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. a What is an “iterated function”?
b. What does it mean for an iterated function to be sensitive to initial conditions?
2. Give 2 examples of chaos in the real world, and explain why they are chaotic in the mathematical sense.
3. If you have 2 stars and a planet--under what conditions is the planet orbit chaotic, and under what conditions is it not chaotic?
4. Edward Lorenz invented chaos theory. What was he studying when he found his situation that was sensitive to initial conditions?
5. What is the butterfly effect?
6. What is an attractor?
7. If a situation is chaotic, does that mean anything at all can happen?