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This paper was written as an assignment for Ian Walton's Math G - Math for liberal Arts Students - at Mission College.

To explore other such papers go to the Math G Projects Page.
 

Math G_Lisa Corbett

Ian Walton_December 6, 1999

Albert Einstein once said in a letter to Max Born: "You believe in a God who plays dice, and I in complete law and order." Chaos gives way to order, which then gives rise to new forms of chaos. Deterministic disorder.

Ancient Theory

In ancient times, people thought of chaos as something immense and creative. Some thought of our beginnings as a state of chaos, or nothingness, from which beings and things came from. A Chinese story says that a ray of light, yin, jumps out of chaos and builds the sky while the heavy dimness, yang, forms the earth. Yin and yang then create everything else. The principles of yin and yang are said to retain the qualities of the chaos they came from. The story finally says that too much yin or yang will bring chaos back.

The Babylonians have a story of chaos that has a god symbolizing the "boundless stretches of primordial formlessness", of representing the intangibility and imperceptibility that exists in chaotic confusion. They believed that new forms came out from chaos and began to give structure to the universe.

Even in Christianity, there are stories related to chaos. In Psalm 74:13-14, it is related that God (who is order) is compelled to "break the heads of the dragons on the waters" and "crush the heads of Leviathan." This follows the notion of creation that emphasizes the struggle of the deity against the powers of chaos. The Biblical universe starts "without form, and void" until God creates, or orders, it. It would take modern science thousands of years to show a different face of chaos, a kind of virtual order.

Until recently, chaos has had two definitions in the dictionary: the disordered formless matter that existed before the ordered universe; and complete disorder or utter confusion. But recently, referring to a scientific conference held in 1986, yet another definition of chaos was proposed: stochastic (random) behavior occurring in a deterministic system (a system ruled by exact and unbreakable law). Deterministic systems that maintain themselves by oscillations, iterations, feedback or limited cycles are vulnerable to chaos and face an unpredictable fate if pushed beyond critical boundaries.

A seemingly orderly, repetitive and predictable motion becomes a chaotic episode and has mystified scientists and mathematicians for hundreds of years. What was thought to be simple becomes complicated. A complicated action becomes simple. And even the simplest equations can generate motion so complex, so sensitive to measurement that this motion appears to be random. Randomness in nature occurs naturally and is to be expected. Reductionism (NewtonÌs cause-and-effect determinism) imagines nature as equally capable of being assembled and disassembled. And reductive scientists believe the most complex systems are made up of many different elements that have been combined by nature in innumerable ways. Chaotic patterns are somewhat unpredictable. They are so complex that scientists in the past could not track them.

One of the fundamental characteristics of a chaotic physical system is its sensitivity to initial conditions. Sensitivity means that if two identical mechanical systems are started at initial conditions x and x+ E, where E is a very small quantity, their dynamical states will diverge from each other very quickly in phase space, their separation increasing exponentially on the average. The motion of a swinging pendulum is an example of this. A chaotic pendulum originating at two nearby points will have trajectories that diverge markedly in less than one forcing period.

In the nineteenth century, scientists wondered why they couldnÌt invert a perpetual motion machine. Whenever they ran a machine, some of the energy fed into it turned into a form that they could not recover and use again. The energy had become disorganized, chaotic.

Chaos seems to have laws of its own. Irregularities in nature have mystified scientists for years, and chaos has brought them to the surface of the scientific world. Chemists, biologists, physicists and mathematicians have all been looking for a connection between the different kinds of irregularities.

One physicist said "Relativity eliminated the Newtonian illusion of absolute space and time; quantum theory eliminated the Newtonian dream of a controllable measurement process; and chaos eliminates the Laplacian fantasy of deterministic predictability."

The relatively new science of chaos has its own language. Words like fractals, bifurcations, intermittencies, periodicities, folded-towel diffeomorphisms and smooth noodle maps are but a few examples of the new vocabulary. There are special techniques of using computers to study chaotic behavior. Some physicists believe that chaos is a science of process rather than state, of becoming rather than being. There are some that believe the fate of the universe depends on the distribution of matter within it.

