Dynamical Systems: Strogatz Chapter 5

This chapter is mainly review of topics from prerequisite courses.  Steve does introduce the (Delta, tau)-plane for classifying fixed points of linear systems.  This chapter is a return to linear systems.

There isn’t a “summary” section in between Chapter 4 and Chapter 5.  That is probably a worthwhile spot to review the differences between the behaviors we see in linear systems vs in nonlinear systems in 1d.  It isn’t noted in chapter 4, but I suppose the only possible linear system for flow on the circle is actually the constant velocity one (because of the periodicity requirement for the vector field).  What are the phenomena that we can encode via a nonlinear model that we can’t get to via a linear model?

  • Section 5.0: Introduction
    • We are back to linear systems, now in 2d.  There is a bit more to this than there was in 1d.  We’ll use this info to classify fixed points in nonlinear systems.  Is it possible to build that out a bit more in 1d?  (To think of classification in 1d as seeing which type of linear fixed point our fixed point is similar to?)  I wonder if building out an analogy there would make the distinction between linear and nonlinear systems more intuitive.
  • Section 5.1: Definitions and examples
    • Students probably need to be reminded how to translate a higher order linear differential equation into a linear system (see Example 5.1.1).
    • Distinguishing hyperbolic from non-hyperbolic fixed points could be helpful again here.
    • Some students confuse “saddle point” and “saddle-node”.
  • Section 5.2: Classification of linear systems
    • The role of eigenvalues and eigenvectors in solutions is something some students remember and others find confusing.  Same with complex eigenvalues.
    • Classifying fixed points both using the (Delta, tau)-plane, and providing the eigenvalue classification information in the (Real, Imaginary)-plane would be a good idea because one can memorize the Delta-tau plane without remembering where it came from.
  • Section 5.3: Love affairs
    • This example actually gets used a lot (so may have been seen in other courses).  For that reason, I skip it.