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. This section introduces the idea of discrete time systems and the relation to iteration in one dimension. Graphical techniques for displaying the iterates are given, including time series, histograms, and the graphs of higher iterates of the function. Fixed points are defined and discussed.
Students observe orbits tending to fixed points, to cycles, to infinity, and so forth.
Many of these exercises can be done by hand. This foreshadows some of our work with bifurcations in Section 8.3. This section studies fixed and periodic points, classifying them as sources, sinks, or nodes. Graphing calculators that possess this capability are most useful here. The student must determine the period of the cycle and then classify it as attracting, repelling, or neutral.