Chaos is all around us. It effects all phases of science, medicine, weather and even the topography of the Earth. It plays a part in just about every aspect of our world. The reversals of EarthÌs magnetic fields, the collision of atoms of gas, irregular heartbeats, even the dripping of a faucet and the bouncing of a billiard ball all have chaotic properties.

Experiments in Chaos

Chaos has been around for thousands of years, but the first true scientific experiments in chaos theory were not performed until 1960 by a meteorologist named Edward Lorenz. He input twelve mathematical equations into a computer program used to predict what the weather might be.

One day, he wanted to repeat a particular sequence. In order to save time, he started in the middle of the sequence instead of the beginning. He entered the figures into the computer, and left the room to let it run. During the next hour, the sequence actually changed from what it had been previously. He learned purely by accident that even a small change in the initial conditions drastically changes the long-term behavior of a system. His chaos theory shows that it is sensitive dependant on initial conditions: where and at what point something starts and where it is at any given point will decide where it will go next. Linearity is a structured, predictable concept. Twice the force equals twice the distance an object will travel. Non-linearity shows that a small force can make an unpredictable reaction; thus, the concept of chaos.

A dynamical system is loosely defined as anything that has motion. There are many different types of motion that can be exhibited by a dynamical system. The simplest is fixed point behavior. An example of this behavior is the movement of a pendulum when friction and gravity bring the system to a halt. Most fixed points can be compared to a ball that rolls downward until it sits on a flat spot and has no momentum to carry it further.

The next simplest type of motion is known as a limit cycle or periodic motion. This motion involves movement that repeats itself over and over. A planet in our solar system orbiting the Earth has a periodic motion.

A more complicated form of motion is found in a quasiperiodic system. These systems are similar to periodic systems, but they do not quite repeat themselves. The moon orbits Earth, which orbits the sun, which, in turn, orbits the galactic center, and so on. In order for the combined motion of the moon and Earth to be truly periodic, they must at some future point return to some previously occupied state. But for this to happen, all of the individual motions must resonate, which means that there must exist a length of time that will evenly divide all of the frequencies.

Fixed Point Limit Cycle Quasiperiodic

One helpful element of analysis in chaos theory involves bifurcation diagrams. In dynamics, a change in the number of solutions to a differential equation as a parameter is called a bifurcation. These diagrams can provide a summary of the essential dynamics involved. The dynamics may also be viewed more globally over a range of parameter values. This allows for simultaneous comparison of periodic and chaotic behavior. An experiment involving the cycle and pattern of a pendulum is an example having bifurcation diagrams. For some values of the parameters, a pendulum will have only one long-term motion. But for slightly different values, two or more motions may be possible. If several of them are stable, the actual behavior can depend on initial conditions.

If the pendulum is lightly driven and the motion is periodic with the same period as the drive frequency, then the velocity has one value at a given time during the drive cycle. If the parameter is increased sufficiently, further components of longer period are added to the motion, and there will now be more than one value at the given phase. This system had undergone a bifurcation.

Chaos and Ecology

Some ecologists use mathematical models to study species and their development. They treat populations as dynamical systems. In mathematics, regular equations can produce irregular behavior. The equations applied to population biology were simple counterparts of the models used by physicists for their pieces of the universe. Physicists can look at a certain system and choose the appropriate equations to solve the problem at hand.

But the life sciences are very complex. A biologist can never simply choose the proper equations and solve the problem by just thinking about an animal population. He must first gather data and try to find the equations that produced similar output. There are questions that formulas must be applied to. Questions such as "what happens when you put one thousand fish in a pond with a limited food supply?" and "what happens if you add fifty sharks that like to eat two fish per day?" are common ones to an ecologist. What happens to a virus that kills and spreads at a certain rate depending on population density? Scientists applied mathematical formulas to these questions that sometimes worked to solve the issue. Population biology learned quite a bit about the history of life, how predators interact with their prey and how a change in the population density of a country can affect the spread of disease.

