ABSTRACT
This chapter introduces the notion of chaos. It shows that there are many dynamical systems that are chaotic but that nevertheless can be completely understood. There are many possible definitions of chaos. In fact, there is no general agreement within the scientific community as to what constitutes a chaotic dynamical system. This definition has the advantage that it may be readily verified in a number of different and important examples. A transitive dynamical system has the property that, given any two points, one can find an orbit that comes arbitrarily close to both. Clearly, a dynamical system that has a dense orbit is transitive, for the dense orbit comes arbitrarily close to all points. The fact is that the converse is also true—a transitive dynamical system has a dense orbit. To summarize, a chaotic map possesses three ingredients: unpredictability, indecomposability, and an element of regularity. A chaotic system is unpredictable because of the sensitive dependence on initial conditions.
