Suppose I have the difference equation $x_{n+1} = f(x_n)$. The point $x^{\ast}$ is called a fixed point of the equation if $x^{\ast}=f(x^{\ast})$.
The fixed point is stable if $\,\left\lvert\, f'(x^{\ast})\right\rvert < 1$ and unstable if $\,\left\lvert\, f'(x^{\ast})\right\rvert > 1$.
This is all from my differential equations notes. But could someone give a proof of these or explain why they are true? Thanks.
Start with a first order difference equation as $x_t = αx_{t−1} + b$ A steady-state $x^∗$ is such that $x_t = x^∗$ at all $t$. The steady-state $x^∗$ of $x_t = αx_{t−1} + b$ is stable if given $\epsilon \gt 0$ there exists $\delta \gt 0$ such that $|x_0 − x^∗| \lt \delta \implies |x_t − x^∗| \lt \epsilon$, for all $t \gt 0$. If $x^∗$ is not stable, then it is called unstable.
link to pdf page 16