If a certain mathematical model went ahead or died out, ecologists could guess something about the circumstances in which a real population or epidemic would do the same. Differential equations describe processes that change smoothly over time, but differential equations are hard to compute. Simpler equations, "difference equations", can be used for processes that jump from state to state. Many animal populations do what they do in one-year intervals. Changes during the short time of a one-year span are very often more important than changes in a longer time-span. Many insects stick to a single breeding season, so their generations do not overlap as humans do. An ecologist might need only corresponding figures from the previous or current year to make predictions for

the coming season. A year-by-year model can produce only a thin layer of information about a systemÌs minute details. However, in many applications, this layer provides enough information for the scientist to complete his research successfully.

The mathematics of ecology is actually quite simple. Basic function equations are common elements. Following a population through time is a matter of taking a starting figure and applying the same function over and over again. The whole history of the population becomes available through this process of functional iteration. Another example of studying population growth is to use a linear function that would show a rise in population a certain percentage each year. A function like this would continuously show a growth every year. But in the real world, that is not always the case.

Ecological mathematicians now realize that they must find a function that more accurately matches the realities of life. Hunger and competition in the animal kingdom are examples of the harsh realities that affect the population of these animals. A more realistic function needs to contain an extra term that restrains growth when the population becomes large. The most natural function to choose would rise steeply when the population is small, reduce growth to near zero at intermediate values, and move downward when the population is very large. By repeating the process, an ecologist can watch a population settle into its long- term behavior.

In the 1950Ìs, scientists were studying the variations of this logistics difference equation. It contained variables representing the reproductive rate, the natural death rate and the additional death rate from starvation or predation among the species. They showed that the population would rise at a certain speed until it reached a certain level of equilibrium. But the scientists wanted to know how the changing parameters in the equation (representing growth rate) would affect the ultimate destiny of a changing population. The most obvious answer was that a lower parameter would cause a population to end up at a lower level. A higher parameter would lead to a higher steady state. This assumption was correct for some parameters, but not all of them. Occasionally, researchers tried parameters that were even higher, and when they did, they witnessed chaos.

The flow of numbers began to misbehave. They continued to grow, but did not converge to a steady level, as was expected. With populations bouncing back and forth, ecologists assumed that it was oscillating around some underlying equilibrium. The equilibrium was the important thing. It did not occur to the ecologists that there might be no equilibrium. During the 1960Ìs, textbooks and reference books did not even acknowledge that chaotic behavior could be expected. There was presented a standard sense of possibilities. Populations could remain constant or fluctuate around a presumed equilibrium point. The researchers knew that, in real life, populations do behave erratically.

But from a scientific point of view, they assumed that erratic behavior had nothing to do with the sort of mathematical models they were describing. And if models started to betray their makersÌ knowledge of the real populationÌs behavior, some missing feature could always explain the discrepancy. They could and did assume simply that numbers on the calculator were not accurate. The stable solutions were the interesting ones. Order was its own reward. Finding appropriate equations and working out the computations was not an easy job. Ecologists did not want to waste time on a line of work that was going awry, producing no stability. They knew that their equations were oversimplified versions of the real phenomena. The point was to model regularity. Why go to all the trouble just to see chaos?

Chaos and Medicine

One application of chaos in medicine is used to study problems with the heart. An irregular heartbeat, known as arrhythmia, can sometimes be fatal. The heart beats at irregular intervals and sometimes with varying intensity instead of the normal, regular (predictable?) heartbeat. A fibrillating heart is never all contracted or all relaxed. The electrical wave that travels throughout the heart is broken up.

It has been established in the medical field that these arrhythmias are manifestations of chaos. They follow fractal laws. They lack the order that should be present.

A doctor listening to a heartbeat hears the sounds of fluid whooshing against fluid, against solid, and solid against solid. Blood flowing, valves opening and closing and muscles contracting are all part of the inner workings of the heart. The muscle contractions themselves depend on a complex three-dimensional wave of electrical activity. Modeling any one piece of the heartÌs behavior would strain a supercomputer; modeling the whole interwoven cycle would be impossible.

