The dimension and divergence rate of the signal's phase-space attractor.
Delay-embeds the time series (at dimensions 2 through 8 for correlation dimension, up to 10 for Lyapunov exponent) using the first zero-crossing of the autocorrelation as the lag. In this reconstructed space, applies Grassberger-Procaccia to estimate the correlation dimension D2 (how many dimensions the attractor fills) and Rosenstein's method to estimate the maximum Lyapunov exponent (how fast nearby trajectories diverge).
How many effective dimensions does the attractor fill? Collatz Stopping Times leads at 4.22: its complex branching dynamics fill a roughly 4D manifold. Neural Net Dense (4.02) and ECG Supraventricular (4.01) are similarly high-dimensional. The Lorenz attractor sits around 2.05 (textbook D2 for the Lorenz system). Constants and Fibonacci Word score 0.0 — degenerate point or 1D attractors.
Does the dimension estimate converge as you increase the embedding dimension? Champernowne (0.997) and Triangle Wave (0.997) saturate immediately — their low intrinsic dimension is captured at the lowest embedding. Collatz Parity scores 0.0 (dimension never converges, suggesting the signal doesn't live on a finite-dimensional manifold). High saturation means you can trust the D2 estimate; low saturation means the attractor is higher-dimensional than the embedding can capture.
What fraction of the embedding space does the trajectory actually visit? Dice Rolls (0.994) and XorShift32 (0.982) fill almost all of it — they're space-filling in delay coordinates. Logistic Period-2 scores 0.002 (the trajectory visits only two points in any embedding). This separates low-dimensional attractors from space-filling noise.
The maximum Lyapunov exponent: how fast do nearby trajectories diverge? Positive means chaos (exponential separation), zero means periodic or quasiperiodic, negative means contracting. Henon Near-Crisis leads at 0.106: it's on the edge of destruction, with maximum divergence. Financial returns (Nikkei -0.003, NYSE -0.0003) are slightly negative — they're mean-reverting on short timescales.
Sum of all positive Lyapunov exponents in the reconstructed spectrum (Wolf-style multi-direction estimate, capped at the embedding dimension). Positive total = expanding directions exceed contracting ones (dissipative chaos). Anderson 1D Localized (+4.20) and Langton's Ant (+4.18) lead; Aubry-André Critical and ECG Beat Conformity follow. Strongly bimodal: ~64/298 sources NaN where the spectrum is degenerate (constants, near-periodic, alphabet-bound symbolic). When valid, separates "many directions expand a little" from "one direction expands a lot."
Kolmogorov-Sinai entropy estimate as the sum of positive exponents from the same spectrum. Fibonacci Tight-Binding (5.00), Langton's Ant (4.60), and Anderson 1D Localized (4.60) saturate the cap — their reconstructed dynamics produce many expanding directions simultaneously. Distinct from lyapunov_max (which sees only the steepest exponent) and from Information Theory's entropy (which sees only the marginal distribution).
Kaplan-Yorke / Lyapunov dimension: a fractional estimate of the attractor's information dimension from the spectrum. MFPT Outer Race / Shuffled Blocks (6.00) saturate at the embedding ceiling; structured low-dimensional dynamics (logistic period orbits, Fibonacci Word) collapse to 0.0. Complements correlation_dimension: D2 measures attractor geometry directly; D_KY infers it from divergence rates, and the two disagree where finite-sample bias hits one but not the other.
| Source | Origin | Value |
|---|---|---|
| Logistic r=3.5 (Period-4) | iterated-maps | 0.9974 |
| Damped Pendulum | continuous-flows | 0.9972 |
| Triangle Wave | signal-synthesis | 0.9969 |
| ··· | ||
| Thue-Morse | symbolic-dynamics | 0.0000 |
| Fibonacci Word | symbolic-dynamics | 0.0000 |
| Periodic Word | symbolic-dynamics | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| Anderson 1D Localized | random-matrix-quantum | 4.2021 |
| Langton's Ant | symbolic-dynamics | 4.1824 |
| Aubry-André Critical | random-matrix-quantum | 3.5017 |
| ··· | ||
| Sandpile | self-organized-criticality | -19.6113 |
| ECG Fusion | physiological | -19.4862 |
| Collatz Flights | number-theory | -19.4048 |
| Source | Origin | Value |
|---|---|---|
| Gaussian Noise | stochastic-process | 4.2978 |
| Network Packet Sizes | algorithmic-bytes | 4.2382 |
| Shuffled Blocks | stochastic-process | 4.2177 |
| ··· | ||
| Pell Word | symbolic-dynamics | 0.0000 |
| Collatz Cycle Word | number-theory | 0.0000 |
| Golden-Mean β-Shift | symbolic-dynamics | 0.0000 |
| Source | Origin | Value |
|---|---|---|
| Dice Rolls | stochastic-process | 0.9938 |
| Euler-Mascheroni γ Digits | number-theory | 0.9830 |
| ChaCha20 | algorithmic-bytes | 0.9829 |
| ··· | ||
| Logistic r=3.2 (Period-2) | iterated-maps | 0.0020 |
| Logistic r=3.5 (Period-4) | iterated-maps | 0.0020 |
| Logistic r=3.83 (Period-3 Window) | iterated-maps | 0.0029 |
| Source | Origin | Value |
|---|---|---|
| MFPT Outer Race | machine-vibration | 6.0000 |
| Shuffled Blocks | stochastic-process | 6.0000 |
| MFPT Normal | machine-vibration | 5.9557 |
| ··· | ||
| Triangle Wave | signal-synthesis | 0.0000 |
| Sine Map (Feigenbaum) | iterated-maps | 0.0000 |
| μ-law Sine | signal-synthesis | 0.2013 |
| Source | Origin | Value |
|---|---|---|
| Fibonacci Tight-Binding | random-matrix-quantum | 5.0000 |
| Langton's Ant | symbolic-dynamics | 4.6816 |
| Anderson 1D Localized | random-matrix-quantum | 4.5984 |
| ··· | ||
| Triangle Wave | signal-synthesis | 0.0000 |
| Sine Map (Feigenbaum) | iterated-maps | 0.0000 |
| Sine Wave | signal-synthesis | 0.0001 |
| Source | Origin | Value |
|---|---|---|
| Fibonacci Tight-Binding | random-matrix-quantum | 0.8698 |
| Tent Map | iterated-maps | 0.5712 |
| von Mangoldt Function | number-theory | 0.5361 |
| ··· | ||
| macOS Mach-O (dyld) | algorithmic-bytes | 0.0000 |
| DNA Phage Lambda | genomic | 0.0000 |
| Bzip2 (level 1) | algorithmic-bytes | 0.0000 |
Attractor Reconstruction provides the classic chaos diagnostic: positive Lyapunov with finite correlation dimension means deterministic chaos. The framework uses it alongside Gottwald-Melbourne (which doesn't need embedding) as a cross-check. In the atlas, correlation_dimension separates the dynamical view's low-dimensional chaos cluster (D2 = 2-4: Lorenz, Rossler, Henon) from noise (D2 saturates at embedding dimension) and periodicity (D2 = 1). The Lyapunov-spectrum triple (lyap_sum / lyap_entropy / kaplan_yorke_dim) adds a dissipativity axis the single max-Lyapunov can't see — useful where many directions expand weakly versus one direction expanding strongly.