Largest Lyapunov Exponent

A map of where the Clifford attractor becomes chaotic. Each pixel is a different pair of parameters, coloured by how quickly two neighbouring starting points fly apart under that setting.

What you are looking at

Take two starting points a hair's breadth apart and run the attractor forward from both. In a calm region they stay together. In a chaotic one they separate exponentially, and the rate of that separation is the largest Lyapunov exponent.

• Positive (bright) means chaos: nearby points diverge, and long-term prediction is impossible.
• Around zero marks the boundary - often where the most intricate attractors live.
• Negative (dark) means the orbit settles into a cycle or a fixed point.
• Grey means the orbit ran off to infinity, so no exponent could be measured.

The attractor has four parameters, so its parameter space is four-dimensional and cannot be shown at once. Two of them become the axes of each heatmap, a third is stepped across the stack of images, and the fourth is fixed by the slider. Moving that slider walks you through the remaining dimension.

Mapping

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Parameter ranges

Compute

Status: idle

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Built with

A pool of Web Workers, one per core, each computing horizontal strips of the parameter grid. The page renders a fast preview on load and while you drag; press Render for the full-resolution version.