> [!NOTE] Definition 1 (Stable Stationary Point) > Let $y_{k}=u(y_{k-1}), \quad \text{for all }k\geq 1$where $u:\mathbb{R}\to \mathbb{R}$ is a given function and $y_{0}=\bar{y}$, be a [[First Order Autonomous Recurrence Relation|first order autonomous recurrence relation]]. Let $y^{*}$ be a [[Stationary Point of First Order Autonomous Recurrence Relation|stationary point]] of the equation. Then $y^{*}$ is stable iff for all $\varepsilon>0$, there exists $\delta>0$ such that $ |y_{0}-y^{*}| < \delta \implies |u^{[n]}(y_{0})-y^{*}|<\varepsilon $for all $n\in \mathbb{N}.$ **Note**: Equivalently, $|y_{0}-y^{*}|<\delta \implies u^{[n]}(y_{0})\to y^{*}$ as $n\to \infty.$ Negating gives: $y^{*}$ is unstable if there exists $\varepsilon$ such that no matter how small we make $\delta,$ we can find $y_{0}$ with $|y_{0}-y^{*}|<\delta \quad \text{but} \quad |u^{n}(y_{0})-y^{*}|>\varepsilon$for some $n>0.$ Note also that $u^{[n]}(y_{0})$ denotes $y_{n}.$ # Properties **Analytic condition for stability**: By [[Condition for Stability of Stationary Point of First Order Autonomous Recurrence Relation]], $y^{*}$ is stable if $|u'(y^{*})|<1$ and unstable if $|u'(y^{*})|>1.$