Lagmental Vicfred

The Derivative Decides Local Fixed-Point Stability by Vicfred

Last updated: Wed 12 March 2025

tags: mathematics, dynamical systems, chaos, Lyapunov exponents

A one-dimensional fixed point attracts when the derivative magnitude is below one and repels when it exceeds one. I want the notation, the mechanism, and the failure mode visible at the same time.

Set-up

A discrete dynamical system iterates \(x_{n+1}=F(x_n)\), while a flow solves \(\dot x=V(x)\). Fixed points, periodic orbits, and invariant sets organize long-term behavior.

$$ F(x^\ast)=x^\ast $$

A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.

$$ |F'(x^\ast)|<1\Longrightarrow x^\ast\ \text{locally attracting} $$

The calculation

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ e_{n+1}=F(x^\ast+e_n)-F(x^\ast)=F'(x^\ast)e_n+O(e_n^2) $$

What survives abstraction

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad F(x^\ast)=x^\ast,\\[5pt] \mathsf{C}\;&:\quad |F'(x^\ast)|<1\Longrightarrow x^\ast\ \text{locally attracting}. \end{aligned} $$

The boundary

Sensitive dependence is not the same as randomness. A deterministic system may be chaotic while remaining exactly specified by its initial condition.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] |F'(x^\ast)|<1\Longrightarrow x^\ast\ \text{locally attracting} \end{gathered}} $$

This is enough machinery for one note: an exact object, a worked case, a structural law, and a clearly marked boundary. Each layer can now be tested independently.

This article was posted on Fri 23 February 2024. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.