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 youre taking the entire state space, and youre 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