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 difference of the continuous potential between consecutive starting conditions. Uses the normalized iteration count (n + 1 - log(log|z|)/log(2)) to remove discrete escape-time steps. English Literature (35.2) and Euler Totient (31.4) score highest — their byte sequences create starting conditions that jump between very different escape regimes. Constants and logistic period-2 score 0.0 (degenerate starting conditions). This measures sequential correlation in the Julia escape landscape. Evolved via ShinkaEvolve.
Variance of the continuous potential across all starting conditions. Devil's Staircase (909) and Berry Random Wave (883) score highest — their broad amplitude distributions sample widely across the Julia set's escape-time landscape. Morse Code and constants score 0.0. Complements escape_entropy by measuring the spread rather than the diversity of escape behavior. Evolved via ShinkaEvolve.
| Source | Domain | Value |
|---|---|---|
| Ambient Microseism | geophysics | 4.5255 |
| Accel Walk | motion | 4.4623 |
| Neural Net (Pruned 90%) | binary | 4.4550 |
| ··· | ||
| Spectral Form Factor | quantum | 2.5153 |
| ETH/BTC Ratio | financial | 2.5381 |
| Logistic r=3.2 (Period-2) | chaos | 2.5671 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | 7045.5463 |
| Logistic r=3.5 (Period-4) | chaos | 7001.6005 |
| Sine Map (Feigenbaum) | chaos | 6957.6571 |
| ··· | ||
| Tank Drain Cascade | exotic | 10.8135 |
| Stochastic Resetting Walk | exotic | 17.2704 |
| Rule 30 | exotic | 28.5342 |
| Source | Domain | Value |
|---|---|---|
| Logistic r=3.2 (Period-2) | chaos | 46408596.8745 |
| Logistic r=3.5 (Period-4) | chaos | 46322561.8590 |
| Sine Map (Feigenbaum) | chaos | 46099674.8410 |
| ··· | ||
| Tank Drain Cascade | exotic | 117.6809 |
| Stochastic Resetting Walk | exotic | 479.4014 |
| Rule 30 | exotic | 1038.6669 |
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). A sister metric stability (variance of integer-escape counts) is class-only — kept in compute_metrics for direct callers but dropped from the atlas because it is r=+1.000 with potential_variance on real data: the continuous escape-time correction sits in [0, 1) per point, so integer-variance dominates and the two columns are indistinguishable.