=x-x^3$

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

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.