Fractals and the Logistic Map - Chaos Theory | Lecture 3

Fractals and chaos theory are often linked in popular imagination, but in this lecture we see them emerge rigorously in the surprisingly simple logistic map. We start by examining its non-chaotic dynamics and then explore what happens when the parameter exceeds 4. In this regime, the only points that remain bounded under all iterates of the logistic map belong to a fractal Cantor set. This set is totally disconnected, closed, has no isolated points, and is uncountable—a remarkable structure arising from such a simple system. This discovery forms a foundation for understanding far more complex dynamical systems and highlights the deep connections between simplicity, fractals, and chaos. Learn more about the Cantor set: https://en.wikipedia.org/wiki/Cantor_set More information on the instructor: https://hybrid.concordia.ca/jbrambur/ Follow @jbramburger7 on Twitter for updates. Image attributions for the thumbnail: 1. “Cantor set (PNG)”, by Thefrettinghand, licensed under CC BY-SA 3.0. Source: Wikimedia Commons, file “File:Cantor.png” (uploaded July 11, 2007). 2. “Logistic Map Bifurcation Diagram, Matplotlib, zoomed.png”, by Morn, licensed under CC BY-SA 4.0. Source: Wikimedia Commons, file “File:Logistic Map Bifurcation Diagram, Matplotlib, zoomed.png”.