Fractals, cellular automata, and more!
The first discrete Painlevé equation is the recurrence equation
\[w_n(w_{n+1}+w_n+w_{n-1})=\alpha n +\beta +\gamma w_n\]with $\alpha,\beta,\gamma\in\mathbb C$ constants.
Given initial conditions $w_0,w_1\in\mathbb C$ we can generate a sequence of numbers $(w_n)_{n\in\mathbb{N}}\in\mathbb C$ using the above recurrence equation. Each iterate of the sequence $w_n$ is assigned a colour based on its order within the sequence (i.e. based on $n$). The colours are, in increasing $n$: red, orange, yellow, green, blue, purple, and back to red (specifically they’re mapped using the hue component of the HSL colour model.
For visualization, the iterates $w_n$ are stereographically projected onto the unit sphere.
| $n$ | $w_n$ | Stereographic projection of $w_n$ |
|---|
The graph above shows a plot of plot $|w_n|$ (the complex magnitude of $w_n$) as a function of $n$.