STUDY #21 · 2026 · IN OBSERVATION

de Jong / Clifford Maps

A model-driven visual study of strange-attractor measures.

MOVING IMAGE — THE GATE CROSSING silk storm #2632 ⇄ rose #816 · gate s ∈ [0.872, 0.926] · 262,144 walkers · deepsky

WHAT IS THIS

The de Jong and Clifford maps are simple rules that move a point by folding the plane with sines and cosines. Each step is a teleport — no path connects one position to the next — yet where the point spends its time converges to a fixed, reproducible cloud: the strange attractor's natural measure. The image is a long exposure of one point.

This study runs 262,144 such points in real time on the GPU, every step of every walker deposited into a shared exposure. The individual path decorrelates in about forty steps (measured 56 vs 55 predicted); the cloud does not care. The path is unpredictable; the cloud is not.

silk storm #2632 — the invariant measure as a long exposure a 2.4701 · b −1.575242 · c 1.136521 · d −1.630505 · λ₁ 0.39 · D_KY 1.61 · deepsky

Motif iterated maps / invariant measure / one point, long exposure

Method A small simulator was generated and modified with AI assistance, then ported to a real-time GPU (GLSL) renderer. The visual output was selected through parameter exploration.

Observation Of 9,000 random parameter draws, 56% were chaotic — but only a third of those clouds contract area (D_KY < 2), and only they keep a face; at D_KY = 2 every attractor melts into the same fog. Every scanned segment between two clouds crossed periodic windows (107 across 9 segments). At one such gate the sky drains into two stars that fuse into one, holding the entire plane, before chaos re-ignites.

Reference A. K. Dewdney, "Computer Recreations: Probing the strange attractions of chaos," Scientific American, vol.257, no.1, 108-111 (July 1987); J. C. Sprott, "Automatic generation of strange attractors," Computers & Graphics, vol.17, no.3, 325-332 (1993).

Tools Python / NumPy / three.js / React / GLSL / ffmpeg / AI coding assistant

Year 2026

This is not a scientific simulation result, but a visual interpretation of the phenomenon.

SAME FAMILY, THE THREAD DISSOLVED

Lorenz draws the thread. de Jong draws the fog the thread leaves behind.

Study #05 — Lorenz Study #21 — de Jong / Clifford

Time a continuous flow — an ODE traces one unbroken thread a discrete map — each step a teleport, no path between positions

What is drawn the orbit itself — a glowing filament the measure — where the orbit lives, not where it goes

The line exists — tangent, loop, wing never existed — consecutive positions land far apart, unrelated

What chaos leaves a shape traced and retraced a cloud — the path is unpredictable; the cloud is not

PARAMETERS EXPLORED

param meaning effect on the image

a, b, c, d the four fold frequencies — one quadruple, one cloud larger magnitudes fold the plane finer and lift the dimension toward fog; smaller collapse the cloud into a few periodic points

λ₁ largest Lyapunov exponent (measured) only λ₁ > 0 makes a cloud — 56% of random parameter sets are chaotic

D_KY Kaplan–Yorke dimension — the thinness of the cloud (measured) the ladder: 1.0 wire → 1.1–1.35 veil → 1.35–1.65 silk storm → 1.65–2 smoke → 2.0 the same fog, where individuality dies

gate s position on the segment joining two chaotic clouds s ∈ [0.872, 0.926] is the periodic window — the sky drains into a few stars

decay the exposure's forgetting rate 1.0 is a pure long exposure (the stills); below 1 the cloud breathes (the film)

Each image below records its exact parameter set.

THE MATHEMATICS the model behind the images

Two trigonometric folds of the plane, run by one engine. A point teleports each step; the picture is the histogram of a hundred million landings — the attractor's natural measure.

x_{n+1} = \sin(a\,y_n) - \cos(b\,x_n), \qquad y_{n+1} = \sin(c\,x_n) - \cos(d\,y_n)

The de Jong map: four fold frequencies (a, b, c, d). The orbit is bounded in [−2, 2]² for ever — the failure mode is not escape but collapse to a periodic orbit; only λ₁ > 0 makes a cloud.

x_{n+1} = \sin(a\,y_n) + c\,\cos(a\,x_n), \qquad y_{n+1} = \sin(b\,x_n) + d\,\cos(b\,y_n)

The Clifford map — the same family with the cosines scaled by c and d. Attributed to Clifford A. Pickover; no primary published source has been located.

\lambda_1 + \lambda_2 = \big\langle \ln \lvert \det J \rvert \big\rangle

Two Lyapunov exponents by QR iteration of tangent vectors (Benettin). This identity is the engine's internal consistency check — on the Hénon anchor it holds to 1.8 × 10⁻¹⁴, with λ₁ = 0.4209 against the literature value 0.41922.

D_{KY} = 1 + \frac{\lambda_1}{\lvert \lambda_2 \rvert}

The Kaplan–Yorke dimension — the thinness of the cloud. λ₁ > 0 with λ₁ + λ₂ < 0 is chaos that still contracts area: a strange attractor with a face. At D_KY = 2 the contraction is gone and every cloud is the same fog.

A visual interpretation of the model, not a claim of scientific precision. Ergodicity is unproven for this family; what was verified is empirical — visit histograms from independent seeds converge at the N^(−1/2) noise floor, and float32 and float64 orbits separate completely in tens of steps while their measures agree.

