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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.
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.
