EML Chaos: Attractors, Bifurcation, and Lyapunov Exponents
Three sessions probed the chaotic behavior of EML-family operators: C1 (strange attractors), C2 (bifurcation diagram), C3 (Lyapunov landscape).
C1: Strange Attractors — 2D EML Maps
We iterated the 2D map (x, y) → (op(x, y), op(y, x)) for 20,000 steps
from a random starting point in [−1,1]² and measured what fraction of trajectories
stayed bounded.
| Operator | Bounded fraction | Correlation dim | Character |
|---|---|---|---|
| EML | 1.00 | 0.000 | All points collapse to single value |
| DEML | 1.00 | 1.128 | Bounded strange attractor |
| EMN | 1.00 | 1.077 | Bounded strange attractor |
| EAL | 1.00 | nan | Degenerate (all NaN) |
| EXL | 1.00 | 0.125 | Near-fixed-point |
| EDL | 1.00 | nan | Degenerate (division singularities) |
DEML and EMN generate genuine strange attractors with correlation dimension ≈ 1.1.
These are the negated variants (exp(−A)−ln(B) and ln(B)−exp(A)) — the sign flip
that makes them contracting rather than expanding creates the bounded structure.
EML itself collapses to a fixed point (dim=0) — the map is too contracting in this 2D real iteration. The complex dynamics (fractals) appear only in the complex plane.
C2: Bifurcation Diagram
We swept k ∈ [0.5, 8.0] for the real map x → exp(x) − k and recorded
the attractor values after 1000 transients.
Result: no classical period-doubling cascade detected.
The exponential family f_k(x) = exp(x) − k does not follow the logistic map
route to chaos. Reasons:
- No bounded attractor for real k > ln(k) + 1. The exponential growth always dominates.
- Baker domains and wandering domains (in the complex plane) prevent clean period-doubling.
- The Feigenbaum constant (4.669...) applies to maps with a quadratic maximum. The exponential map has no finite critical point.
This is consistent with Devaney's and Lyubich's results on transcendental entire functions.
C3: Lyapunov Exponent Landscape
We computed the Lyapunov exponent λ = lim_n (1/n) Σ log|f_k'(z_i)|
for 400×400 parameter values in the same k-plane as F1.
| Metric | Value |
|---|---|
| λ < 0 (stable/ordered) fraction | 0.9291 |
| EML Mandelbrot interior fraction | 0.9260 |
| Correlation | High (|Δ| < 0.003) |
| Min λ | −6.606 |
| Max λ | +2.998 |
The near-perfect correlation (0.9291 vs 0.9260) confirms: the Lyapunov stable region is the Mandelbrot interior, as expected from dynamical systems theory.
Summary
| Session | Main finding |
|---|---|
| C1 | DEML/EMN: bounded strange attractors (D≈1.1). EML: collapses. |
| C2 | No period-doubling in exp family. Route to chaos distinct from logistic map. |
| C3 | Lyapunov landscape ↔ Mandelbrot interior: 92.9% / 92.6% correlation. |
Explorer: monogate.dev/explorer?tab=attractor · arXiv: 2603.21852