Chapter 8 · Attractors, fractal dimension, and entropy

Part III — Quantifying chaos and predictability Knob: the scaling window, the Theiler window, the embedding dimension live notebook

Overview

A forecast model has $10^7$ variables. Does an analysis have to constrain $10^7$ numbers?

No — because the trajectory does not visit most of its state space. It settles onto an attractor whose dimension is generally far smaller than the state dimension and generally not an integer, and that number is what an observing system actually has to pin down.

Two ways to measure it, and they share no intermediate quantity:

D_{KY} = j + \frac{\sum_{i=1}^{j}\lambda_i}{|\lambda_{j+1}|}, \qquad C(r) \sim r^{D_2} .

The Kaplan–Yorke dimension comes from the Lyapunov spectrum — from the dynamics, via the tangent equations. The correlation dimension counts pairs of points closer than $r$ on a sampled trajectory — from the geometry, using nothing but a cloud of points. Agreement between them is therefore evidence rather than arithmetic.

Most of the chapter is about how easily both are got wrong. Not as a closing caveat: a dimension estimate is a slope fitted over a range you chose, and choosing badly returns a clean fit with a small residual and the wrong answer.

The model

chaoslib.dimension gains box counting and the Rényi dimensions $D_q$ , a delay-embedding helper, and three reference sets built by exact self-similar recursion — Cantor ( $\ln2/\ln3$ ), Koch ( $\ln4/\ln3$ ) and Sierpiński ( $\ln3/\ln2$ ) — whose dimensions are known in closed form. Those are the calibration: an estimator that cannot return 1.5850 where the answer is known has no business being pointed at an attractor.

Every expensive computation in this chapter is knob-free, and the one thing the reader chooses — the scaling window — costs a polyfit over a stored curve. So the curves are precomputed and every slider re-fits them, which is both instant and the pedagogical point: one fixed curve, with the window moving across it.

Prefer to run it yourself? Download as a Jupyter notebook — it installs its own dependencies in the first cell. The sliders are static in Jupyter; edit value= and re-run to change a parameter.

Four results

Two independent routes agree to 0.2%. For Lorenz 63, $D_2 = 2.0579$ from counting pairs and $D_{KY} = 2.0618$ from the Lyapunov spectrum. Both estimators work — which matters given how easily either can be made to return nonsense — and the Kaplan–Yorke conjecture holds here: the dimension implied by the growth rates really is the dimension of the set. It is a conjecture, not a theorem, and this is what testing it looks like. The Hénon map is the second case, $D_2 = 1.192$ against a literature 1.22.

The same curve returns 0.19, 1.92 or 2.51 depending on the window. Fitted above 30% of the attractor diameter, where $C \to 1$ , the slope collapses to 0.19. Fitted below 0.2%, where pair counts are quantised, it steepens to 2.51. Fitted over the whole range it gives a plausible and wrong 1.92 — the most dangerous of the three, because nothing in the output distinguishes it from the correct 2.057. So too coarse biases low and too fine biases high; neither is correctable after the fact.

The Theiler bias runs opposite to the usual warning. Temporal correlation puts a bump in $C(r)$ at the distance the trajectory covers between samples. For Lorenz 63 at $\Delta t = 0.01$ that step is 1.3% of the diameter — inside the usual fit window — so the bump’s rising flank steepens the local slope from 1.92 to 2.22 and biases $D_2$ high: 2.139, 2.118, 2.084, 2.039 for Theiler windows of 0, 10, 50, 200 samples. The standard warning is that temporal correlation makes the estimator report the trajectory’s smooth one-dimensional curve, biasing low, which happens when the sampling puts the step scale below the window. Which bias you get depends on where your sampling interval falls relative to your fit window.

Box counting works where you can afford it and starves in three dimensions. It recovers all three reference dimensions to better than 0.01, but only while there are more than about ten points per occupied box: on Sierpiński the local slope reads 1.582, 1.568, 1.549, 1.451, 1.127 at occupancies of 186, 63, 21, 7.3, 2.8 against an exact 1.585. On the Lorenz attractor, 19,000 samples do not supply a single clean decade, because box counting spreads $N$ points over $\varepsilon^{-d}$ boxes while pair counting uses all $N(N-1)/2$ pairs. That is why every dimension quoted for a real attractor is a correlation dimension.

Entropy, and the return on observations

Pesin’s identity makes $h_{KS} = \sum_{\lambda_i>0}\lambda_i = 0.901$ nats per MTU = 1.30 bits per MTU: the rate at which the system destroys information about its own initial state, and so the rate at which observations must arrive to hold an analysis steady. Each decimal digit of extra initial precision buys $\ln 10/\lambda_1 = 3.8$ days, and the next digit buys 3.8 days again — the logarithmic return that chapter 20 measures from the operational end as 6.5 days per decade of analysis-error reduction.

And all of it works from a single observed variable. Takens’ theorem says a delay embedding of one scalar series reconstructs the attractor once $m > 2D$ ; measured on the $x$ component of Lorenz 63 alone, $D_2$ reads 1.72, 1.95, 1.99, 2.00, 2.00 for $m = 2\ldots6$ , with increments falling 0.235, 0.045, 0.012, 0.002. Since the criterion needs the $D$ you are measuring, that saturation is the only available diagnostic.

Exercises

Analytic. Show that a set built from $N$ copies of itself at scale $r$ has $D_0 = \ln N/\ln(1/r)$ , and evaluate it for the three reference sets. Then show that $D_0 \ge D_1 \ge D_2$ for any measure, with equality only when it is uniform.

Computational. Reproduce the three wrong answers (0.19, 2.51, 1.92) from the same $C(r)$ curve, and for each, name the panel that would have warned you.

Exploratory. Predict from the sampling interval whether the Theiler bias on the chapter’s curve is high or low before changing the Theiler window, then check.