This chapter is not included in Steve’s youtube videos.
- Section 4.0: Introduction
- The connection between putting the vector field on a circle and oscillation is not obvious. Showing time series x(t) or y(t) for a uniform oscillator might help (the time series figures in the text have to do with bottlenecks).
- Section 4.1: Examples and definitions
- The mechanics of checking whether a vector field is well-defined on the circle are fine. I think problems arise when this is approached via intuition instead of calculation.
- Section 4.2: Uniform oscillator
- The idea of a phase angle, and what it represents isn’t obvious.
- We are sometimes making a model of a phase angle and sometimes making a model of a phase difference. These two contexts lead to different interpretations of the phase portrait, so being explicit about which one we’re talking about is important.
- Section 4.3: Nonuniform oscillator
- That nonuniform oscillators have a number of applications is stated. Intuition for the functional form isn’t provided though.
- The calculation to find the period of oscillation, and how it changes as the bifurcation is approached, isn’t so easy to follow. This type of period calculation comes back in the van der Pol.
- Section 4.4: Overdamped pendulum
- An overdamped pendulum isn’t so intuitive (because we tend to picture the swinging case), so I’m not sure about this example.
- Section 4.5: Fireflies
- Firefly phase locking is a fun example! In this model, the fireflies reach a fixed phase difference (they phase-lock), but they don’t flash in unison. That is a bit unintuitive: since they are not in unison, students sometimes struggle with the idea that they can reach a non-unison steady state (the math works out cleanly, but translating it to the model can be hard).
- It may be worth looking up the Ermentrout 1991 paper about a species that shifts its frequency.
- Section 4.6: Superconducting Josephson junctions
- I skip this section. Other possible applications: jet lag, rat brain grid cells, ??