EXAMPLE 11 Chaos in Manhattan

You may already know from experience that getting around Manhattan can be a chaotic experience, in the ordinary sense of the word. How is Manhattan’s subway system also chaotic in the mathematical sense?

  1. Orbitally dense: Subway trains “orbit,” periodically visiting subway stops, and everybody lives near a subway stop.
  2. Transitive: You can get close to anywhere else in Manhattan by taking the subway.
  3. Sensitive: If you get on the wrong train, you could wind up miles from where you want to be.

Because the system covers the island of Manhattan, #1 is actually a consequence of #2. Also, anyone who has gotten on the wrong subway train or bus realizes that #3 is an inevitable consequence of #1 and #2. So, in fact, #1 and #3 both follow from #2—a conclusion that is true not just of this Manhattan example but of a large class of dynamical systems.