Cycle Practice

Warning: You must understand the concept of graphical iteration (the web diagram) BEFORE trying to use this applet. You should also be familiar with the Target Practice applet before attempting to use this applet. Go to Target Practice Applet now.

As in the Target Practice applet, you are given an itinerary and must find an orbit that follows the given itinerary. This time, however, you must find a cycle that has the given itinerary. More precisely, when you open the applet, you see a collection of red and green dots in the upper left corner of the screen. Note that the last dot has empty interior and is of the sam color as the original dot. For example, suppose that you see the four dots green, red, green, and green in order. The final green dot is hollow. This means that you must find an orbit that begins in the green interval, moves to the red interval after one iteration, then moves to the green interval after the second iteration, and then returns to its starting point on the third iteration. This means that you have found a three-cycle with itinerary green, red, green. Here is a picture of a successful solution.

Note that the orbit does not return to the exact location of the original dot---accomplishing that is almost impossible! So you win if you find an orbit that is very close to a cycle.

Further Options: Clck on Medium, Hard, or Master to play Cycle 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. In fact it is almost impossible at the Master level! 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.

Note that this game shows that the above functions have many cycles.

Go to Cycle Practice Applet.

For more information about the web diagram, consult the book Chaos, particularly Lessons 3 and 5.

Created by Adrian Vajiac and Robert L. Devaney.

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