In this chapter we introduce one-dimensional dynamical systems and analyze some elementary examples. A study of the iteration in Newton's method leads naturally to the notion of attracting fixed points for dynamical systems. Newton's method is emphasized throughout as an important motivation for the study of dynamical systems. The chapter concludes with various criteria for establishing the stability of the fixed points of a dynamical system.

Iteration of Functions and Examples of Dynamical Systems

Chaotic dynamical systems has its origins in Henri Poincaré's memoir on celestial mechanics and the three-body problem (1890s). Poincaré's memoir arose from his entry in a competition celebrating the 60th birthday of King Oscar of Sweden. His manuscript concerned the stability of the solar system and the question of how three bodies, with mutual gravitational interaction, behave. This was a problem that had been solved for two bodies by Isaac Newton. Although Poincaré was not able to determine exact solutions to the three-body problem, his study of the long term behavior of such dynamical systems resulted in a prize winning manuscript. In particular, he claimed that the solutions to the three-body problem (restricted to the plane) are stable, so that a solar system such as ours would continue orbiting more or less as it does, forever. After the competition, and when his manuscript was ready for publication, he noticed it contained a deep error which showed that instability may arise in the solutions. In correcting the error, Poincaré discovered chaos and his memoir became one of the most influential scientific publications of the past century [10]. Aspects of dynamical systems were already evident in the study of iteration in Newton's method for approximating the zeros of functions. The work of Cayley and Schroeder concerning Newton's method in the complex domain appeared during the 1880s, and interest in this new field of complex dynamics continued in the early 1900s with the work of Fatou and Julia. Their work lay dormant until the invention of the electronic computer. In the 1960s the subject exploded into life with the work of Sharkovsky and Li and Yorke on one-dimensional dynamics, and with that of Kolmogorov, Smale, Anosov and others on differentiable dynamics and ergodic theory. The advent of computer graphics allowed for the resurgence of complex dynamics and the depiction of fractals (Devaney and Mandelbrot).

Many of the most recent International Congresses of Mathematics have awarded Fields Medals to researchers in chaotic dynamics and related fields, indicating the importance of these areas. Dynamics has blossomed in the past 50 years, making it useful as a tool for demonstrating techniques to mathematics majors and for developing their general mathematical maturity. It is my hope that this book will provide interested students with an introduction to onedimensional dynamical systems, giving them the tools necessary to succeed in more advanced courses on this topic. The early chapters of this book can be used as a stepping stone from the non-rigorous courses of freshman calculus to the more advanced topics of real analysis and topology.

Towson University is a liberal arts college and is part of the University of Maryland System. In my first years of teaching a course on dynamical systems, I based my lectures on the material of some of the existing textbooks which were then currently available, such as [122], [41], [65] and [32]. Each semester, I found myself changing the course content and exercises (frequently to meet the needs of my students). This led to the production of my own lecture notes (these notes owe a debt to the above mentioned books).

The content of this text arises primarily from lecture notes that I created over many years of teaching senior seminar-type courses to final year students at Towson University, and also from courses in the Towson University Applied Mathematics Graduate Program and the Graduate Program in Mathematics Education. In the senior seminar course, students were taught the basics of one-dimensional dynamics and were required to present a project at completion of the course. The later chapters of this book include many topics from these projects, for example, Sharkovsky's Theorem, as well as topics resulting from the independent study of some of my Master's students. With students in the Master's Program, I was generally able to move quickly through the earlier material and spend more time on advanced topics, the choice of which changed from semester to semester.