Question 1 asks if both projections and sections reduce the dimension of an object
And this is true
This is precisely what these two techniques are used for in the study of nonlinear dynamics
And while both projections and sections reduce the dimension of an object, they do so in different and useful ways
For example, a medical x-ray, which is a projection, tells you something very different than a CT scan, which is a section
This leads us into Question 2: a medical x-ray performs a projection operation, and this is true
This was described very nicely in the lecture video
Question 3 asks if measuring the value of a single state variable of a dynamical system is like performing a section
This is not like performing a section, this is like performing a projection
So youre taking the entire state space, and youre projecting it onto a single axis
So this question is false
Question 4 states that sections can be slices in time or slices in state space, and this is true
A section in time is called a temporal section, and this is like flashing a strobe light at a pendulum
Alternatively, you could slice in space
This would be like taking a plane and sticking it through one of the wings of the Lorenz butterfly
Which, as you saw in lecture, would result in a Cantor set
So this question is true
Poincare sections, or sections, can be slices in both time and space
The next four questions involve taking Poincare sections, or spatial sections, of this orbit of a dynamical system
This is the typical method for determining the period of a trajectory from a flow
For part a, we want to take a plane of section at x = 3, so here
If we do this, the Poincare section will have two points on it: here and here
So the answer to part a is Two points
If you take a spatial Poincare section and you have two points on it, this type of section is associated with a two-cycle, so this is the answer to part b
However, if we instead take a spatial Poincare section at y = 2 so here we will get four points: here, here, here, and here
So the answer to part c is Four points
And similarly, just as two points result in a two-cycle, four points on a Poincare section is defined as a four-cycle
This should illustrate to you the difficulty of assigning a value to the period of an orbit of a flow
Depending on where you take the plane of section, your conclusions can be different, like in this case
One plane of section said it was a two-cycle, and another plane of section described this as a four-cycle