Fixed-Point Stability

Core Sentence / 核心句

A fixed point is locally stable when repeated application of the function contracts small errors around it.

中文理解：不动点本身满足 $f(p)=p$；稳定性进一步问，受到微小扰动后，迭代是回到它还是远离它。

Formula / 公式或规则

For $x_{n+1}=f(x_n)$ and $f(p)=p$:

$$e_{n+1}\approx f^{\prime }(p)e_n$$ $$|f^{\prime }(p)|<1\Rightarrow \text{locally attracting},|f^{\prime }(p)|>1\Rightarrow \text{locally repelling}.$$

When To Use / 什么时候用

用于 repeated function iteration、fixed-point algorithms、recursive sequences 和 dynamical systems 的局部收敛判断。

Watch Out / 易错点

必须看 $|f^{\prime }(p)|$，不是只看 $f^{\prime }(p)<1$。这个 test 是 local 的；$|f^{\prime }(p)|=1$ 时不能下结论，还需 higher-order analysis。

Search and Source Mentions / 搜索与出现位置

Mentioned In / 出现位置

• Differential Equations 01 · Continuous-Time Stability
  • phase flow asks whether nearby states contract toward or expand away from an equilibrium
• Chapter 12 · Stable Fixed Point
  • local linearization turns derivative magnitude into an error-contraction factor
• Chapter 12 · Unstable Fixed Point
  • the negative fixed point of $1+1/x$ expands and flips nearby errors
• Chapter 12 · Roots vs Limits
  • solving $f(p)=p$ finds candidates, while stability decides attracting limits

Connected Notes / 连接

• 12 - The Other Way to Visualize Derivatives
• Transformational View of Derivatives
• Derivatives
• Limits
• Function Composition
• Higher-Order Derivatives
• Phase Space
• 01 - Differential Equations, a Tourist’s Guide