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.
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reporting any difficulties you may have experienced with the applet.
- 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
- Now increase k alittle past 0.75. Again iterate the map several times. What behaviour
- 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?