Explanation of simple harmonic motion as an example of problem based learning

After exams were over this term, students doing further maths in Abingdon have the opportunity of doing a maths project. Given two weeks, we are asked to find a topic related to maths that interests us, and that we can give a presentation on afterwards. I decided to choose a simple example of a dynamical system, because they are often used to model biological phenomenon. As it happens, the topic I chose wasn't as simple as I thought, but here's how it works, from basic maths upwards:
(warning: huge post)