Instructions:

The applet opens in a separate window. You can move this window aside, resize it, or iconize it when you want it out of the way. To close it, use the Close option from the File menu, not the Quit or Back commands.

The applet illustrates the iterative map

xn = 4k xn-1 ( 1-xn-1 ) .
Use the slider or the text area to enter an initial value of x to begin the iteration. In addition, you may choose the value of k. It is particularly interesting to start with k just below 0.75 and to increase this value in subsequent runs. Press the iterate button repeatedly to iterate the map.

If the applet fails to render properly, as can happen if you scroll the browser window rapidly (especially if using Netscape 3.0 for the Mac), simply choose Reload from the browser menu.

If you receive a message informing you that your browser "does not understand the applet tag," examine the Security Preferences Menu of your browser. There should be a checkbox which you can use to enable Java; absence of such a box probably indicates you are using an older web-browser and you must upgrade it in order to run the applet. Please take a moment to send me an e-mail reporting any difficulties you may have experienced with the applet.

Questions:

• Notice that each successive iteration of the map is achieved by "bouncing off the line y=x." Why is this so?

• There are two values of the starting point x0 that generate sequences such that x0=x1=x2=...; i.e. all the iterates have exactly the same value as the starting point. What are these two values? They are known as ``fxed points.'' One of them depends on k and the other doesn't. Can you work out exactly how it depends on k?

• Choose k just less than 0.75, say at 0.70. Apply the iterative map repeatedly several times. Try this again, varying the initial value of x. What qualitative behaviour results in all such cases? Can you explain this behaviour (Hint: What is the the slope of the graph y = 4kx(1-x) at the fixed points, and how does this slope change as k is varied?)
• Now increase k alittle past 0.75. Again iterate the map several times. What behaviour results?
• By clicking the iterate button twice every time, you are effectively applying a map that takes xn as input and yields xn+2 as output. Can you write down an explicit expression for this map? This expression may help you to analyse the previous question.
• As you continue to increase k, what new phenomena occur?