The Attractor Demonstration display plots the curve of the logistic function and another line representing all the points on the graph where y=x. Remember that in the process of iteration a value for x is put into the function, a value of y is the result, and this value of y is fed back as a new x. For x greater than 0 and x less than 1, if there is an attractor for the function it does not matter what the starting value of x was, ultimately y will reach the attractor values. For x out of the defined domain, y eventually runs away to infinity.
Beginning on the x-axis at 0.001 the program draws a line straight up to the function curve where the corresponding y is located. From there we draw a horizontal line at a constant y level until we hit the line where y=x. These two lines, one ending on the curve and another ending on the y=x line, represent one iteration. To get the next iteration we draw a line vertically from the new x position to the curve and horizontally to the y=x line to find the next x. The "Action" button draws a new iteration.
Each pair of lines will form a square corner where they intersect the function curve. This intersection marks the function value for that iteration. As the number of iterations increases we are looking for those y values where further iterations produce no change. Those values are on the attractor.
As the process of drawing iterations goes on the screen will become cluttered. So as to be able to see the results of new iterations, you may change drawing colors by clicking on the Color button.
Push the iterations as far as you wish, either to settle out on an attractor or to convince yourself that it will never settle out (the chaos case).