The first question asks if Smales horseshoe is a map of the unit square into itself, and this is true

In general, Smales horseshoe is actually defined on any square in the plane, or any region of the plane that can be continually deformed to the unit square

But regardless, this question is true

Question 2 asks if Smales horseshoe maps some close points far apart, and vice versa

So points that are close go far apart, and points that are far apart go close

And this is true

If you go back to the kneading bread analogy, which actually works very good, this is why, if youre baking bread (if anyone is familiar with baking), you can stick all the chocolate chips in one place in the dough, and then knead the bread, and the chocolate chips will evenly spread across the entire dough, or any other ingredient or spice for that matter

Its not necessary to evenly space the spices all over the dough, because once you knead the bread, spices that were close together, or in some small cluster that you dumped on top of the dough, will spread apart evenly across the dough

And this is effectively what Smales horseshoe does on the unit square

It can be thought of as taking the unit square and then kneading the unit square through three topological mappings

And the result of which is that some close points are far apart, and some far apart points are then close

For Question 3, were given the following picture of a small ball of initial conditions around theta = pi in the undamped pendulum

With the stable and unstable manifolds drawn in, and we want to assume that the ball is very tightly clustered around theta = pi

We want to know what these points will look like as time evolves

If we just think about this intuitively, we know that along the stable manifolds well have shrinking, so these points will shrink in

And along the unstable manifolds well have stretching

So what we should see is a long line of points along the unstable manifolds

For this reason, a cannot be correct, because you have points spreading along the stable manifold

This would only occur in backwards time

Part b looks the most probable

You have stretching along the unstable manifold, and you have no growth along the stable manifold

The fact that you cant really see shrinking that well has more to do with the artists ability  that is, my ability to use Sketchpad  than anything else

I think this one is the most probable, but lets take a look at the other ones

c would imply that they spread out in every direction equally

This is not the case

We know that well have shrinking along the stable manifold, so this cannot be the case

Part d can also not be the case, because you only have growth along the stable manifold, which is exactly the opposite of what occurs

So part b  even though the shrinking is not really very good because of my artistic ability  we see this stretching along the unstable manifolds like we should see

So b is the answer to this question

Question 4 is a series of questions about a generalized horseshoe

Part a asks if horseshoes only turn up in the dynamics of the Smale horseshoe map, and this is false

For example, horseshoes show up in the dynamics of the pendulum

That actually answers part c as well

Part b states that horseshoes are important because they play a role in proofs of chaos, and this is true as well

See the lecture for more details on this

Question 5 asks whether dissipation is a necessary condition for the existence of attractors, and this is true