---
tag: RTG32790CW
kind: definition
name: Devaney chaos
msc: [37D45, 37B02]
defines: 21B8AHPD4P
uses: [0VKD1TP7F3, 0VGFHTM75Y, FR49Y9GQHR, T75CZ5H47N, 0BSJSX7ME2]
sources:
  - url: https://en.wikipedia.org/wiki/Chaos_theory
---

## Definition

Let $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $f \colon X \to X$ be a [[continuous map|0VGFHTM75Y]].
The map $f$ is *chaotic in the sense of Devaney* if:

1. $f$ is [[topologically transitive|FR49Y9GQHR]].
2. The set of [[periodic points|T75CZ5H47N]] of $f$ is [[dense|0BSJSX7ME2]] in $X$.
3. $f$ has *sensitive dependence on initial conditions*: there is $\delta > 0$ such that each $x \in X$ and $\eta > 0$ admit $y \in X$ and $n \ge 0$ as follows.
   They satisfy $d(x, y) < \eta$ and $d(f^n(x), f^n(y)) > \delta$.

## Remarks

Banks, Brooks, Cairns, Davis and Stacey showed that if $X$ is infinite, then conditions 1 and 2 imply condition 3.
Example: the doubling map $f(x) = 2x \bmod 1$ on the circle $\R / \Z$.
It is transitive, since $f^n$ maps any arc of length greater than $2^{-n}$ onto the whole circle.
Its periodic points are the points $k / (2^n - 1)$, which are dense.
It is sensitive with $\delta = 1/8$: for $y \neq x$ close to $x$, the distance between $f^n(x)$ and $f^n(y)$ doubles at each step while it is below $1/4$, so it eventually exceeds $1/8$.
Non-example: the rotation $x \mapsto x + \alpha \bmod 1$ with $\alpha$ irrational is transitive but has no periodic points, so condition 2 fails; it is also not sensitive, being an isometry.

## Explanations

### L1 human

A map is chaotic in the sense of Devaney if it links all regions, has repeating orbits near every point, and amplifies small differences in starting points.