Math 2.0

Dynamical systems and ergodic theory Definition

Devaney chaos

Builds on Metric spaceContinuous mapTopological transitivityPeriodic pointDense set

Definition

Let (X,d)(X, d) be a metric space and let f ⁣:X→Xf \colon X \to X be a continuous map. The map ff is chaotic in the sense of Devaney if:

  1. ff is topologically transitive.
  2. The set of periodic points of ff is dense in XX.
  3. ff has sensitive dependence on initial conditions: there is δ>0\delta > 0 such that each x∈Xx \in X and η>0\eta > 0 admit y∈Xy \in X and n≥0n \ge 0 as follows. They satisfy d(x,y)<ηd(x, y) < \eta and d(fn(x),fn(y))>δd(f^n(x), f^n(y)) > \delta.

Remarks

Banks, Brooks, Cairns, Davis and Stacey showed that if XX is infinite, then conditions 1 and 2 imply condition 3. Example: the doubling map f(x)=2x mod 1f(x) = 2x \bmod 1 on the circle ℝ/ℤ\R / \Z. It is transitive, since fnf^n maps any arc of length greater than 2−n2^{-n} onto the whole circle. Its periodic points are the points k/(2n−1)k / (2^n - 1), which are dense. It is sensitive with δ=1/8\delta = 1/8: for y≠xy \neq x close to xx, the distance between fn(x)f^n(x) and fn(y)f^n(y) doubles at each step while it is below 1/41/4, so it eventually exceeds 1/81/8. Non-example: the rotation x↦x+α mod 1x \mapsto x + \alpha \bmod 1 with α\alpha irrational is transitive but has no periodic points, so condition 2 fails; it is also not sensitive, being an isometry.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Continuous map, concept, no evidence levelContinuous mapconceptDense set, concept, no evidence levelDense setconceptMetric space, concept, no evidence levelMetric spaceconceptPeriodic point, concept, no evidence levelPeriodic pointconceptTopological transitivity, concept, no evidence levelTopological transitiv…conceptDevaney chaos, definition, no evidence levelDevaney chaosdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Continuous map concept
  2. Dense set concept
  3. Metric space concept
  4. Periodic point concept
  5. Topological transitivity concept
  6. Devaney chaos definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Chaos_theory
About this page
TagRTG32790CW
URIm2://RTG32790CW
Kinddefinition
WritingW0 stub
MSC37D45, 37B02
DefinesDevaney chaos
UsesMetric space, Continuous map, Topological transitivity, Periodic point, Dense set

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.