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 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 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 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. |