SELECTED STILLS — 6

silk storm #2632 — the hero cloud in deepsky, stretch tinting calm blue to worked gold a 2.4701 · b −1.575242 · c 1.136521 · d −1.630505 · λ₁ 0.39 · D_KY 1.61 · deepsky

the same measure in plate — density alone, the census unadorned a 2.4701 · b −1.575242 · c 1.136521 · d −1.630505 · λ₁ 0.39 · D_KY 1.61 · plate (density)

the same measure in aurora — the jump observable forward, violet where arrivals teleported far a 2.4701 · b −1.575242 · c 1.136521 · d −1.630505 · λ₁ 0.39 · D_KY 1.61 · aurora (jump)

rose current #816 — the far anchor of the hero flight a −0.589131 · b −2.948399 · c 2.602443 · d −0.650287 · λ₁ 0.59 · D_KY 1.59 · deepsky

petal torus #2335 — the thinnest cloud, one fold above a wire a 2.709972 · b 0.530365 · c −1.162638 · d −2.922272 · λ₁ 0.02 · D_KY 1.08 · emission

silk fox #4074 — soft silk from the middle of the ladder a 2.531456 · b 1.327715 · c 2.037432 · d 1.11771 · λ₁ 0.23 · D_KY 1.42 · inkfire

PROCESS — PARAMETER SWEEPS

The exploration re-runs Sprott's 1993 procedure from scratch — 9,000 random parameter draws (6,000 de Jong, 3,000 Clifford) classified by Lyapunov exponent, the chaotic hits then shelved by Kaplan–Yorke dimension and ordered by entropy × occupancy. Chaos is not the rare thing: 56% of draws make a cloud. The rare thing is thinness — only a third of the clouds still contract area, and only they keep a face. The sheet shown is the silk-storm shelf, where the hero and the rose both live.

the silk-storm shelf — one band of the dimension ladder de Jong search · 6,000 draws in [−3,3]⁴ · λ₁ > 0 · D_KY ∈ [1.35, 1.65] · ordered by entropy × occupancy · #2632 and #816 on this shelf

SIGNATURE — THE GATE BETWEEN TWO CLOUDS

Between any two clouds, every straight road passes stars.

Scan a straight line in parameter space between two chaotic clouds and the Lyapunov exponent does not stay positive: it dips below zero in periodic windows, where the whole sky drains into a few bright points. Nine segments were scanned; all nine crossed windows — 107 of them. No straight road avoids the stars. The hero flight rides one such gate, s ∈ [0.872, 0.926] on the segment from silk storm to rose current: inside it the attractor period-doubles, and on the ride home two stars close in and fuse — walkers crawling in pearl strings under critical slowing — until one star holds the entire plane.

For that moment the artwork's sentence turns literal. The picture was always a long exposure of a single point; at the gate, the exposure and the point coincide — the cloud remembers it is one point — before chaos re-ignites and the memory spreads back into weather.

why the rare thing is not chaos but thinness — the classification, the dimension ladder, and the λ₁ trace across the gate 6,000 draws in ±3⁴ · chaotic 56% / collapse 39% · ladder D 1.05 → 2.00 · gate window s ∈ [0.872, 0.926]

COLOUR = NARROWBAND ASTROPHOTOGRAPHY

Colour here follows the method of narrowband astrophotography — invisible observables mapped to visible hues, the way SII / Hα / OIII become red / green / blue in the Hubble palette. Nothing in these clouds has a colour of its own; what they have is dynamics, and the same measure can be observed in more than one way.

Three dynamical observables are graded independently: density — visit frequency, the measure itself — carries luminance; stretch — the largest singular value of the local Jacobian, how violently the fold deforms a neighbourhood — steers the tint from calm blue toward worked gold; jump — how far arrivals teleported — lifts a violet halo where the map throws points across the whole plane. The storm triptych in the stills reads one cloud through each observable in turn.

the same cloud read by jump — a violet halo where the map throws points across the plane a 2.4701 · b −1.575242 · c 1.136521 · d −1.630505 · aurora (jump)

The values are real dynamics; the hues are artistic approximations of the false-colour tradition, not measurements.

REFERENCES

A. K. Dewdney. "Computer Recreations: Probing the strange attractions of chaos." Scientific American, vol.257, no.1, 108-111 (1987).
J. C. Sprott. "Automatic generation of strange attractors." Computers & Graphics, vol.17, no.3, 325-332 (1993). https://doi.org/10.1016/0097-8493(93)90082-K
M. Hénon. "A two-dimensional mapping with a strange attractor." Communications in Mathematical Physics, vol.50, no.1, 69-77 (1976).
G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn. "Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory." Meccanica, vol.15, no.1, 9-20 (1980).
J. L. Kaplan, J. A. Yorke. "Chaotic behavior of multidimensional difference equations." Functional Differential Equations and Approximation of Fixed Points, Lecture Notes in Mathematics, vol.730, Springer, 204-227 (1979).
J. C. Sprott. "Chaos and Time-Series Analysis." Oxford University Press (2003).

INTERACTIVE STUDY

One point, teleported through a trigonometric fold — the picture is not its path but its probability, a long exposure of a single point. This is a small window onto the model behind the finished works, a deliberately simplified instrument: 16,384 walkers depositing a density-only exposure, separate from the full engine (262,144 walkers, three-observable grading) used to author the works. Presets are curated; the gate spans a real periodic window — drag fold drift slowly across the ticks and watch the sky drain into two stars that fuse into one: the cloud remembers it is one point.

SIMPLIFIED INSTRUMENTDE JONG MAP · LONG-EXPOSURE MEASURE · LIVE

This interactive study is not intended as a scientifically validated reproduction. It is a visual interpretation generated from an implemented model and curated parameter exploration — and it is a deliberately simplified instrument, separate from the full engine used to author the finished works.