Hospitals today rely upon professionals to interpret a patientÌs EEG, with respect to the "randomness" of the heartbeat, and offer a solution that will help restore the "predictable regularity" to the heartbeat. Irregular heartbeats have been categorized and investigated. There are dozens of names for the various irregular rhythms associated with the heart. Doctors use this information for diagnosis of heart problems. By using the tools of chaos, researchers have now discovered that traditional cardiology has been making the wrong generalizations about irregular heartbeats, inadvertently using superficial classifications to obscure deep causes.

At this point in time, training for cardiologists does not include much mathematics. But now that research has shown that nonlinear mathematics can help to understand these rhythms and their orderings, future cardiologists-in-training will most definitely use more mathematics to solve medical problems.

Chaos and Gasses

The motion of gaseous substance is chaotic. In an experiment, one can control the motion of the gas with the use of heat and cold. Heated gas inside of a box will rise up one side and seem to "roll" to the other side and lower in the box as it cools, effecting a "rolling pattern". But nature is not so predictable. Heated gasses will roll around in one direction, but will stop and reverse directions at random. This "randomness" is repeated in a seemingly "random" pattern; but at unpredictable speeds and unpredictable times.

Chaos and Weather

There are many variables that go into a weather forecasting equation. Air temperature, air pressure, wind velocity, wind direction and humidity are a few of these variables. Too many elements can seem to make an outcome unpredictable. We might think that physical laws govern the changes in the weather from one moment to the next. These laws would seemingly make prediction possible. But nature is full of "unexplainable" phenomena and many "unpredictable" changes, making it difficult to accurately predict the weather more than a few days in advance.

Yet, we are able to predict the high and low tides quite accurately for an entire year. In predicting the tides, consider that the ocean and the atmosphere are both large fluid masses that almost completely envelop the earth. They have similar physical laws. They both have fields of motion that tend to be drawn out by internal processes; yet, are driven by varying external influences. They are two components of a larger dynamical system, since each exerts a considerable influence on the other at the surface where they come into contact. The winds produce most of the oceanÌs waves. Evaporation from the ocean supplies the atmosphere with most of the moisture that eventually condenses and falls back into the ocean as rain or snow. The heat from the sun and gravitational pull of the sun and moon are also external forces that drive the position of the tides.

A regular response to these forces is the height of the tides. An irregular response might be an unusually high wave caused by excessive winds.

Weather variations are often thought of as not periodic. But they have periodic components (the cooling and warming during seasonal changes) that have been stated scientifically to several decimal places. If we subtract out the verified components, we are left with irregular signals. But the huge migratory storms that cross the oceans and continents are still out in full force. These are manifestations of chaos. However, we may be in a periodic state that lasts longer than the length of our weather records.

So, in attempting to forecast the tides, we are trying to predict the regular responses (which are already "predictable"). But in attempting to forecast the weather, we are met with many chaotic episodes involving the irregular (often "unpredictable") external forces; making accurate advance weather forecasting very difficult.

Chaos and the Stock Market

Stock markets are another example of non-linear, dynamic systems. Chaos analysis has determined that stock market prices are highly random, but with a trend. The amount of the trend varies from market to market and from time frame to time frame. One concept in chaotic systems is fractals. They are objects that are "self-similar" in the sense that the individual parts are related to the whole. It has been shown that major recessions mimic both monthly and daily price fluctuations; showing that it is self-similar from its largest to its smallest scales. The daily, weekly and monthly market statistics are all related to the entire market. It is, in a sense, a tree showing more and more detail as it "branches out". And the predictions of the stock market, like the ever-changing weather, are also dependent upon initial conditions. That is why these markets are so often difficult to predict.

The complexity of the components of the stock market changes so rapidly, that accurate predictions are almost impossible. The short- term data is random and very hard to predict. However, the long-term data is not random and is more easily predicted. But the short-term data accumulates to make the long-term data; so one would think that the short-term data would be predictable like the long-term. But this is a paradox that shows a system can be random in the short-term and deterministic in the long term. Chaos at work.

