Circle Map

Channel:

Brad Smith

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7,300

Published on February 16, 2019 7:08:39 PM ● Video Link: https://www.youtube.com/watch?v=-l12chlb31M

Duration: 2:01

1,989 views

60

An animation smoothly varying parameters of the Circle Map.

x ← x + Ω - K * sin(2πx) / 2π mod 1

It began as a simple iterative system modelling an interaction between a spring and a rotating motor, but when you vary the speed and strength parameters, interesting regions of stability and chaos appear.

http://mathworld.wolfram.com/CircleMap.html

In this rendering, each frame represents a single Ω, varying from 0 to 1 across the animation. Each vertical line represents a single K, varying from 4π to 0 from left to right.

For each vertical slice, many starting values for x are chosen and iterated, accumulating a count of how many times each pixel is visited. Bright spots in the image represent a point of stability where they tend to converge.

Horizontally you can see it tends to split into pairs, recursively as K increases, eventually approaching total chaos, but then surprisingly returning to stability again periodically.

It's a form of the classic bifurcation pattern seen in many chaotic systems:
https://en.wikipedia.org/wiki/Bifurcation_diagram

More info and example program:
https://www.patreon.com/posts/24826459

An animation of the Circle Map's evil twin can be seen here:
https://youtu.be/3kBwqK-9Ckc


Music is something I wrote a few years ago for a one hour compo:
http://battleofthebits.org/arena/Entry/lava_man_and_lava_woman_lava_love_time.mp3/8944/