---
tag: 21B8AHPD4P
kind: concept
name: Devaney chaos
msc: [37D45, 37B02]
sources:
  - url: https://en.wikipedia.org/wiki/Chaos_theory
---

## Summary

Devaney chaos is a precise definition of a chaotic map, proposed by Robert Devaney in his textbook on chaotic dynamical systems.
A continuous map of a metric space is chaotic in this sense when three things hold.
It is [[topologically transitive|FR49Y9GQHR]], its periodic points are dense, and it has sensitive dependence on initial conditions.
It turns the informal ideas of [[chaos theory|DBKXCGDKTD]] into conditions that can be checked.
For infinite spaces the third condition follows from the first two.

## Explanations

### L1 human

A chaotic system mixes regular and irregular behaviour in a specific way.
It stirs every region into every other region, so it cannot be split into separate parts.
Regular, repeating orbits are found arbitrarily close to every point.
Yet any tiny change in the starting point can lead to a large difference later, which is the butterfly effect.
The doubling map $x \mapsto 2x \bmod 1$ of the circle has all three features.