Dynamics, Where Do We Start?

Originally Posted 12-23-2011

Dynamical Systems for the Newbie

To structure my basic course on dynamical systems, I have to think very carefully about the introduction. Some of the class will have a background consisting of only single varialbe calculus courses–no multivariable calculus and no linear algebra. But a few of them will be advanced students who have had courses in basic differential geometry, point-set topology, and real analysis. Somehow, I have to get them all learning and enjoying together.

Objectives for the First Unit

What is the basic level of acquaintance with one-dimensional discrete dynamics? I think we need to carefully handle the following concepts.

  • Notion of iteration
  • The idea of an orbit
  • Fixed points, periodic points, and “eventually periodic points”
  • orbits which “escape”
  • the notion of an attracting of repelling periodic orbit
  • basic graphical analysis techniques like cobweb plots and phase portraits.

To introduce these ideas, I want to have systems where they occur naturally. I think I can get many of the examples for the class introduced this way.

Examples to Use

  • simple exponential growth/decay model
  • The tent map
  • The quadriatic (logisitic) family x \mapsto ax(1-x) for several values of the parameter a.
  • The continued fraction map
  • The Newton-Raphson method for different polynomials

My goal is to somehow sequence (three to) four weeks of work that uses the examples above to clearly introduce the concepts in my objectives list. And if I do it right, then is should occur to the students to ask the (historically important) question

"What is going on to the family of quadratic maps? 
As we change the parameter, the behaviors seem to 
change in interesting and understandable ways for a 
while, but then all hell breaks loose! What picture 
can we draw to help us understand this family."

And that will set us up for the second unit on bifurcation diagrams and the period doubling route to chaos.