1. a What is an “iterated function”?

An iterated function is a function where you plug in an input, and then you take the output, and you plug it back into the function, so you are doing the same function rule over and over. Each time you do it is called an iteration.

b. What does it mean for an iterated function to be sensitive to initial conditions?

It means that if you change the first number you plug in just a little bit, then, after several iterations, the output changes a lot, and it changes pretty much unpredictably

2. Give 2 examples of chaos in the real world, and explain why they are chaotic in the mathematical sense.

There is chaos in the weather (you can't make good predictions very far in advance). This is chaotic because if the measurements/numbers for what the weather is now are changed a little bit, then after you iterate the weather predicting function for long enough, the predictions (far in advance) get wildly different.

There is chaos in the 3-body problem: you can't predict what will happen if you have 3 stars/planets that are all interacting with each other. The orbits do very strange things, and changing the initial conditions a bit changes the orbits a lot.

There is chaos in a vibrating piece of metal if you make it vibrate fast enough: Once the vibrations are large, it no longer does predictable things, but bounces around in unpredictable ways.

A double pendulum toy is chaotic: the first pendulum is regular, but the second pendulum (which is swung by the first pendulum) has an unpredictable path--even though the first pendulum always does the same thing, the second pendulum changes what it does all the time--little changes in where it is when the first pendulum starts swinging, make a large difference in where it goes next.

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?

It is chaotic if the planet is in between the stars in such a way that the gravity of both stars has a significant effect on it.

It's not chaotic if it only one star has much effect. This might happen if the it were very close to one of the stars, or if one star were much larger than the other. Check it out here http://www.upscale.utoronto.ca/PVB/Harrison/Flash/Chaos/ThreeBody/ThreeBody.html (try it with the default settings, and then try making the sun on the left smaller, and moving the planet as far to the right as you can)

4. Edward Lorenz invented chaos theory. What was he studying when he found his situation that was sensitive to initial conditions?

He was studying a simple model for the weather.

5. What is the butterfly effect?

It means that if there is a small change (such as could be made by a butterfly flapping its wings), then over time that can turn into a large change in the overall weather.

6. What is an attractor?

It's a set of outcomes that a chaotic function will stay near to.

(It's a strange attractor if the set is a fractal!)

7. If a situation is chaotic, does that mean anything at all can happen?

No, there's usually a range of things that can happen, and also a bunch of things that can't happen. That range can include things that are very different from each other however.