The repetition of the word chaos in the title is to indicate that this word is used with quite different meanings. The famous artist Escher once drew a picture with the title Order and Chaos. In the middle of that picture we may see a beautiful regular crystal that is surrounded by all kinds of trash, such as broken bottles, opened cans, and so on. The interpretation of the words order and chaos is obvious. According to this interpretation, we are dealing with states that do not change in the course of time. In science, especially physics and mathematics, the word chaos has a quite different meaning; it is connected with changes. But here again we have to distinguish between two different kinds of chaos: the microscopic and the macroscopic or deterministic chaos. We came across an example of microscopic chaos when dealing with the emission of light by lamps. The light field consists of many individual waves that are emitted quite irregularly by the atoms. Another example of microscopic chaos is provided by the motion of the molecules of a gas (Fig. 13.1). They collide again and again and change their directions and velocities of motion at every instant in an entirely irregular fashion. Thus microscopic chaos stems from the irregular motion or behavior of very many individual parts. By contrast, macroscopic chaos originates from the irregular behavior of very few degrees of freedom or variables. For instance, the motion of a pendulum can be described by two variables, namely its displacement and velocity. When we couple the pendulum to a rotating motor by means of a spring, it is as if we are adding a third variable. If the pendulum is driven strongly enough, it may show an entirely irregular motion that cannot be described as motion on a limit cycle or torus (cf. Sect. 5.3).