Lagmental Vicfred

A Lyapunov Exponent Measures Exponential Separation by Vicfred

tags: mathematics, dynamical systems, chaos, Lyapunov exponents

The average logarithmic derivative records the growth rate of an infinitesimal perturbation. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

The data

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.

$$ \delta x_n\approx(F^n)'(x_0)\delta x_0 $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in \(\mathcal A\) may change ambient meaning, so I keep \(\mathsf D\) separate from \(\mathsf C\).

$$ \lambda(x_0)=\lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}\log|F'(x_k)| $$

Derivation

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ |\delta x_n|\approx e^{n\lambda}|\delta x_0|,\qquad\begin{cases}\lambda<0&\text{contraction},\\\lambda>0&\text{sensitive growth}.\end{cases} $$

Invariant content

The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.

$$ \begin{aligned} \mathsf{D}\;&:\quad \delta x_n\approx(F^n)'(x_0)\delta x_0,\\[5pt] \mathsf{C}\;&:\quad \lambda(x_0)=\lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}\log|F'(x_k)|. \end{aligned} $$

Scope

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \lambda(x_0)=\lim_{n\to\infty}\frac1n\sum_{k=0}^{n-1}\log|F'(x_k)| \end{gathered}} $$

The notation is dense, but it is doing honest work: every delimiter records scope and every index records dependence. Removing one should require a mathematical reason.

This article was posted on Sat 05 November 2016. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.