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 picture for an animation.

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

Above are the time series (left) and return map (right) of the average of two chaotic tent maps. Later we'll see why the return map takes this shape.

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