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Before we start with the subject proper, it is perhaps useful to look at a concretephysical example, which can be easily built in the laboratory. It is a pendulum with a magnet at the end, which oscillates above three symmetrically arranged fixed magnets, which attract the oscillating magnet, as shown in Fig. 1.1. When one holds the magnet slightly eccentrically and let it go, it will dance around the three magnets, and finally settle at one of the three, when friction has slowed it down enough.

While we assume some familiarity with elementary mechanics, we begin here with the nonlinear pendulum in 1 dimension, to have some basis of discussion of phase space and the like.

In any experiment, one has to use a measuring apparatus which naturally has only a finite precision. In this section, we take the first steps to formalize this. One formalization will be based on partitioning the phase space into pieces and considering that the experiment will only tell us in which piece of the partition (called an atom) a point currently is, but not where inside this piece it happens to be. This is analogous to classical coarse-graining in statistical mechanics. (Later, we will also encounter partitions with different weights on the pieces.) Thus, we know only a fuzzy approximation of the true orbit of the system. As we shall see, one of the miracles appearing in hyperbolic systems is that this information alone, when accumulated over long enough time, will in fact tell us many details about the orbit. In physical applications, one often can observe only one orbit and the information one obtains is considered to be typical of the whole system. We discuss this in more detail in Chap. 9.

This chapter deals with the sources of chaotic behavior. The theory relies on two fundamental concepts. The first, called hyperbolicity, deals with the issue of instability. It generalizes the notion of unstable (or stable) fixed point, to points which are neither fixed nor periodic.

We start by giving a brief overview of some ideas and results to which we will come back later in detail.

In Sect. 3.2 we already addressed the subject of fuzzy knowledge from a purely topological point of view. Here, we come back to this idea, but now, we also connect it to the notion of invariant measure. That is, we account (in particular in the natural case of the Physical measure) for how often (or how long) a trajectory stays in a particular region. One can ask how much information this gives us about the long-time dynamics of the system and the variability of the set of orbits. The main difference with the topological entropy is that we are not going to consider all trajectories but only those “typical” for a given measure.

This chapter is somewhat more technical than the earlier ones. Its aim is to discuss some more recent results in dynamical systems which refine our knowledge of statistical properties. These results follow from a combination of methods from statistics and statistical mechanics.

A number of questions of probabilistic origin have been investigated in the class of stochastic processes generated by dynamical systems. We describe some of the results below.

Our aim in this chapter is mostly to describe measurement techniques which have a solid theoretical background. As usual, in Nature, one has no way of knowing whether the necessary mathematical assumptions are satisfied by a given system. But it turns out that it is very useful, in concrete examples, to act “as if” they were true. We do not enter into the many details which have been studied in the experimental-theoretical literature, such as optimal methods to work with not so abundant data, nonlinear fits to discover the evolution equations, and the like. There is a large literature on this subject; see e.g. (Grassberger, Schreiber, and Schaffrath 1991) and references therein.