Does the signal ever return to places it's been before — and when it does, does it follow the same path?
Embeds the signal in 3D via delay coordinates, then asks: for every pair of points in the trajectory, are they close? The answers form a binary recurrence matrix whose texture reveals the dynamics.
Of the recurrent points, what fraction form diagonal lines in the recurrence matrix? Diagonal lines mean: when the signal returns to a previous state, it follows the same trajectory it followed last time. All logistic periodic orbits score 1.0 (perfectly deterministic — they retrace their paths exactly). White noise scores 0.71 (some diagonal structure by chance). Constants score 0.0 (everything is recurrent, but there's no trajectory to follow).
What fraction form vertical lines? Vertical lines mean: the signal gets stuck near one state for extended periods. Thue-Morse, Rule 30, and Symbolic Henon all score 1.0. Logistic period-4 scores 0.0 — it recurs but never lingers.
When the signal gets stuck, how long does it stay? Thue-Morse and Rule 30 score 249.5 (maximal trapping — the symbolic dynamics creates long laminar stretches). Exponential chirp scores 0 (never stays anywhere).
How varied are the diagonal line lengths? Thue-Morse and Rule 30 maximize this at 6.2 bits: they revisit their trajectory at many different timescales. Simple periodic orbits have low entropy (one dominant recurrence period).
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.5 (Period-4) | chaos | 250.0000 |
| Logistic r=3.74 (Period-5 Window) | chaos | 250.0000 |
| Symbolic Henon | exotic | 249.5000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Exponential Chirp | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.5 (Period-4) | chaos | 1.0000 |
| Logistic r=3.83 (Period-3 Window) | chaos | 1.0000 |
| Logistic r=3.2 (Period-2) | chaos | 1.0000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Exponential Chirp | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Thue-Morse | exotic | 6.2066 |
| Rule 30 | exotic | 6.2066 |
| Symbolic Henon | exotic | 6.2066 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Exponential Chirp | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Thue-Morse | exotic | 1.0000 |
| Rule 30 | exotic | 1.0000 |
| Symbolic Henon | exotic | 1.0000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.83 (Period-3 Window) | chaos | 497.0000 |
| Rule 30 | exotic | 497.0000 |
| Symbolic Henon | exotic | 497.0000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Exponential Chirp | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Quantum Walk | quantum | 1.0000 |
| Symbolic Lorenz | exotic | 1.0000 |
| Morse Code | waveform | 1.0000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Exponential Chirp | exotic | 0.0000 |
| Source | Domain | Value |
|---|---|---|
| Thue-Morse | exotic | 249.5000 |
| Rule 30 | exotic | 249.5000 |
| Symbolic Henon | exotic | 249.5000 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Logistic r=3.5 (Period-4) | chaos | 0.0000 |
Recurrence Quantification drives the gap between the ordinal view's C3 (symbolic dynamics: high recurrence, high laminarity) and C2 (smooth correlated noise: low recurrence). In the solar eclipse VLF investigation, laminarity and determinism were significant only during the eclipse-active interval — the D-layer collapse changed the signal's recurrence structure in real time.