DYNAMICAL SYSTEMS

Worksheet 1--Homework

NOTE: You will need a calculator to do this homework. You may write your conclusions on this sheet, but put your data on a separate sheet. As you record your data, you can round off to the nearest tenth.

Use this recursive rule:

P_n+1 = .8P_n + 100

1) Experiment with a variety of seeds (your P₁ value). Try P₁ =100; P₁ =-200; P₁ = .2; P₁ =1000; and P₁ = 0. Run each sequence through 24 iterations. As you record your data, you can round off to the nearest tenth. You might want to work with a partner and share data.

The purpose of running the sequence through so many iterations is so that you can make a guess about what the sequence of values will do in the long run. Will it grow to be incredibly large? Will it just bounce around? Will it head toward a certain number and stay there?

2) Look at your data and decide if it seems like every seed has the same behavior in the long run (known as the "ultimate fate"), or if different types of seeds do different things.

3) What prevents the terms of the sequence from getting huge? You might start out with a really large seed, after all-say ten thousand. Won't the sequence just get bigger after that? In your explanation, refer specifically to the arithmetic operations (the multiplication and addition) of the rule: P_n+1 = .8感_n + 100. What impact do those operations have on a big number like ten thousand?