Lyapunov fractals
A Lyapunov fractal assigns two parameter values to the axes, alternates them according to a symbolic sequence and colors each pair by the orbit’s average sensitivity.
A two-parameter experiment
The logistic rule xₙ₊₁ = r xₙ(1 − xₙ) normally varies one parameter r. A sequence such as AABAB chooses between horizontal parameter A and vertical parameter B at each iteration, producing a two-dimensional stability field.
A symbol sequence such as AABAB determines which of two parameters is used at each step of the logistic map. Each pixel reads both parameters from the axes, discards a transient and then averages the logarithmic derivative. Changing the symbol sequence reorganizes the stability islands. It is therefore an essential part of the scene, not merely an interchangeable preset name.
Exponent as the color source
The Lyapunov exponent averages the logarithm of the derivative magnitude along the orbit. Negative values indicate local contraction and stable cycles; positive values indicate sensitive chaotic behavior. Undefined or divergent samples require explicit rendering policy.
The Lyapunov exponent λ averages log|r(1−2x)| along the orbit. Negative values indicate contraction and stable long-term behavior in the sampled regime; positive values indicate sensitive separation. Near λ=0 the estimate converges slowly, and short runs create speckled boundary bands. A color scale should mark zero clearly and normalize positive and negative ranges separately rather than mixing them in an arbitrary cyclic gradient.
Feathers are stability boundaries
The recognizable wings and feathers trace transitions among periodic windows and chaos. A gradient can make the result dramatic, but it is mapping a measured dynamical signal—not generating the geometry by itself.
The feather-like forms occur where periodic windows and chaotic regions interlock in a two-dimensional parameter plane. They are contours of related stability, not painted texture. Observation length must grow under magnification because fine windows can have longer transients. An immediate preview may be coarse; the final image must continue refining the exponent estimate.
Sources and further reading
This article summarizes the following specialist sources in original wording. Accessed and editorially reviewed 12 August 2026.
- Lyapunov ExponentWolfram MathWorld
- Simple mathematical models with very complicated dynamicsNature
Continue the idea
The bifurcation diagram
Read the logistic-map bifurcation diagram from stable point through period doubling, chaos and periodic windows.
Chaos is deterministic
Why deterministic equations can become unpredictable, how sensitivity differs from randomness and where fractals enter phase space.
Fractal coloring beyond iteration bands
A guide to smooth iteration, potential, distance, angle, orbit traps, interior coloring and perceptual gradients.