Deterministic Chaos

6.D.5. Return Map

A return map plot of a sequence x₀, x₁, x₂, x₃, ... is a plot of the points

(x₀, x₁), (x₁, x₂), (x₂, x₃), (x₃, x₄), ...

Here is an illustration of the return using graphical iteration. Here is an explanation of the parts of this diagram.

Click the animation to stop.

To illustrate the sorts of things we can learn from a return map, here are some time series (left) and return maps (right).

uniform random chaotic tent map average of chaotic tent maps

Here are the time series (left) and return map (right) of a uniform random sequence of 200 points.