!header Lyapunov Fractals
The logistic map is one of the most famous and applicable bits of math, used to model population growth and quite a bit else. It depends on a single parameter \(r\) that corresponds to the growth rate of the population when it’s not near the enironment’s capacity. If we instead flip \(r\) between two different values, we get something quite a bit more complicated, called a Lyapunov fractal. Here, every point with \(x\) and \(y\) coordinates between \(0\) and \(4\) (the same range that the standard logistic map can accept) cooresponds to a pair of values for \(r\). The manner in which we flip between those two values depends on the generating string — for example, a string of ABBAB will use \(x\) for the first iteration, \(y\) for the second and third, \(x\) for the fourth, and then \(y\) for the fifth, before repeating this pattern indefinitely. This means that for any string, the diagonal from bottom-left to top-right matches the logistic map. The color is chosen so that the red and green channels reflect the degree to which the point is influenced by \(x\) and \(y\), respectively, and the blue component is derived from the stability of the solution overall — regions with little or no blue tend to be closer to chaos. As always, drag, pinch, and scroll to move around.
resolution 500 Resolution
generating-string AB Generating String
generate l Generate
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