Graphic display of iterated equation instability, deterministic chaos

If you have an equation that calculates a value of Y based on an input value of a variable X, you can take the value of Y you just calculated and use that as the input value (X) in another evaluation of the equation. Then take that Y and use it as X again, over and over. That is called iteration.

If you graph the successive values of Y, you discover that they form one of the following patterns:

1. Become larger and larger, running away to infinity.
   Example: Y = X * X where starting X is greater than 1.
2. Become smaller and smaller, running towards 1/infinity.
   Example: Y = X * X where starting X is between 0 and 1.
3. Converge quickly to a single value, which repeats over and over.
   Example: Y = X.
4. Settle to a cycle of two or more values, repeating the cycle forever.
   Example: Y = 1 / X.
5. Generate unique values forever that never repeat. This is called deterministic chaos. Example below.

This program graphs the successive values of an equation that shows the interesting behaviors 3 through 5,  depending on the value of a variable R (see below). Each loop sets a new value of R and begins an iteration sequence, graphing the result. As R increases, the behavior goes from type 3 (single value) to type 4 (cyclical) with the number of points in the cycle rising from 1 to 2 to 4 to 8, etc. This is called period doubling. At higher R values, chaos appears, but even within the chaotic regions are areas where the behavior goes back to being cyclical. This phenomenon is best viewed using the BIFURCAT.BAS program and its descendants.