>[!abstract] >**Smale’s horseshoe** is a classic concept in in chaos theory and dynamical systems, used to explain how complex and chaotic behavior can arise from simple deterministic systems. It is a transformation map applied to an object (e.g., a square piece of dough) that goes as follows: >1. Stretch it vertically, making it taller. >2. Compress it horizontally, making it narrower. >3. Bend and fold it into a horseshoe shape (like a “∩”) so that it is a square again. >4. Repeat the process a few times. > >Following this deterministic procedure, points that were close may end up far apart, and points that were far apart may end up close together. More importantly, certain points keep returning again and again, although never exactly in the same place. They follow complex and non-repeating orbits characteristic of chaotic behavior. >[!example] Additional references >- [[Gleick, 1998|Gleick (1988, p. 51)]] >[!related] >- **North** (upstream): [[Nonlinear system]], [[Topological dynamics]] >- **West** (similar): [[Baker’s map]] >- **East** (different): — >- **South** (downstream): [[Strange attractor]]