Attractors

A Rust project to generate and explore strange attractors visually and easily. The emphasis is on using strange attractors as an artistic medium, and not explicitly on the maths behind them (although the maths is unavoidable).

See below for example images of strange attractors to give you a feel for what they look like. After that is some info about getting started with the project and running the code to find your own. Following that is some more background info about strange attractors and the different types.

Examples of strange attractors

See more images in the imgs directory. They're all very high resolution (about 7000 pixels on each side) so you can print them out at An size at 300 DPI.

Get started and making your own strange attractors

Install the rust compiler and cargo by following the instructions at install rust or by running the command:

curl --proto '=https' --tlsv1.2 -sSf https://sh.rustup.rs | sh

Then clone this project:

git clone https://github.com/beyarkay/attractors.git

Change directory and run the project in --release mode:

cd attractors && cargo run --release

The extra optimisations given by the --release flag are really required for a smooth experience.

Two windows open: the attractor itself, and a square Map window to its left. The terminal lists every key; press ? in the attractor window to list them again. The attractor window's title shows the current parameters, palette and step size.

Click the attractor window to focus it, then change the parameters with the two rows of keys under your right hand. The keys act on a window of four parameters at a time, which for a Clifford attractor is all of a, b, c and d:

u i o p    increase the 1st, 2nd, 3rd, 4th parameter in the window
j k l ;    decrease the 1st, 2nd, 3rd, 4th parameter in the window
[ / ]      slide the window left / right, for attractors with more than four
t / T      multiply / divide the step size by 10

Hold a key to keep changing a parameter.

Kinds of attractor

Press a / A to switch to the next / previous kind of attractor, or start with one directly:

cargo run --release -- --attractor tinkerbell

There are two families. Maps jump from one point on the plane to the next:

• clifford, dejong, kingsdream, svensson, tinkerbell
• henon and lozi (2 parameters), duffing, ikeda (1 parameter)
• hopalong, gumowskimira, bedhead (Sánchez's "Bad Hair Day")
• sprott, Sprott's general quadratic map with 12 parameters

Flows move smoothly through 3D space, and are drawn as a view from one side:

• lorenz, rossler, chen, aizawa (6 parameters)
• thomas and halvorsen (1 parameter each)
• arneodo, burkeshaw, yuwang

Each kind keeps its saved images and special attractors in its own cache/<name>/ directory. With a single parameter, the Map window's top left plot becomes a slider.

To add a new kind, write a struct holding its parameters in src/attractors.rs, implement the Attractor trait for it (its name, parameter names, default parameters, a range for the map, and the formula), and add it to the list in all_attractors! in src/lib.rs. For a flow, next takes one step with the rk4 helper, and PROJECTION picks which two of x, y and z to draw. If you don't know a box that contains the attractor, leave out bounds and it will be estimated from a sample of points. The tests check that every attractor's default parameters are chaotic.

Colours

Key	Effect
c / C	next / previous palette
d / D	more / less contrast (higher darkens the faint areas)
n / N	blend frames together less / more

The palettes are Inferno, Aurora, Ice, Gold, Neon, Magma, Cubehelix, Turbo, and two light-background ones for printing on paper, Ink and Terracotta.

The preview is coloured so that it looks like the saved image shrunk down to the window's size, so what you see is what you'll print. Blending frames together (N) makes changes fade in like echoes, which reduces flashing and is gentler for people with photosensitive epilepsy.

The Map window

If you look at the Map window, you'll see four 2D plots. Each of the four plots shows two of the parameters in the current window, and together they can guide you around the parameter space. Click and drag on a plot to set its two parameters. Attractors with fewer than four parameters show fewer plots. For a Clifford attractor the plots are oriented like this:

       b            b
       |            |
  -----+----a  -----+----c
       |            |
       |            |

       d            d
       |            |
  -----+----a  -----+----c
       |            |
       |            |

And the window looks like this:

As you change parameters, the white cross-hair on each plot moves to show the current values of that plot's two parameters.

The dots on the map, like constellations, are special attractors. A dot glows amber as the parameters the plot doesn't show get close to that attractor's, and turns green once you're right on it. Press m to mark the current attractor as special. Special attractors are saved to cache/<name>/special.txt and persist between runs.

Other commands

Key	Effect
r	jump to a random special attractor
R	random parameters, retried until the attractor is chaotic
s / S	save an A3 / A0 png at 600 DPI into cache/<name>/
z / Z	zoom in / out
x	reset zoom and pan
y	run more simulation steps for a smoother preview
m	mark the current attractor as special
Esc	quit

You can also save every attractor in a file of parameter lines like a=..,b=..,c=..,d=.. without opening a window:

cargo run --release -- cache/clifford/special.txt A3
cargo run --release -- --attractor henon henon_params.txt A3

What are Strange Attractors

Strange attractors are (usually) a recursive formula which take in a point in 2D or 3D space and (using a set of parameters) return a different point in that space. An example equation might be:

x_new = sin(a * y_old) + c * cos(a * x_old)
y_new = sin(b * x_old) + d * cos(b * y_old)

That new point is then sent back into the same recursive formula to generate another point. And another, and another, etc. Those points are then coerced into a pixel grid, and the resulting 2D or 3D histogram sometimes looks spectacular.

See these links for examples of strange attractors.

• Clifford Attractors
• De Jong Attractors
• Lyapunov Exponent Attractors
• Sprott Polynomial Attractors
• Juan Attractors
• Den Tsucs Attractors
• Sánchez 'Bad Hairday' Attractors, also see here
• Arneodo Attractors
• Catrián Attractors
• Burke-Shaw Attractors
• Yu-Wang Attractors
• Tinkerbell Attractor
• The King's Dream Attractors
  • float xnew = sin(y*b) + c*sin(x*b); float ynew = sin(x*a) + d*sin(y*a);
• Others defined on wikipedia/chaotic_maps
• Others defined on Chaoscope

TODO

• Add Diagnostics: fractal dimension greater than 1.5 => Chaotic?
• Add Diagnostics: positive Lyapunov exponent => Chaotic
• Add a way to create videos
• Add a way to line up parameter changes with timestamps so that the moving attractors will 'react' in time with the music
• Use crate meval to auto-parse maths formulas for new attractors
• Add in commands to pan and zoom the display window
• Add in better colour palettes (maybe https://crates.io/crates/palette, https://crates.io/crates/color-art, or https://crates.io/crates/colorous?)