Chaos and the Consumer

Chaos theory even plays a part in todayÌs consumer world market. In 1993, The Goldstar Company created a "chaotic washing machine". It was the worldÌs first consumer product to exploit "chaos theory". The washing machine was supposed to produce cleaner and less tangled clothes. (By way of identifiable and predictable movements within the nonlinear system of the washing machine.) A small pulsator that rises and falls randomly as the main pulsator rotates, is the key to the chaotic motion. The company expected to make a huge profit and up their share of the world market when they released this product. But, as usual, in the world of consumer sales, another company made a similar appliance a few years earlier. The Daewoo Company used "fuzzy logic circuits" to control the amount of bubbles and turbulence in their washing machine. Fuzzy circuits make choices between the numbers zero and one, and between true and false. This idea was used to control the chaotic conditions of the washing machine.

Chaos and the Solar System

Astronomers have known for a long time that the solar system does not run with great precision. There are certain kinds of instabilities that occur throughout the solar system. Some examples of these instabilities are the motions of SaturnÌs moon Hyperion, the gaps in the asteroid belt between Mars and Jupiter, and the orbits of the systemÌs planets themselves.

To astronomers, chaos means an abrupt change in some property of an objectÌs orbit. One of SaturnÌs moons, Hyperion, is in a chaotic state. It is not defying NewtonÌs laws of gravitation and dynamics. Its orbit is precise and regular, but its attitude in orbit is not. It is tumbling over in a very complex, irregular pattern.

HyperionÌs position in orbit and its attitude are determined by the same physical laws and mathematical equations. Its position corresponds to a regular solution of those equations; but its attitude corresponds to an irregular solution. The attitude is the directions in which itÌs three axes point. Hyperion is not the typical spherical shape of most moons. It is elongated and irregular, much like a potato. Its shape does not affect its chaotic position.

Every solid body, no matter what its shape or density, has a corresponding ellipsoid of inertia. It is a ghost-like anomaly that is rigidly attached to the body, but has no mass. The lengths of each axis of the inertial ellipsoid are proportional to the inertia of the body when spun about that axis; so long axes correspond to greater inertia. The ellipsoid moves along with the body. When the body tumbles, the ellipsoid tumbles. But suddenly, the ellipsoid absorbs the material essence of the body, so we now have a solid ghost and a spectral body still attached to it. But the body and its ghost have the same inertial properties, so their motion is identical.

In order to make the best possible prediction of HyperionÌs future motion; scientists would have to formulate a complex mathematical equation for accurate predictability. The Voyager 1 satellite would make some calculations when it passed over Hyperion. A few months later, Voyager 2 would do the same; and the calculations would be compared. The predictions were not comparable. The motions seemed random, yet, were not. Hyperion is in dynamical chaos.

Chaos and Fractals

The name ÎfractalsÌ comes from the Latin ÌfractuaÌ, which means irregular or broken. Benoit Mandelbrot, a French mathematician, started using the phrase in 1975 to describe the pictures he was seeing that showed the infinite complexities of real rugged systems. Fractals, short for "fractional dimension", embrace the forms of geometry in nature that have been studied for thousands of years, yet were unable to be described scientifically.

Many of the forms of nature do not fit the mold of the typical geometric shapes that we are familiar with. For instance, clouds are not spheres, bark is not smooth, mountains are not cones and coastlines are not circles. They do not

have a natural number dimension, as in conventional Euclidean geometry, but have a "fractional dimension". Fractal geometry is the concept that came about because of this phenomenon.

The theory of fractals applies to many areas of science and technology. Fractals visually describe and analyze the physical forms of coastlines, trees, mountains, clouds, galaxies, rivers, weather patterns and even the human bodyÌs organs such as the heart, brain, and lungs.

A fractular pattern looks the same whether or not it is viewed from a distance or close and magnified. A coastline is an example of a self-similar shape, a shape that repeats itself over and over on different scales. How does one actually measure a coastline? The true, actual length of a coastline is infinite. In measuring with a larger scale map, the process of repeating leads to a greater estimate of length. Directly measuring the coast would result in even greater estimates. This fact shows that as the scale decreases, the estimated length

increases without limit. Therefore, if the scale of the measurements were to be infinitely small, the estimated length would become infinitely large.

