For want of a nail, the shoe was lost; For want of a shoe, the horse was lost; For want of a horse, the rider was lost; For want of a rider, a message was lost; For want of a message the battle was lost; For want of a battle, the kingdom was lost!'
As a relatively new and exciting science, chaos science grew very slowly during its infancy. Yet in the last decade, due to active research in many areas, it has became one of the hottest topics in academia as well as the popular science press. Many books have been published, and millions of Internet pages have been designed full of fractal pictures.1 Given that chaos is associated with disorder or confusion in a system or condition, why does it continue to attract so many people?
History of chaos
To understand chaos, one first has to understand the Newtonian worldview. Sir Isaac Newton's (1642-1727) development of the calculus and laws of classical mechanics began a scientific revolution in seventeenth-century Europe that caused all subsequent scientists to view nature from a profoundly different perspective. Now that they finally could determine the dynamics of bodies by simple equations, they believed that they had found the ultimate eternal rules that shape the universe.
French physicist Pierre-Simon Laplace (1749-1827), who based his work upon Newton's work, is credited with the following famous quotation (often referred to as Laplace's Demon): 'We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.'2
Laplace's Demon states the idea of determinism, that the past completely determines the future. One can clearly see why determinism was so attractive to scientists at that time. However, in Laplace's word everything was predetermined: no chance, no choice, no uncertainty. A solid, inevitable destiny had frozen the events in every corner of the past and is continuously spreading out to the future to do same there. Determinism apparently invokes the idea that whole universe is like a clock. God set it in motion at the beginning of creation and then removed Himself, for everything had been planed before. The ideas that there was no place for free will and that God could not interfere killed the belief in a soul and, consequently, in spirituality. Philosophers and scientists have discussed this for many years. Determinism affected many philosophies and triggered the major ideological movements of during eighteenth century, especially in Europe.
Toward the end of the 1800s, mathematicians and scientists began encountering some very difficult equations, some of which we know today are unsolvable. The most troublesome are various nonlinear differential equations. Even though it looks like such a simple and totally deterministic system, the problem of three bodies attracting each other with purely gravitational forces (e.g., the sun, Earth, and moon triple) turns out to be missing an exact solution. At first, such problems were cast-off as special cases and largely ignored.
One reason for this also might come from the fascinating world of quantum mechanics, which dazzled even the great physicists, and the lack of fast computers at that time. When these equations finally were studied in detail, a fundamental change that would ultimately overthrow determinism began to occur in mathematics and science. An indication of the science that would be come to be known as 'chaos' began to appear.
Why does chaos interest people?
In contrast to its common usage, chaos does not actually mean disorder or confusion. By definition, it should occur in well-defined orderly systems. However, most natural physical systems often can exhibit an unpredictable or intractable behavior in the long run, even though the system is defined by clear-cut orderly mechanisms. In this sense, chaos can be defined as unpredictability rather than disorder.
For example, meteorologists use 12 sets of well-defined equations to forecast the weather. They relate such atmospheric parameters as pressure, temperature, flow speed, and time to each other. One can make a computer program that calculates the parameters' final values by taking any initial conditions as the run's starting point. In principle, therefore, if we know the initial temperature, pressure, and time values that describe today's weather conditions, it is possible to derive tomorrow's weather conditions by running a computer program, which is nothing more than a chain reaction of numerical iterations.
However, in practice, initial conditions cannot be measured exactly and so contain a degree of uncertainty. But since the system's governing laws are known, one may estimate the effect of errors on future results. Hence, instead of giving the exact results, one can provide an approximate range of possibilities. This range can still be very useful, provided that the deviations do not stray too far from the actual values. In addition, knowing how the error grows in the system might help us understand and control the systems. But if we apply these error estimates to weather forecast equations, we will encounter a large problem, for errors grow exponentially in such systems. Even a tiny deviation at the beginning can create huge deviations from the actual values. It also can provide unrelated or nonsensical results.
An example of chaotic systems
This numerical behavior was first observed by the meteorologist Edward Lorenz, a pioneer in modern chaos work. Fascinated by the results he obtained, in the early 1960s he gave an interesting metaphor to explain the situation of high sensitivity to initial conditions: A butterfly's slight wing movement (i.e., a little deviation from the initial conditions) can change the future in a way that causes some chain reaction that ultimately result in a hurricane.
Such systems that exhibit a very high sensitivity to initial conditions are called chaotic systems. Chaos comes from the mathematical properties hidden in the equations defining the system, and such unpredictability cannot be removed. Even if the measurements' quality could be improved by minimizing errors, chaos never disappears from a chaotic system.
One may suppose that chaos occurs in complicated systems, such as weather forecast systems having 12 sets of equations. But even much simpler systems, such as billiards, can exhibit a very high degree chaos. A usual billiard system consists of many balls and a rectangular shaped table. I challenge master billiard players by requesting them to play the game in a stadium-shaped table. I am sure that they will find it difficult to do so, because such billiard tables would be chaotic systems.
If a system is defined as chaotic, this does not necessarily mean that its behavior is totally undefined all the time. As in stadium billiards, a ball has to be inside the billiards, so it should be somewhere on the table, even though sometimes we cannot foretell its exact position because of chaos. Besides, if you send the ball with a velocity perpendicular to a straight side, it will bounce back and forth between the two sides forever. Therefore, depending on which initial conditions are taken, chaotic systems also can show characteristics of regular motion.
A 3-body problem (in general n-body problems) such as the sun, Earth, and moon system, is a chaotic system. But since we can predict the motions of these celestial objects with great precision for many years in the future, why do we call this system chaotic? This triple system possesses a very special set of conditions: distance between bodies, their masses, and their velocities. These parameters cause it to exhibit near-regular behavior. It is analogous to the example of stadium billiards given above, for this triple system bounces back and forth between the table's sides.
Conditions for chaos
Chaos also can be caused by other factors than just uncertainties measured in the initial conditions. Scientists generally define hypothetical systems by isolating them from the outside world in order to simplify them as much as possible. However, as even objects in the real world that are far apart interact with each other, no system in the real world can be isolated. Given this, a closed (isolated) system might be defined as a fluctuating approximation to its real counterpart, which is changing in an unpredictable manner all the time. In short, even though we would know the exact initial conditions, the actual system could be chaotic due to changes in the approximate system. In this sense, many physical systems have an inclination toward being chaotic.
Due to its maximum complexity, the universe is the largest chaotic system. Observable regular patterns in special parts of that system repeat themselves in time. While the rest of the flows are unpredictable, they are not totally irregular, abrupt, or disordered. Just like whirls in a flowing river, they are in a kind of free motion searching for convenient conditions in which to give birth to organized structures.
The future of chaos
Chaos gives today's scientist a new worldview, for Newton's concrete, cold, and deterministic one has been shaken by the uncertainty principle of quantum mechanics. No one ever thought that the Newtonian worldview could be replaced. However, now scientists are more likely to be open to chance and choice than their predecessors. The question is whether chaos theory will cause large revolutions in how we understand the universe. However, the existing excitement, expanding research and growing number of articles, and its numerous applications from economy to biology, seem to indicate that a surprise improvement might not be so far off. '
- Fractal: A geometric pattern that is repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry. Fractals are used especially in computer modeling of irregular patterns and structures in nature.
- 'Chaos and Fractals: Laplace's Demon.' Online at: www.pha.jhu.edu/ ldb/seminar/laplace.html.