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The applet illustrates the iterative map

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- 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
*x*that generate sequences such that_{0}*x*; 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_{0}=x_{1}=x_{2}=...*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*x*as input and yields_{n}*x*as output. Can you write down an explicit expression for this map? This expression may help you to analyse the previous question._{n+2}

- As you continue to increase
*k*, what new phenomena occur?