- Section 3.0: Introduction
- I need to help students distinguish between parameters and variables.
- The beam bending example is ok, but the intuition isn’t so clear. If I back up on the load does the beam straighten (is this a supercritical pitchfork)?
- It would be nice to introduce an intuitive example of a saddle-node bifurcation.
- Section 3.1: Saddle-node bifurcation
- Example 3.1.1 is algebraic, which is great. Example 3.1.2 is geometric, which is also great. For each of those examples, adding procedural instructions for creating an associated bifurcation diagram would be helpful.
- The explanation of normal forms, including the Taylor expansion and Figure 3.1.7 (where f(x) has a local minimum and is being shifted up and down) is clear in the text. It comes across less clearly in Steve’s youtube videos.
- Figure 3.1.4 is the only example of a saddle-node bifurcation diagram. Because it is associated with the normal form it might give the impression that the bifurcation happens at (0,0) and that saddle-node bifurcations always require a perfect parabola. Adding more examples of bifurcation diagrams would be helpful.
- Section 3.2: Transcritical bifurcation
- Instructions / examples of making the bifurcation diagram are also missing here.
- A “bifurcation curve” in parameter space is an idea that is introduced here. The idea of parameter space can be confusing (we have phase space, parameter space, and the mixed space that is used for bifurcation diagrams). It is probably worth introducing this example and the idea of a stability diagram explicitly at this point.
- Section 3.3: Laser threshold
- I skip this section. I should find a different application example of a transcritical bifurcation to replace it with.
- Section 3.4: Pitchfork bifurcation
- In the text, the idea of an equation being invariant under a change of coordinates is introduced. We should do the coordinate replacement in class… (x to -x). It would be worth building more intuition around the idea of “symmetry”
- I should assign 2.4.9 on critical slowing down. I think I assign a modified version that isn’t very helpful. The idea that solutions decay more slowly than exponential is hard to convey.
- The geometry associated with a pitchfork bifurcation doesn’t come across very well (that f(x) = g(x) – h(x) and the bifurcation happens when g(x) and h(x) become tangent).
- There are instructions for plotting a bifurcation diagram here. The trick to find r in terms of x but then plot in rx-space is introduced.
- Steve’s youtube videos skip the subcritical pitchfork. I have a video introducing it, but it could probably be better.
- The idea of slowly varying a parameter is introduced here. When varying a parameter, the fixed point will change. That the fixed point is chained to the parameter is a point of confusion for some students (they sometimes think of the parameter value responding to the fixed point value). Explicitly defining hysteresis and talking about jumps would be a good thing to do here.
- Section 3.5: Overdamped bead on a rotating hoop
- I skip this section and replace it with an introduction to dimension and nondimensionalization. Understanding this example requires being able to follow a nondimensionalization argument, as well the discussion about neglecting a term.
- Section 3.6: Imperfect bifurcations and catastrophes
- Some years I have explicitly introduced the imperfect bifurcation content. Other times I have skipped it.
- Providing a reminder of the definition of a bifurcation curve and the distinction between parameter space and other coordinate planes that we use is important here.
- The definition of a cusp point and the term codimension-2 bifurcation both appear here.
- Stability diagrams, parameter space, etc are all introduced better in the text than they are in Steve’s youtube videos.
- The catastrophe example of a bead on a tilted wire seems fun to play with: could we work with it in simulation/animation or do I need a physical version?
- Section 3.7: Insect outbreak
- Separation of time scales comes up here and is worth emphasizing.
- This is a nice modeling example.
- Making sure to present the terms cusp, bifurcation curve, stability diagram, and catastrophe in advance of this example might help make it easier to follow.
- Building more intuition for thinking about bifurcations via the intersection of two curves, f(x) = g(x) – h(x), could also help.
- The idea of bistability comes up in section 3.4 but is introduced here.