Combining Fractals and Chaos

The Intersection of Two Forms of Modern Mathematics

Mar 27, 2008

Labeled Bifurcation Diagram, http://www.calresco.org/beckermn/bifurcat.gif

Fractal geometry and chaos theory are two very interconnected forms of mathematics. It is difficult to fully understand one without also knowing the other.

Fractal geometry (described more thoroughly in the article "A Definition to Fractal Geometry") is the study of self-similar and endlessly complex, seemingly paradoxical patterns - some from the abnormal minds of mathematicians, others from "natural fractals;" patterns discovered in nature.

Chaos theory (described more thoroughly in the article "What is Chaos Theory?") can be described as the study of the behavior of unpredictable processes. It analyses and attempts to discover patterns in the seemingly patternless.

In learning the basic tenants of either of these relatively new forms of mathematics (both developing within the past couple centuries), one invariably finds them cross-referenced. In chaos there are fractals. Fractals can be used to define chaos.

But where exactly is this intersection, and how is it used?

Patterns of Chaos

As mentioned before, chaos is, by nature, seemingly devoid of pattern. It is the product of a combination of variances from an initial condition, and a compounding (like interest, but much more chaotic) of these variations to a degree which is essentially "incalculable."

One of the best ways to demonstrate the complexity of what is happening in a chaotic process is by the use of a "bifurcation diagram." This rather simple tool is merely a plotting of lines - a single line, representing an object or process, travels forward until it splits (the split shows a place where the object may be "diverted" from its course, and represents two possible paths from this point, either of which the process could potentially take). These two "hypothetical" paths travel forward a bit more, then split again. Then these split, then again and again...

One can imagine such a diagram being used to model just about any chaotic process - the cause/effect reaction of billiard balls, the gravitational attraction of bodies in space (notably those of the 3+ body problem), the cycling of whether patterns, etc.

After a few splits, bifurcation diagrams truly demonstrate that chaos is indeed the right word used to describe the number of possible outcomes. It should also be noted, however, that these diagrams are much more scientific and precise than what is described here. While this description may lead one to imagine a symple tree diagram, such as one modeling geneologies, a chaotic bifurcation diagram is much more complex (as the images below hopefully convey).

One of the most fascinating things that these diagrams show about chaos, is that even in this "unpredictability" it is not predictable. One cannot even predict that a process will remain chaotic.

After a certain amount of time, in fact, there is a possibility that the chaos of a system will suddenly give way to moments of calm, where all processes return to normal (these placid moments are denoted as "gaps" in the images below).

Fractal Chaos

And it is here, in these visual representations of chaos, that fractal geometry finds one if its applications. For one of the important definitions of a fractal is that it contains elements of "self-similarity," meaning that magnified portions of the figure tend to show themselves merely as smaller portions of the whole (a process which can be repeated ad nauseum).

When one magnifies considerably any portion of "calm" within a bifurcation diagram, it suddenly becomes clear that what is being viewed is simply another bifurcation diagram. All of the chaotic elements suddenly come together to form a single "path" once again, which soon splits and devolves into further chaos.

In this sense, a bifurcation diagram can be seen as just another example of a naturally occurring fractal. At the same time, however, many "synthetic" fractal images, such as the famous "Mandelbrot Set" can be viewed in the opposite light - chaotic mathematics which represent a fractal image.

In essence, it has become clear to mathematicians (and physicists and meteorogists and climatologists and every other scientist who finds themselves suddenly faced with trying to understand chaotic behavior) that fractal geometry is a fundamental aspect of the natural, chaotic world, and by understand one, a better understanding of the other must necessarily follow.

These two subjects have become fundamental to mathematicians in recent decades, and with just a brief look at some of their applications, it is very clear why.