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).
| Source | Domain | Value |
|---|---|---|
| Rössler Hyperchaos | chaos | 0.9997 |
| Ocean Wind (Buoy) | climate | 0.9985 |
| Gzip (level 1) | binary | 0.9985 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| L-System (Dragon Curve) | exotic | 0.8836 |
| Champernowne | number_theory | 0.8306 |
| De Bruijn Sequence | number_theory | 0.7694 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 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.