In 1971 Ruelle and Takens came up with a new theory, based on the abstract concept of a strange attractor, for the onset of turbulence in fluids. Several years later May found an instance of chaos in the iterative map (logistic map) that was used to study population dynamics in biology.

 

Feigenbaum discovered that there are certain universal routes, which systems will take in transitioning from regular to irregular motion. This discovery provided the link between chaos and its transitioning phase.

 

In the late 1970's there were two additional important developments in the field. Mandelbrot discovered fractals and showed that they can be applied to other subjects.  And in the field of mathematical biology, Winfree used the geometric idea to study biological oscillations such as circadian rhythms and heart rhythms.

 

By 1980 the widespread interest in chaos, fractals, oscillators and their applications had taken a firm root for this emerging field of dynamics.

 

This resulted in the rapid growth of developing theories and other mathematical tools to deal with the systems. One of these would lead to the development of the control of chaos in the early 1990's.

 

**Chaos Theory

 

 

"It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible."
----Henri Poincare

 

The study of Chaos is part of a bigger program of study of a supposed “strongly” nonlinear system (Springer-Verlag 1). A nonlinear system is a system that is not in the first degree.  Multi-dimensional systems and systems that are not in a straight line are prime examples of a nonlinear system (1).

 

An example in the world of physics of such a system would be a fluid in turbulent motion (1). 

 

If this were not exactly the study of chaos, then the image of the turbulent motion would serve as a powerful symbol to remind a physicist of the sorts of problems he would like to understand (1).

 

Fluid turbulence certainly presents us with extremely erratic and only a somewhat predictable phenomena (1).  Since the historic times of La Place say, physical scientists have turned to the statistical methods when presented with problems that concern the joint behaviors of innumerably large number of pieces (1).

 

For the majority, chaos theory already belongs to the greatest achievements in the natural sciences in the twentieth century (9).

 

Indeed, it can be said that very few developments in natural science have gotten so much interest from the public (9).  Sometimes, people even hear of changing images of reality or of a revolution in the natural sciences (9).

 

Opponents of chaos theory have been inquiring about whether this popularity could have something to do with the witty choice of catchy terms (9).  Other critics say that an excuse could be the very human need for a theoretical explanation of chaos (9).

 

Chaos theory is once and awhile at risk of being overtaxed by being associated with everything that can be even superficially related to the concept of chaos (9).  Yet, what is it that makes chaos theory so fascinating?  

 

In particular cases one inferred that chaos was an unwanted anarchic quality.  In the past people have incorporated this idea of disorder into chaos. 

 

Dictionaries defined chaos as turmoil, turbulence, and primordial abyss.  Biblical references to Tohu and Bohu had the same referential character of undesired randomness.

 

Technically, Chaos implied the existence of the unwanted randomness, but the self-organization concept at the edge of chaos denoted the order we get out of chaos.

 

The American novelist and historian Henry Adams (1858-1918) wrote his scientific meaning of "chaos" succinctly: "Chaos often breeds life, when order breeds habit".

 

In a manner of speaking chaos theory came in the back door of the researcher's world.  It was not a law like thermodynamics or quantum physics, but it did allow the researcher to analyze events or areas with many problematic intricacies.

Several explanations for Chaos theory called for the words synthesis, cross-discipline, edge of chaos, dynamical, cellular automata, or neural networks, but all take with them the concept of complex systems.

The implications of Chaos are profound, for who could know the absolute conditions of any system for a complete prediction to be made of the behavior of that system?

To ancient humans, Chaos represented the unknown, nightmarish visions that reflected man’s fear of the irrational and the need to give shape and form to his apprehensions (Pickover 29).

 

Today Chaos usually involves the study of a range of phenomena exhibiting a sensitive dependence on initial conditions (29).  Which means that if you very slightly change a parameter in an equation or system, very different behaviors can result (29).

 

From chaotic toys with randomly blinking lights to wisps and eddies of cigarette smoke, chaotic behavior is generally irregular and disorderly (29).

 

Other examples include weather patterns, some neurological and cardiac activity, the stock market, and certain electrical networks of computers (29).

 

Our daily encounter with Chaos is seen in traffic flow, weather changes, population dynamics, organizational behavior, shifts in public opinion, urban development and decay, cardiological arrhythmias, epidemics.

 

It might be found in the operation of the communications and computer technologies on which we rely, the combustion processes in our automobiles, cell differentiation, immunology, decision-making, the fracture structures, and turbulence.

 

Chaos, fractals and dynamics, these are topics that are pretty much in the news today.  What do these teams mean though?  These are terms that come from the area of mathematics known as dynamical systems.

 

Dynamics is a natural topic to expose students from high school and college to and is a fascinating topic in mathematics. 

 

One reason it is so fascinating is that you will see is the computer graphic images that come out of the field of dynamics are so beautiful.  There’s something in these computer graphics that strikes the human eyes as quite alluring. 

 

Another reason for the fascination for this field of mathematics is the fact that is so exciting, even for people who despise mathematics.  Much of what you see in this field of study today is less than a quarter of a century old.  This is new mathematics.

