ABSTRACT

The main goal of the study of Dynamical Systems is to understand the long term behavior of states in a system for which there is a deterministic rule for how a state evolves. The systems often involve several variables and are usually nonlinear. In a variety of settings, very complicated behavior is observed even though the equations themselves are not very complicated (only “slightly nonlinear”). Thus, the simple algebraic form of the equations does not mean that the dynamical behavior is simple: in fact, it can be very complicated or even “chaotic.” Another aspect of the chaotic nature of the system is the feature of “sensitive dependence on initial conditions.” If the initial conditions are only approximately specified, then the evolution of the state may be very different. This feature leads to another difficulty in using approximate, or even real, solutions to predict future states based on present knowledge. To develop an understanding of these aspects of chaotic dynamics, we want to find situations which exhibit this behavior and yet for which we can still understand the important features of how a solution evolves with time.