Iteration and Chaotic Dynamics

Have you ever played with your calculator by applying one of the functions over and over again? On some calculators this is done by hitting the function button. On other calculators such as the TI-82, 83, etc., you can do this with the ANS key.

After setting things up, you tap ENTER button to repeatedly apply the function. Did you notice that repeating functions such as the ex or 10x a few times leads to an error message? This happens because the answer becomes larger than your calculator can display-even with scientific notation. Why not try repeated application of one of the other function keys. If you have done this, then you have been experimentally investigating a dynamical system.

In this paper we will do some experimental mathematics in a very active field of mathematical research, where new mathematics is being discovered every day. Dynamical systems is one of the mathematical disciplines that appeals to researchers and is accessible to students. In fact this article can be read by any student who can do basic algebra.

This lesson is hands on and requires experimentation by the reader. So get your calculator, pencil and paper! Any scientific calculator will suffice for now. If it is programmable, does graphing, and has an ANS key that's even better. Later you may want to use the programming features of your calculator or a personal computer.