Math 131

Spring 2000

This is due Monday, April 3 at the beginning of class. It is worth 10 points. Papers turned in late lose 3 points per day. Write neatly or use a word-processor. Your grade will depend on the correctness of your work and how well you explain it. Don't just answer questions! Write a report that makes sense. You don't need to list every calculation, but describe what happens. You may discuss your work with me but not with others in the class.

The scientific field of chaos has been called the most interesting scientific development of the last third of the twentieth century. One of the fundamental principles of chaos is that small changes in numbers can have huge effects on future calculations. Physically, this concept has been illustrated as the butterfly effect. The butterfly effect states that the air stirred by a butterfly in South America can create or disperse a tornado a few days later in Texas. In this assignment, you will see a numerical example of the butterfly effect.

We will work with the recurrence relation x_n = x_n-1 ( c- x_n-1 ) with x₀ = 0.5 for various values of the constant c. This equation is a variation on the logistic equation used to model population growth (we will talk about this in class some). The constant c is related to the growth rate of the species being studied in ideal conditions. An important question is what effect do small changes in the growth rate have on the population.

Start with c=2. That is, use the recurrence relation x_n = x_n-1 ( 2- x_n-1 ) with x₀ = 0.5 to compute x₁, x₂, x₃, etc. (you will probably want to use a calculator or computer). You should find that x_n approaches and gets "stuck" at a particular number; write this number down. Now repeat this process with c=2.5. You should again get stuck at one number. Write this number down. Now, compare your results: what effect did the small change in c have?

Now, try c=3.2. This time, you should get "stuck" alternating back and forth between two numbers. Write these numbers down. So, this time changing c had a definite effect on the behavior of the sequence. Next, try c=3.48. You should find a 4-number cycle; write the numbers down. Briefly discuss the effect that a small change in c made this time. Next, try c=3.555, report your results and comment on the effect a small change in c has.

Finally, try c=3.57 and you will find no repetitions at all. This result (bounded but not repeating) is called mathematical chaos and represents a surprising possibility for simple systems.

Based on your calculations, which of the following statements seems most reasonable: (a) small changes never make much of a difference, (b) small changes sometimes make a big difference, (c) small changes always make a big difference.