There are “simple” systems whose parts are governed by well‐understood physical laws, yet these systems behave unpredictably. If vagueness is indeed a fundamental property of this world, then what is its role in this unpredictable behavior?

B.1 Chaos Theory in a Nutshell

It is a fact that weather forecasting is not easy. It is also known that we cannot predict how the weather will be in 10 or 20 days from now. What is not widely known is why this happens. Moreover, it is not known whether we can make our predictions more accurate and if we can predict the weather for long periods.

Edward Norton Lorenz [195] was a mathematician and meteorologist who wanted to use computers to predict the weather. He created a simple model, but soon he realized that when he slightly changed the input parameters, he was getting completely different results. For example, he noticed that, when his computer simulation was fed with decimal numbers having three decimal digits instead of decimal numbers with six decimal digits, the results were not the same as he would expect. One does not need to reimplement his simulation, which involved 12 ordinary differential equations, instead, one can observe the same effect using a much simpler mathematical expression. For example, one can use an iteration of the quadratic expression with initial value and . If, at some point, the iteration stops and is truncated to three decimal digits and the operation resumes with this value and goes to the end without any other interruption, the result will be quite different from what we will get if no interruption happened. This effect, that is known in the literature as the sensitive dependence on initial conditions effect, is a manifestation of a chaotic system. The theory that studies such phenomena is known as chaos theory (see Ref. [239] for an accessible overview).

Recently, in a study [34], it was discovered that there are systematic distortions in the statistical properties of chaotic dynamical systems when represented and simulated on digital computers using standard IEEE floating‐point numbers (i.e. the most common representation for real numbers on computers today). However, this is not a limitation of modern computers or modern programming languages, but a limitation imposed by people who implement their simulations using this specific approach. In fact, modern programming languages such as Java and Perl support arbitrary precision decimal numbers that can be used to compute anything to any accuracy. Of course, the calculations are much slower, but this is the price one has to pay when not using IEEE floating‐point numbers.

Chaos theory is usually studied alongside fractal geometry (again, see Ref. [239] for an accessible overview of fractal geometry). Fractals are geometric figures, where each part of such a figure looks like the whole figure. This property is known as self‐similarity. In general, objects encountered in Nature do not have the precise properties of the objects of Euclidean geometry. Nevertheless, they can be “approached” by fractals. In a layman's terms, the word dimension is associated with the numbers 1, 2, or 3 if we are talking about points, plane objects, or solid objects, respectively. However, each fractal object has its own fractal dimension, which is a rational or, more generally, a real number. Most of the time, when we say fractal dimension, we actually mean Hausdorff dimension.¹

The Hausdorff dimension of a subset is then defined as

In order to give a formal definition of fractals, we need to be familiar with the notion of topological dimension (also known as Lebesgue covering dimension) [118].

Using these notions, Benoit B. Mandelbrot [209], the father of fractal geometry, gave the following definition:

In addition, every set with an noninteger is a fractal.

Unfortunately, the definition of the Hausdorff dimension is such that it is causing difficulty to calculate on computers. Thus, we typically use the capacity dimension that can be computed with the box‐counting algorithm. The description of the algorithm that follows is based on its presentation in [128].

Suppose the space is such that one can define sensible distances and its integer dimension is . In this space, we are able to cover a fractal with ‐dimensional boxes (i.e. squares, cubes, etc.). Their sizes are regulated by one dimensionless parameter . This is always possible by choosing nondimensionless basic units so that becomes just a scaling factor. Next, we count the number of boxes that cover some part of the fractal. is connected to the capacity or box‐counting dimension, , by

By plotting against for many values of , we get a straight line whose slope is the fractal dimension. When the resulting curve is not a straight line, we employ the least square method to compute an approximation of the fractal dimension.

