Things to know about Chaos:
Big ideas
Iterated functions are functions that are intended to be used over and over to find out what happens over time (each round of calculation tells you a what should happen during the next second/minute/day/etc.
Iterated functions are used to predict the outcomes of complex real-world processes.
Some iterated functions are "sensitive to the initial conditions", which means that if the input varies just a little bit, then after a few iterations, the output can vary a lot. Situations like these are called chaotic, and we can't make predictions about them very far in advance.
Some iterated functions aren't sensitive to initial condiutions: if the inputs vary a little bit, then the outputs don't vary too much, and they vary in a predictable sort of way. Situations like this are not considered to be chaotic.
Practice problems
1. Below are tables showing values for two iterated functions. For each function, tell whether it appears to be a chaotic function or not, and why you think so (all of the numbers are percents of the maximum capacity of the environment)
.700
2. A. Using the population simulation equation:
(1-N)*N*3+N, starting with the value N = .250. Find the first 5 iterations of this function (round to 3 decimal places). Does it appear to be staying near a particular value? If so, tell what value it is getting close to.
B. Using the population simulation equation:
(1-N)*N*2+N, starting with the value N = .250. Find the first 5 iterations of this function (round to 3 decimal places). Does it appear to be staying near a particular value? If so, tell what value it is getting close to.