ABSTRACT
This chapter introduces many of the basic techniques from the theory of dynamical systems in a setting that is as simple as possible. It introduces topics such ashyperbolicity, symbolic dynamics, topological conjugacy, structural stability, and chaos. The chapter explores some elementary notions from single variable and multivariable calculus. One of the most abstract and seemingly useless theorems from multivariable calculus is the Implicit Function Theorem. The basic goal of the theory of dynamical systems is to understand the eventual or asymptotic behavior of an iterative process. Maps with hyperbolic periodic points are the ones that occur typically in many dynamical systems and, moreover, they provide the simplest types of periodic behavior to analyze. Bifurcation means a division in two, a splitting apart, a change. In dynamical systems, the object of bifurcation theory is to study the changes that maps undergo as parameters change. These changes often involve the periodic point structure, but may also involve other changes as well.
