ABSTRACT
This chapter introduces a geometric procedure that will help people to understand the dynamics of one-dimensional mappings. This procedure, called graphical analysis, enables them to use the graph of a function to determine the behavior of orbits in many cases. Graphical analysis sometimes allows people to describe the behavior of all orbits of a dynamical system. One succinct method for depicting all orbits of a dynamical system is the phase portrait of the system. This is a picture on the real line of the orbits. In the one-dimensional case, the phase portrait gives people no more information than graphical analysis. In dimension two, however, when graphical analysis is no longer possible, people will rely solely on the phase portrait to describe the behavior of orbits.
