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.

There will be some console output explaining the available commands and what they do, and two windows should pop up: a small diagnostics window (which you can ignore) a square Map window (which you can ignore), and a larger square window.

Click on the large square window to focus it. The attractor will automatically be drawn, and you can change the 4 parameters fed to the attractor (named a, b, c, d) with vim-like key bindings:

j -> decrease a; k -> increase a; 
J -> decrease b; K -> increase b;
h -> decrease c; l -> increase c; 
H -> decrease d; L -> increase d;

The Map window

If you look at the Map window, you'll see four 2D plots. Each of the four plots shows two parameters, and together they can guide you around the 4D parameter space. The plots are: a-b, a-d, c-b, and c-d. They're oriented like this:

       b            d     
       |            |     
  -----+----a  -----+----a
       |            |     
       |            |     

       b            d     
       |            |     
  -----+----c  -----+----c
       |            |     
       |            |     

And the window looks like this:

As you press hjklHJKL the 4 cross-hairs on the Map window will move around the four plots so the cross-hair on the a-b plot will show the current values for the a and b parameters, the cross-hair on the a-d plot will show the current values for the a and d parameters, and so on.

There are also lots of white dots on the map, like constellations. Each dot represents a special or interesting strange attractor, and you can mark a particular attractor as special by pressing the m key. Special attractors are saved to cache/clifford/special.txt and will persist between runs of the program. Try navigate to different special attractors. It's takes getting used to, since all four cross-hairs need to line up different white dots but it's a fun exercise.

Other commands

Additionally, you can select a random set of parameters by pressing R:

`[R]` => Randomize the Clifford parameters and re-run the attractor with these new parameters

You can print off an attractor by pressing p which will save it as a .png in cached/clifford/.

`[P]` => Save the attractor in high resolution to disc as png (enabled: true)

You can also change the hue of the attractor by pressing e or E:

`[E]` => Increase or decrease the LCH hue intercept by 0.01 (enabled: true)

Pressing n or N will increase/decrease the decay factor between 0 and 1. Values closer to zero will cause previous attractors to blend into future attractors like echoes, which is a nice effect but primarily is good at reducing the strobing and stuttering effect dangerous to those with photosensitive epilepsy.

`[N]` =>  Change how quickly one attractor merges to another (helps with photosensitive epilepsy)

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