B.2 Fuzzy Chaos

A form of fuzzy chaos was proposed by Oscar Castillo and Patricia Melin [60]. Their starting point is dynamical systems that can be expressed as a nonlinear differential equation:

Or as a nonlinear difference equation:

In order to define and then to study fuzzy chaos, the authors considered a dynamical system on the real line given as follows:

The idea is to associate chaotic behavior with the number of period doublings (or bifurcations, that is, a separation of a structure into two branches) that occur when the parameter is varied.

An alternative definition that relies on fractal dimensions follows:

As was explained earlier, a numerical simulation of a dynamical system is realized by a feedback process, where the output produced by a function at some stage is given as input to the same function at stage . The output generated by such a simulation is called a time series: , ,…, . Next, we need to make a behavior identification for the dynamic system. For example, we can define the typical dynamic behaviors of a three‐dimensional system with the following fuzzy rules:

Here the fractal dimension can be calculated using the box‐counting algorithm and the time series. In addition, the fractal dimension and the behavior are linguistic variables and one needs to define their membership functions.

Now, let us consider a dynamic system that is described by the following mathematical model:

Here and , , and are positive numbers. However, there is a problem here: How do we choose these parameters? Fortunately, there is an algorithm that can be used to choose the best set of parameters. This algorithm is presented in Figure B.1. Also, the genetic algorithm mentioned in Figure B.1 is presented in Figure B.2.

Step 1	Read the mathematical model .
Step 2	Analyze the model so to “understand” its complexity.
Step 3	Generate a set of admissible parameters using the understanding of the model.
Step 4	Choose the best set of parameter values using a specific genetic algorithm.
Step 5	Perform the simulations by solving numerically the equations of the mathematical model. At this time, the different types of dynamical behaviors are identified using a fuzzy rule base.

It is possible to build a set of fuzzy rules for dynamic behavior identification based on the analytical properties of the mathematical models and using known results from dynamical systems theory. In particular, we should be able to get this dynamic behavior identification from the time series that we get from the simulation of a dynamical system. More specifically, we can calculate the Lyapunov exponents³ (see Ref. [299] for the description of an algorithm that solves this problem) of the dynamical system and also the fractal dimension of the time series. Having this information, we can easily identify the corresponding behaviors of the system.

B.3 Fuzzy Fractals

Fractal images have entered the pop culture mainly because they are stunning and relatively easy to create. The algorithms used to draw Julia sets (named after Gaston Maurice Julia), the Mandelbrot set, and images generated by iterated function systems (IFSs), which were conceived in their present form by John E. Hutchinson [164] (the term IFS has been introduced in [21]), are the tools that most people use to draw these impressive images.

Carlos A. Cabrelli et al. [56] presented a variant of the IFSs that can be used to draw “fuzzy” fractals. An IFS consists of a set of contraction maps⁴ , and associated probabilities , , , where is a compact metric space. Now, we can use the chaos game and an IFS to actually draw fractals. The chaos game is a method that can be used to generate the attractor (usually a fractal) of an IFS. We start with a random point that is supposed to belong to the fractal, then we randomly apply any of the contraction maps and get a new point . We continue this process and get a sequence of points that form a fractal.

A grayscale digitized picture is actually a table , where corresponds to the screen pixel ( and ). Each represents the amount of light that is emitted by the pixel . We can assume that , where 0 denotes no emission (i.e. black) and 1 denotes maximum luminosity (i.e. white).

Thus, a grayscale digitized picture is described by an image function. Similarly, the image generated by the chaos game can be represented by a function , where for each point , the value represents a normalized gray‐level value.

The collection of maps is used to define an operator , where is a subclass of the class , which is contractive with respect to a metric on . This metric is induced by the Hausdorff distance on the nonempty closed subsets of . Starting with an (arbitrary) initial fuzzy set , the sequence produced by the iteration converges in the metric to a unique and invariant fuzzy set , that is, . The unique, invariant fuzzy set is an attractor for the IFZS.

Iterated fuzzy function systems have been used to study fuzzy turbulence [206]. In addition, IFZS have been used for the study of fuzzy hyperfractals [10]. Also, a similar path was used to “fuzzify” the Mandelbrot set and Julia sets [318].