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Given a twice continuously differentiable function $f$ used for a difference equation $x_{n+1} = f(x_n)$, we can show that a stationary point $f(s) = s$ is asymptotically stable (see e.g. here for definition) with the criterion that $ | f'(s) | < 1$, if we ignore (second order) terms $O(z_n^2) = O((x_n - s)^2)$.

Now the question is: how can we show that no case exists, in which $ | f'(s) | < 1$ is fulfilled and simultaneously $O(z_n^2)$ is big enough to make the stationary point $s$ unstable?

NightRain23
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    No matter how big the hidden constant $C$ in $O(z_n^2)$ is, $C z_n^2$ will be microscopic in comparison to $z_n$ if only $z_n$ is close enough to zero. So it's not really that you ignore higher-order terms, it's rather that you use this idea to prove that they can be controlled by the first-order terms. Here is a similar question, by the way. – Hans Lundmark May 01 '22 at 09:57

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