Types and Importance of Exponent Dimensions

An exponent dimension often remains constant over a range of scale sizes. To that extent, it's an invariant quantitative measure. As we'll see in the following several chapters, it tells us how complex a system is. It's a quantitative numerical way of evaluating or comparing the geometric complexity of objects or systems of different size, shape, and structure. Therefore, it's new and potentially useful information. Aside from being a useful concept, dimensions as scaling exponents are important in chaos theory for two reasons.
• Measuring the dimension (the exponent, or slope of a straight line on a log plot) can help distinguish between chaos and randomness. An object's dimension characterizes its geometry and is a fundamental physical trait. As explained in Chapter 19, a chaotic attractor has a shape in phase space. That shape has a finite and measurable dimension. In contrast, if a system operates randomly, it theoretically can visit the entire phase space sooner or later, and there probably won't be any definable configuration (attractor) within the phase space. The dimension then equals the embedding dimension and so becomes infinite as we continue to increase the embedding dimension (although there are exceptions, according to Osborne & Provenzale 1989). In practice, noise and scanty data often complicate this approach to identifying chaos.
• An attractor's dimension reveals the number of variables or measurable quantities we'll need to describe or model the system. For instance, a dimension of zero indicates a point in phase space (a constant value). To describe a line, we need one dimension; to model a rectangle, we need two variables-length and width. Modeling any system or object therefore requires at least as many variables as its dimension.
The Los Angeles temperature and wind-velocity attractors, discussed in Chapter 11, exemplify these qualities of dimensions. The point attractor (same temperature and wind velocity all the time) is the simplest possible type of attractor. It has a dimension of zero—we don't need any variables to describe it, because the weather is always the same. The next simplest type of attractor is the loop attractor (same daily cyclicity, day after day). It has a dimension of one, because a single variable, namely the time of day, can characterize such weather. Most weather evolution, of course, is fairly complex. Describing or modeling that evolution takes quite a few variables. In general, a chaotic attractor's dimension is a quantitative measure of its "strangeness" or of how chaotic it is.
There are many types or variations of dimension as a scaling exponent. The most useful type in chaos theory hasn't yet been determined. Some that are mentioned quite commonly, in no particular order of importance, are:
• similarity dimension
• capacity
• Hausdorff-Besicovich (or simply Hausdorff) dimension
• information dimension
• correlation dimension
• fractal dimension.
The following chapters discuss the first five on that list; we'll talk about the fractal dimension later in this chapter. Some candidates that, for brevity, I have left off the list are the generalized, cluster, pointwise, Lyapunov, and nearest-neighbor dimensions. References that give further information include Farmer et al. (1983), Holzfuss & Mayer-Kress (1986), Mayer-Kress (1987), Rasband (1990), Theiler (1990a), and Moon (1992).
The various types of exponent dimensions are all interrelated. Some even have the same numerical value for certain conditions. As a result, terminology tends to be loose, especially if the technical paper discussing "dimension" deals more with fractals than with chaos. The only way to be sure of the particular exponent dimension authors use is to look carefully at their mathematical definition and computational procedure. That's especially true if they mention the Hausdorff or fractal dimensions.

Most exponent dimensions fall into one of two categories. The first category measures only the attractor's geometry. That is, it takes no account of how often the trajectory visits various neighborhoods in the phase space. Examples are the similarity dimension, capacity, and Hausdorff-Besicovich dimension. The other category considers not only geometry but also probabilistic or informational aspects of the attractor. It takes into account that a trajectory may visit some phase space neighborhoods more often than others. Examples in this group are the information and correlation dimensions.
The correlation dimension presently seems to be the most popular in chaos theory. Actually, more than one of the above types might be helpful for a particular problem. Abraham et al. (1989) point out that some people use several measures of dimension to describe a chaotic time series.
As with many aspects of chaos, determining an attractor's dimension is an inexact procedure (Peitgen et al. 1992: 744) and is the subject of much research and rapid development. Glass & Mackey (1988: 52) point out that some topics particularly in need of more attention are the effects of noise, the requirements for the size of the dataset, and the effects of attractor geometry.