Target Practice

Warning: You must understand the concept of graphical iteration (the web diagram) BEFORE trying to use this applet.

The goal of this applet is to show you how to perform graphical iteration in the setting of a game. We give you a "target itinerary." Your goal is to find an orbit that follows the given itinerary via graphical iteration.

More precisely, when you open the Target Practice applet, you will see displayed the graph of the function F(x) = 4x(1 - x). You will also see three dots colored red or green in the upper left hand portion of the screen. This is the target itinerary.

Now notice that we have colored the left half of the domain of the function red and the right half green. This is displayed on the horizontal axis.

Suppose, for example, that the target itinerary is Red, Green, Red in order from top to bottom. You must find an orbit whose seed lies in the red interval, whose next iterate lies in the green interval, and whose third iterate lies back in the red interval. To do this click on a point on the screen over the x-value you decide to use as seed. Of course, this x-value had better lie in the red interval. You see a (hopefully red) dot on the diagonal line over the chosen x-value. Now click Iterate. This computes the next point on the orbit and displays it via graphical iteration. Hopefully, this point lies over the green portion of the axis. Click Iterate again. You see the next point on the orbit displayed via graphical iteration. If this point lies over the red interval, then you win. You have found an orbit with the prescribed itinerary. If at any point your orbit disagrees with the target itinerary, you receive an error warning and you must begin again. Simply click on a new x-value to try again.

Here is an example of a choice of orbit that "works," that is, has the chosen itinerary. We clicked first on the x-value corresponding to the rightmost red dot then clicked iterate twice to get this orbit.

Further Options: Clck on Medium, Hard, or Master to play Target Practice with 4, 5, or 6 points in the itinerary. You will see that the game gets progressively more difficult as you lengthen the itinerary. You will see that small changes in the original seed lead to large changes in the behavior of the later stages of the orbit. This is an example chaos, one of the main topics of the book.

You may also play the game with other functions. Click on 2 in the upper right to change the function to x² - 2. Or click 3 to play using the doubling function 2x mod 1.

Go to Target Practice Applet.

For more information about the web diagram, consult the book Chaos

Created by Adrian Vajiac and Robert L. Devaney.

For comments and suggestions write to Robert L. Devaney at bob@math.bu.edu