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