How the signal's values behave as starting conditions for a complex dynamical system.
Maps pairs of consecutive data values to complex starting positions z₀ in the plane, then iterates z → z² + c (with a fixed parameter c). Some starting conditions escape to infinity; others are trapped in bounded orbits. The fractal boundary between escaping and trapped regions is the Julia set. The data's distribution of escape times and orbit stabilities reveals its "dynamical texture" as seen through this particular lens.
Shannon entropy of the escape-time distribution. High entropy means the data samples many different escape times — the starting conditions are spread across different dynamical basins. Coupled map lattice (4.45) scores highest: its spatiotemporal chaos creates a rich diversity of starting conditions. Gaussian noise (4.39) is close behind. Logistic period-2 and Collatz parity score 0.0 — their starting conditions are degenerate (only 1-2 distinct z₀ values), so there's only one escape time.
Mean absolute value of the final iterate for non-escaping orbits. Devil's staircase (938) has the highest stability: its starting conditions land in far-flung bounded orbits that stay large without escaping. Berry random wave (911) and seismograph (911) are similar — their broad amplitude distributions create starting conditions near the boundary of the Julia set, where orbits are large but trapped. Constants score 0.0 (degenerate single-point orbit).
| Source | Domain | Value |
|---|---|---|
| Coupled Map Lattice | chaos | 4.4501 |
| Gaussian Noise | noise | 4.3863 |
| Tohoku Aftershock Intervals | geophysics | 4.3109 |
| ··· | ||
| Collatz Parity | number_theory | -0.0000 |
| Logistic r=3.2 (Period-2) | chaos | -0.0000 |
| Collatz Gap Lengths | number_theory | -0.0000 |
| Source | Domain | Value |
|---|---|---|
| Devil's Staircase | exotic | 938.4536 |
| Berry Random Wave | quantum | 911.2338 |
| Seismograph (ANMO) | geophysics | 911.0774 |
| ··· | ||
| Constant 0xFF | noise | 0.0000 |
| Constant 0x00 | noise | 0.0000 |
| Collatz Gap Lengths | number_theory | 0.0000 |
Julia Set geometry acts as a "nonlinear filter" — it transforms the data through a fixed dynamical system and asks what the output looks like. This is conceptually different from geometries that analyze the data's own dynamics. It's most useful for separating signals with similar linear statistics but different amplitude distributions: two signals with the same mean and variance can have very different Julia escape profiles if one has heavy tails (creating starting conditions near the Julia set boundary) and the other is Gaussian (starting conditions far from the boundary).