A mountain range is another good example of this concept. From a distance, mountains look like very large, rugged areas with similar, smaller areas within. A closer look will also show a similar pattern, just on a different scale. The mountains look much larger, and the smaller areas appear to be a smaller version of mountains and rocks. The closer you get, the more shapes you begin to see within the physical makeup of the mountains. The rocks have formations on them, and the smaller stones have yet smaller formations. These smaller patterns still resemble the original patterns viewed from a distance, only on a smaller scale.

Benoit Mandelbrot thought that fractals were a way to deal with problems of scale in the real world. He defined a fractal to be "any curve or surface that is independent of scale, characterized by their built-in self-similarity in which a motif keeps repeating itself on an ever-diminishing scale". This property means that

any portion of the curve, if blown up in scale, would appear identical to the whole curve. The same property applies to a "picture", rather than a curve. The

picture, enlarged to scale, would appear identical to the smaller picture. The transition from one scale to another is represented as iterations of a scaling process.

Fractals are a way to visualize chaotic behavior. Benoit Mandelbrot again discovered a chaotic pattern in a study of line interference. He found that the bursts of interference had similar, tinier bursts of interference within it. Within each pattern, more similar patterns were detected on each level. He had found the pattern, or order, underlying the chaos.

The Future of Chaos Theory

Another mathematician, Pierre-Simon de Laplace, believed that if we had an accurate measure of the state of the universe and knew all of the laws that govern the motion of everything, then we would be able to predict the future with near perfect accuracy.

But we now know that this is not true. Science was mistaken in its assumption that everything is either a fixed point or a limit cycle. Chaotic systems are not just exceptions to the norm, but are, in fact, more prevalent than anyone could imagine.

The subject of chaos is very exciting to scientists and mathematicians. The movers and shakers of the scientific world want to be the first to discover or prove something new in the world of chaos. It is fashionable. It moves very fast. It cuts across numerous disciplines. These circumstances indicate that we must be careful of misunderstandings. Good ideas can become discredited if attempts to apply them are made by people who donÌt understand the pitfalls. This happens often in Îbandwagon scienceÌ. A research team definitely needs a well-informed mathematician as a team member.

But too much unbridled speculation (trying to forecast stocks, races, etc) could damage the truly useful understanding of chaos. The true description is deterministic chaos, that is, hidden structure in apparently random systems. This theory does not imply that everything that looks ÎchaoticÌ must have a hidden explanation. Only those things that have an underlying deterministic cause are actually chaotic. The base is still mathematics.

Chaos is everywhere and in everything. From the air that we breathe, to the wobble of planets, to the motion of a child swinging on a playground swing, chaos is in abundance.

We now live in an age of great change. Modern scientific research is at an all-time high. Who would not want to be able to more accurately predict the movement of the Earth in relation to natural disasters? Hundreds of thousands of lives would be saved if we could accomplish this feat. More lives could be saved with the study of chaotic episodes in medicine involving the human body. Future study is definitely in the works for this relatively new science. Hopefully, we have only good for mankind to obtain from this research.

Book References

Does God Play Dice?, The Mathematics of Chaos by Ian Stewart

The Essence of Chaos by Edward Lorenz

Chaotic Dynamics by G. Baker and J. Gollub

Chaotic Dynamics of Non Linear Systems by Neil Rasband

Turbulent Mirror by John Briggs and David Peat

Chaos, Making a New Science by James Gleick

Mathematical Ideas by Miller, Heeren and Hornsby

Internet References

//tqd.advanced.org/3120/text/gas.htm (Accessed 11/13/99)

//tqd.advanced.org/3120/text/s-market.htm (Accessed 11/13/99)

//tqd.advanced.org/3120/text/washmach.htm (Accessed 11/13/99)

www2.kanazawa-it.ac.jp/t-sutani/pofchaos-e.htm (Accessed 11/22/99)

//members.tripod.com/~SPConnection/nature.html (Accessed 11/30/99)

//members.tripod.com/~planet_e/thirdeye.html (Accessed 11/30/99)

//members.tripod.com/~xinsun/index.html (Accessed 11/30/99)