Is the signal chaotic or regular? A binary classification with a confidence estimate.
Constructs 2D translation variables p(n) and q(n) by accumulating the signal modulated by cosine and sine at random frequencies. For chaotic signals, (p, q) performs a random walk whose mean square displacement grows linearly, giving K = 1. For regular signals, (p, q) stays bounded, giving K = 0. The median over 10 random frequencies provides robustness. No embedding dimension or lag estimation required.
The chaos indicator: 0 = regular, 1 = chaotic. Rossler Hyperchaos scores 1.0 (textbook chaos). Ocean Wind (0.999) and Gzip (0.999) score nearly 1.0 — both appear chaotic to this test. All logistic periodic orbits score exactly 0.0 (perfectly regular). The key limitation: noise also gives K = 1 because uncorrelated random walks have linearly growing MSD. This test distinguishes chaos from periodicity, not chaos from noise.
Interquartile range of K across the 10 random frequencies. Low variance means the classification is confident. L-System Dragon Curve scores 0.884 (highest variance — the test can't make up its mind because the signal is at the boundary between order and chaos). Champernowne (0.831) and De Bruijn (0.769) are similarly ambiguous. Constants and periodic orbits have zero variance (confidently regular).
Spectral structure of the angular component of the (p,q) translation variables. L-System Dragon (1.0) scores highest — its angular dynamics have concentrated spectral energy. Random Steps (0.0) has no angular spectral structure. This captures periodic or quasi-periodic structure in the rotational component of the 0-1 test variables that the scalar K statistic misses.
Spectral structure of the radial component of the (p,q) translation variables. Similar interpretation to angular_spectral_structure but for the growth component. Together, angular and radial spectral structure decompose the 0-1 test's 2D dynamics into rotation and expansion, providing richer information than the single K statistic.
| Source | Origin | Value |
|---|---|---|
| ln(2) Digits | number-theory | 0.9987 |
| LFSR (16-bit) | algorithmic-bytes | 0.9986 |
| Wind Speed | atmospheric | 0.9985 |
| ··· | ||
| Logistic Edge-of-Chaos | iterated-maps | 0.0000 |
| Sine Map (Feigenbaum) | iterated-maps | 0.0000 |
| Logistic r=3.5 (Period-4) | iterated-maps | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| L-System (Dragon Curve) | symbolic-dynamics | 0.8832 |
| Champernowne | number-theory | 0.8397 |
| Gray Code Counter | symbolic-dynamics | 0.7746 |
| ··· | ||
| 2-Torus Quasiperiodic | coupled-oscillators | 0.0000 |
| Sine Map (Feigenbaum) | iterated-maps | 0.0000 |
| Logistic r=3.5 (Period-4) | iterated-maps | 0.0000 |
Gottwald-Melbourne is the framework's only embedding-free chaos test. Its strength is that it requires no parameter tuning — no embedding dimension, no lag, no threshold selection. Its weakness is that it cannot distinguish chaos from noise (both give K = 1). In practice it's most useful paired with Attractor Reconstruction: K = 1 with finite D2 means chaos; K = 1 with unsaturated D2 means noise. The k_variance metric is uniquely informative for edge-of-chaos signals — high variance at K = 0.5 detects intermittency and crisis transitions that the binary K statistic misses.