One goal of the chapter is to introduce mathematical topics that are both accessible to students at this level and quite modern. Very few of our students have ever seen a theorem of a living mathematician and have no idea that mathematical research is ongoing. Topics such as chaos and the bifurcation diagram are easy ways to overcome this myth. (See also Sections 2.5 and 5.5 on the Lorenz system and Section 4.5 on recent work on the Tacoma Narrows Bridge.)

Section 8.1 gives definitions and develops graphical techniques for plotting orbits. Sections 8.2 and 8.3 deal with the calculus of fixed and periodic points and bifurcations respectively. A discussion of chaos is given in Section 8.4. In Section 8.5, we discuss the behavior of the Lorenz equations using a one-dimensional model. The discrete logistic map appears repeatedly in Sections 8.1-8.4.

Available Technology

There are many programs available on the web that help illustrate this material. For the PC, one of our favorites is Rick Parris's Winfeed. The web site for the Dynamical Systems and Technology Project at Boston University has java applets that compute orbits and histograms, perform graphical iteration, and compute the bifurcation diagram.

8.1 The Discrete Logistic Equation

This section introduces the idea of discrete time systems and the relation to iteration in one dimension. We discuss graphical techniques for displaying the iterates, including time series, histograms, and the graphs of higher iterates of the function. We also define and discuss fixed points.

Comments on selected exercises

Many of the exercises in this section can be done by hand, but others (iteration of cos x and sin x, for example) require the use of a calculator or computer.

Exercises 1-8 involve computation of the orbit of 0 for various functions. Students observe orbits tending to fixed points, to cycles, to infinity, and so forth.

In Exercises 9-21, students are asked to find fixed points and periodic points of period 2. Many of these exercises can be done by hand. Occasionally the student must resort to looking at the graph of the function or its second iterate to answer this question (as in the case of e^x, sin x, or -e^x).

Exercises 24-35 involve experimentation with the tent map.

Exercises 36-39 consider the role of a parameter by studying the one-parameter family F(x)=x² + c. These exercises foreshadow some of our work with bifurcations in Section 8.3.

8.2 Fixed and Periodic Points

In this section, we study fixed and periodic points. We classify them as sources, sinks, or nodes. We also introduce the process of graphical iteration (the "web diagram"). Graphing calculators that possess the ability to produce web diagrams are very useful.

Comments on selected exercises

In Exercises 1-11, the students find and classify certain fixed points.

In Exercises 12-17, the number 0 lies on a cycle. The student must determine the period of the cycle and then classify it as attracting, repelling, or neutral.

In Exercises 18-25, students try to determine whether certain fixed points whose derivative is either 1 or -1 are attracting, repelling, or neutral. The students generally use the graphs of the function to classify the cycle.

In Exercises 26 and 28, the fixed points of certain one-parameter families are considered.

Newton's method is studied as a discrete dynamical system in Exercise 30. Newton's method also appears as Lab 8.1.

8.3 Bifurcations

In this section, we consider one-parameter families of maps and bifurcations of fixed and periodic points. The logistic map and the period doubling route to chaos are considered in some detail. Toward the end of the section, the level of sophistication demanded of students is higher than the preceding sections.

Comments on selected exercises

Exercises 1-7 involve the analysis of basic bifurcations. Students find these problems difficult because they must explain what happens to orbits "at, before, and after" the bifurcation. Students at first do not know how close to take nearby parameter values in order to be "before" or "after" the bifurcation. They often simply choose the nearest integer which may be too far away.

Exercises 8 and 9 involve other families of functions where the students must determine the specific bifurcation value.

8.4 Chaos

The phenomenon of chaos is observed in the logistic map and studied for the shift map on decimal expansion (multiplication by 10 modulo 1). The second example gives an introduction to symbolic dynamics, dense orbits, and sensitive dependence on initial conditions. Again, the level of mathematical sophistication required is somewhat higher than in previous sections.

Comments on selected exercises

Exercise 1 studies chaos in the logistic map numerically.

Exercises 2-14 consider the tent map, including an introduction to symbolic dynamics for this map. This series of exercises could be a good project for students in an honors class.

8.5 Chaos in the Lorenz System

In this section we return to the Lorenz equations and, using the template construction, develop the one-dimensional model for the dynamics on the attractor. If the Lorenz equations have been studied earlier in the course (Sections 2.5 and 5.5), this section can be covered after Section 8.1.

Review Exercises

Exercises 1-10 are "short answer" exercises. The answers are (usually) one or two sentences. Most (but not all) are relatively straightforward.

Exercises 11-16 are true/false problems. We always expect our students to justify their answers.

In Exercise 17, the student is asked to compute cycles of period two for the chopping function.

Exercise 18 requests a bifurcation diagram for a two-parameter family of linear maps.

Exercise 19 asks for the number of points that lie on cycles for F(x)=x²-2.

Exercise 20 involves a one-parameter family of tent functions.

Comments on the Labs

Lab 8.1. Newton's Method as a Difference Equation

Newton's method on a cubic is studied using the terminology of discrete dynamical systems. A programmable calculator is sufficient technology for this lab, and it can be assigned after Section 8.1. It provides another approach to complicated behavior of one-dimensional maps

Lab 8.2. The Delayed Logistic and Iteration in Two Dimensions

This lab requires the ability to plot phase planes of functions of two variables. While this idea is natural given our previous work, students will need some guidance in viewing orbits in this manner.

Lab 8.3 The Bifurcation Diagram

This lab necessitates that students either write a simple program or use some available software to sketch the bifurcation diagram(s) requested.