Dynamical Systems: Strogatz Chapter 2

Following this text, students study 1d, then 2d, then 3d flows.  In 1d, we find stability, construct phase portraits, and in chapter 3, make bifurcation diagrams.  We loop back to these topics with more complexity in 2d.  This creates natural “spacing”.

A few notes on spacing:

Spacing improves induction/generalization from examples (but learners perceive otherwise): https://journals.sagepub.com/doi/abs/10.1111/j.1467-9280.2008.02127.x

The gap between perception by a learner of what is effective for them, and the actual performance of learners, is reasonably well documented for retrieval practice.  I wasn’t aware that it also occurs with spacing.
“Across experiments, spacing was more effective than massing for 90% of the participants, yet after the first study session, 72% of the participants believed that massing had been more effective than spacing”.  https://doi.org/10.1002/acp.1537 (From Kornell. “Optimising learning using flashcards: Spacing is more effective than cramming”, Applied Cognitive Psychology 2009)

It is also worth noting that spacing isn’t a panacea if info needs to be retained for a long time.  In a study where it was a long time between learning and testing (about a year), subjects in one study retained about 20% (even with spacing in their learning):
https://www-jstor-org.ezp-prod1.hul.harvard.edu/stable/pdf/40064895.pdf?refreqid=excelsior%3A3cf8dd6aa66f4bf0ab7b6ba6428a14e0

Back to my notes on chapter 2:

  • Section 2.0: Introduction
  • Section 2.1: A geometric way of thinking
    • The idea of interpreting the differential equation via a vector field does not come through sufficiently.  I think more care in distinguishing the vector field itself from the phase portrait (phase line) would help.
    • Sketching the qualitative shape of solutions occurs here.  I’ll call those plots “time series” plots, so that we have a way to refer to them.
  • Section 2.2: Fixed points and stability
    • When “phase portrait” is defined, the f(x) plot is present (but it doesn’t need to be: the phase portrait is just what is happening along the x-axis).
    • Writing f(x) = g(x) – h(x) and comparing g(x) and h(x) to construct the phase portrait is a method introduced with a single example.  Writing out the procedural steps for the general procedure could be helpful.
  • Section 2.3: Population growth
    • Figure 2.3.1 doesn’t make it into the lecture videos and is great for explaining the logistic model (that we’re just choosing a simple way to have a carrying capacity).
    • It would be nice to find some of the data for these population models.
    • The “per capita” growth rate is something students have found confusing some semesters.
  • Section 2.4: Linear stability analysis
    • This section is the heavy lift in this chapter…  Adding a visual of the small perturbation may help.  I’m not sure how intuitive the idea of a small perturbation is…
    • The big-O notation needs to be introduced.  When we talk about “higher order terms”, students need to be reminded about the meaning of the word “order” in this context.  In addition, that O(eta^2) encompasses all the higher order terms should be made explicit (https://en.wikipedia.org/wiki/Big_O_notation#Infinitesimal_asymptotics).
    • The connection to exponential growth and decay is relying on students having the solution to x’ = a x very accessible in their memories.  Without this being second nature, a lot of the intuition in this section is lost.
    • Needed background is the definition of a separable diff eq and then how to solve it (this could be introduced in the population growth section).
    • The f ‘ (x*) = 0 case always leads to a number of questions.  I think introducing the terms “hyperbolic” and “nonhyperbolic” would actually help.
    • Most of my students mix up f ‘ (x) and d^2 x/ dt^2.  I need to think about how to avoid that confusion…
    • I’d like to emphasize that we can solve the linearized system.  We can even use it to get the timescale of decay and to sketch solutions near the equilibrium solution.  Is it worth using this to piece together time-series plots of trajectories?  The distinction between linear and nonlinear systems doesn’t currently come through very well.
    • I’d like to compare the solution to the linearized system for small perturbations to numerical approximations via RK4 / Mathematica.
  • Section 2.5: Existence and uniqueness
    • I should provide some intuition for what we mean by the word “smooth” and the term “smooth enough”.
  • Section 2.6: Impossibility of oscillations
    • Reintroducing the definition of “monotonic” could be helpful here.
    • The analogy to the “over-damped” limit is probably not illuminating for students without physics/engineering interests/background.
  • Section 2.7: Potentials
  • Section 2.8: Solving equations on the computer
    • Also not in Steve’s youtube videos.  I haven’t been introducing it explicitly, but probably should.  It converts a flow to a map…
    • They should know about slope fields from Math 1b, so I could remind them and anchor on that.
    • This is a good place to introduce computers.
      • Find fixed points symbolically (Solve) and with root finding (FindRoot).
      • Take the derivative symbolically (and perhaps numerically?).
      • Create time series plots.

2.0 – 2.6 is the prep for a single class meeting.  In the following class, the goals would be to gain procedural fluency with

  • finding fixed points (either given a diff eq or a plot of f(x))
  • determining their stability from a graph of f(x)
  • determining their stability by finding f ‘ (x*) and interpreting it
  • sketching approximate time series of trajectories
  • making phase portraits on the x-axis

Some extra stuff that would be good too:

  • recognizing and setting up an integral for the solution of separable diff eqs
  • distinguishing between linear and nonlinear equations
  • identifying phenomena that occur in linear vs nonlinear autonomous diff eqs
  • doing the procedural stuff above in Mathematica