ABSTRACT
In this chapter the authors consider only functions of one variable as encountered in elementary calculus. They discuss the quadratic functions where a real number is a constant. One of the important questions the authors address is how the dynamics of these functions change as these parameters are varied. There are many kinds of problems in science and mathematics that involve iteration. The authors discuss the simple orbits—fixed, periodic, and eventually periodic orbits as well as orbits that tend to a specific limit—might tempt mathematician to think that dynamical systems typically have very simple behavior. One of the major discoveries in mathematics is that many simple functions—such as quadratic functions of a real variable—may have many orbits of incredible complexity.
