Return Maps

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

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

Click on the picture below to see how the graphical iteration (left window) of the Tent Map generates the return map (right window). Recall in our presentation of graphical iteration, the pair of thick green lines connects (xn, xn) to (xn+1, xn+1). To generate the return map, we want to plot a point with y-coordinate xn+1 and x-coordinate xn. The (light blue) horizontal line from (xn+1, xn+1) (the end of the horizontal green graphical iteration line) consists of points all with y-coordinate xn+1. The (orange) horizontal line from (xn, xn) (the beginning of the vertical green graphical iteration line) consists of points all with y-coordinate xn. This orange line intersects the diagonal in the return map square at the point (xn, xn). The vertical (orange) line from this point consists of points with x-coordinate xn. Consequently, the vertical orange line and horizontal light blue line intersect at the point (xn, xn+1), a point of the return map.

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

Above are the time series (left) and return map (right) of a uniform random sequence of 200 points.
Above are the time series (left) and return map (right) of a chaotic tent map. Does the return map surprise you? later we'll see why this is the shape it takes.
Above are the time series (left) and return map (right) of the average of two chaotic tent maps.

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