## ABSTRACT

Fixed points and cycles are among the most important kinds of orbits in a dynamical system, so it is important to find them easily. Sometimes this involves solving equations or drawing accurate graphs. These methods, however, are not always easily carried out. One of the simplest criteria for finding fixed points is an immediate consequence of the important fact from calculus. There are two markedly different types of fixed points, attracting and repelling fixed points. Just as in the case of fixed points, periodic points may also be classified as attracting, repelling, or neutral. The calculus is only slightly more complicated. A periodic point of period n is attracting (repelling) if it is an attracting (repelling) fixed point for F^{n}. This immediately brings up the question of whether periodic orbits can contain some points that are attracting and some that are repelling.