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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 . 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).
This undergraduate textbook is a rigorous mathematical introduction to dynamical systems and an accessible guide for students transitioning from calculus to advanced mathematics. It has many student-friendly features, such as graded exercises that range from straightforward to more difficult with hints, and includes concrete applications of real analysis and metric space theory to dynamical problems. Proofs are complete and carefully explained, and there is opportunity to practice manipulating algebraic expressions in an applied context of dynamical problems. After presenting a foundation in one-dimensional dynamical systems, the text introduces students to advanced subjects in the latter chapters, such as topological and symbolic dynamics. It includes two-dimensional dynamics, Sharkovsky's theorem, and the theory of substitutions, and takes special care in covering Newton's method. Mathematica code is available online, so that students can see implementation of many of the dynamical aspects of the text.