Smale's Horseshoe Map - Chaos Theory | Lecture 18

In this lecture, I introduce Smale’s horseshoe map, one of the most beautiful and influential examples in dynamical systems theory. The horseshoe map provides a concrete mechanism for how simple geometric operations—stretching, folding, and reinserting—can generate infinitely complex behavior. Through this example, I show how deterministic systems can produce chaos, sensitive dependence on initial conditions, and intricate invariant sets with fractal structure. I then introduce the basics of symbolic dynamics, explain the structure of stable and unstable sets (particularly in relation to fixed points), and discuss the abundance of periodic orbits generated by the map. We also examine the presence of chaotic transients, highlighting how intricate dynamics can appear even before trajectories settle into long-term behavior. This lecture offers a structured and accessible introduction to one of the foundational models of chaotic dynamics. Learn more about the history of Smale's horseshoe map and its dynamics: https://en.wikipedia.org/wiki/Horsesh... More information on the instructor: https://hybrid.concordia.ca/jbrambur/ Follow @jbramburger7 on Twitter for updates. Thumbnail image attributions: 1. “Cantor3D3.png”, by User:A Horsfall, licensed under Creative Commons Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0). Source: Wikimedia Commons, file “File:Cantor3D3.png” (illustration of a 3D Cantor set). 2. “Markov_Partition_for_Smale_Horseshoe.svg”, by User:Opensourceway, licensed under Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0). Source: Wikimedia Commons, file “File:Markov_Partition_for_Smale_Horseshoe.svg” (diagram of a Markov partition for the Smale horseshoe map). 3. “Smale_Horseshoe_Map.svg”, by User:Opensourceway, licensed under Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0). Source: Wikimedia Commons, file “File:Smale_Horseshoe_Map.svg” (illustration of the Smale horseshoe map).