Hi everyone! I hope you enjoy my first video. I've known about Markus-Lyapunov Fractals for a few years now, and it surprised me that I couldn't find any video explaining how they work - so I thought I'd take a stab at it myself! This is also my submission for Summer of Math Exposition 2.

Basic Lyapunov Fractal Demo in Shadertoy:
https://www.shadertoy.com/view/fldBWr

Further Reading:
A 1990 paper by Mario Markus that introduces the fractal: https://aip.scitation.org/doi/pdf/10....

Chapters:
0:00 Intro
1:00 Maps
3:20 The Logistic Map
5:44 The Bifurcation Diagram
7:29 The Lyapunov Exponent
11:35 Markus's Modified Logistic Map
16:51 The Markus-Lyapunov Fractal
18:55 Overlapping Branches
20:37 3-D Bifurcation Diagram
23:18 3-D Lyapunov Fractals
23:59 Beyond the Logistic Map

Image credits:
Mario Markus picture: https://commons.wikimedia.org/w/index...
Lyapunov picture: https://commons.wikimedia.org/w/index...
Lyapunov picture (in thumbnail): https://commons.wikimedia.org/wiki/Fi...