# The Standard Map

Submitted by michael on Mon, 02/04/2019 - 17:40

The standard map is the first chaotic thing I remember seeing, probably in an AfterDark screen saver. It has an interesting physical interpretation involved rotors, but that isn't important for the pretty pictures.

$$x_{n+1} = x_n + K \sin(y_n) \mod 2\pi$$

$$y_{n+1} = y_n + x_n + K \sin(y_n) \mod 2\pi$$

The result is a bunch of points the unit square scaled up by a factor of $2\pi$. $K$ is randomly chosen on $(0, 4)$. Each step of the computation takes 200 points and maps them into 200 other points following the formula above. The colour of each point is set based on its initial $y$ coordinate.

Start and Stop work as indicated. Step draws one round of pixels in the computation. Reseed generates a new set of initial values and continues the computation. Clear blanks the canvas and continues drawing. Restart will clear the canvas, pick a new $k$ and reseed the initial values. Save will download an image to your computer.