 

One might think that this area of mathematics might be somewhat esoteric, since it’s so new, but in fact much of this mathematics is very accessible. 

 

We’re talking about quadratic functions, such as sin and cos, the kind of things that students in high school know quite a bit about.   

 

This area of mathematics involves in an essential way, the use of a computer.  Computers, computer graphics, and computer experiments are essential to understanding and working with dynamical systems. 

 

This area of mathematics promises to be quite applicable.  Anyone who has read the book by James Glick called Chaos: The Making of New Science will see the tremendous power of this mathematics in terms of applications in all areas of science.

 

One of the complicated objects in mathematics is the Julia set.  The Julia set is considered to be one of the most beautiful of these dynamical systems.  A process called iteration produces the images.

Julia Sets

Closely associated to the Mandelbrot set is the collection of fractals known as Julia sets. These sets were first examined by Gaston Julia (1893-1978) and Pierre Fatou (1878-1929).

Mathematicians originally characterized Julia sets as being "pathologically complicated," but it is their complexity that gives Julia sets their astonishing beauty.

Like the Mandelbrot set, Julia sets are the creation of dynamical systems based on the function f(x) = x² + c, where x and c are complex numbers. But for Julia sets each spot in the complex plane becomes an initial value of x, and the same fixed value for c is used for a complete set.

When the function is iterated, individual points in the plane will have open areas that either converge to some particular point or diverge to infinity. Those points in convergent regions form a Fatou set. The boundary of this set is a Julia set.

For each initial value x₀, the recursion x_n+1 = f(x_n) describes a sequence of points called the forward orbit of x₀. A set of points whose forward orbits move toward the same limit point is called a basin of attraction, and this limit point is known as the set's attractor.

A Julia set is a set of extraordinary points that separate different basins of attraction. It can be thought of as the repeller set of the iterative function, or as the attractor of the opposite relation. 

For the quadratic function f(x) = x² + c, the Julia set is the limit between the set of points that iterate to infinity and those that do not.

Assigning a value of 0 to c creates the function f(x) = z², which provides a helpful example. When this function is iterated, every point whose space from the origin is less than 1 converges to the origin.

Every point bigger than 1 unit from the origin iterates to infinity. Thus the Julia set of f(x) = x² is the unit circle centered at the origin. For nonzero values of c the matching Julia sets are fractals.

Two essentially different algorithms for producing Julia sets exist. One plots the boundary between basins of attraction; the other graphs the attractor of the inverse relation.

They construct somewhat different results, but neither is really superior to the other. The first method is more successful than the second for definite values of c, but less satisfactory for others. And both algorithms verify to be somewhat insufficient for a large number of Julia sets.

The first method is capable of generating color images by representing different convergence rates with different colors. The approach is very alike to that used to generate Mandelbrot set graphics.

For each pixel in the chosen domain, the function is iterated for the equivalent value of x. The iteration ceases when the size of x reaches 2 or the number of iterations arrives at a predetermined maximum. The last iteration count determines the color assigned to the pixel.

Infinity can be considered as an attractor for the iterative process defined by x_k+1 = x_k² + c. The pixels in its domain of attraction can be colored according to the how rapidly their corresponding points escape to infinity.

Most Julia sets are controlled within the square centered at the origin and having sides 3 units long, so this is the region usually selected for examination. A value for the complex parameter c must also be selected.

The maximum number of iterations to be permitted is assigned to the integer constant K.

Those points that do not converge in the direction of infinity after K steps will be shaded with one solid color. If the number of available colors is limited, they may be used in a periodic manner.

Furthermore, the computation time can be sliced in half by taking advantage of the Julia set's symmetry about the origin; the iteration counts for x = a + bi and -x = -a - bi will always be the same.

The value of c completely determines the figure of the associated Julia set. For values of c with a relatively small complete value, the Julia set is a simple closed curve.

As can be observed, the fractal curve is self-similar.  The location of c with reverence to the Mandelbrot set determines the general form of the corresponding Julia set. When c belongs to the Mandelbrot set, its Julia set is related.

But if c lies exterior the Mandelbrot set, its Julia set is "broken into infinitely many pieces."

When c lies within the huge atom of the Mandelbrot set's continental molecule, the related Julia set is a fractally distorted circle. If c is contained inside one of the continental molecule's smaller atoms, the Julia set consists of an infinite number of fractal loops, each surrounding a different basin of attraction.

If c is selected from the interior of one of the island molecules, the Julia set contains infinitely many copies of the Julia set that would be determined by the corresponding position in the continental molecule, all connected by a dendrite.

A value for c that lies on one of the filaments creates a similarly shaped Julia set, because infinity is the only attractor.  And if c lies outside the Mandelbrot set, the Julia set falls apart into a cloud of points called Fatou dust.  These points become sparser as c becomes more isolated from the Mandelbrot set.

What makes chaos theory so fascinating?  What do the supposed changes in the image of reality consist of?  To these subjects the philosophy of nature might help in the attempt to answer them and also impose some more questions.

 

And to do so, the topic of dynamical systems will be discussed.