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Continuação e final

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When you “turn on” quantum mechanics, one must now deal with Heisenberg’s uncertainty principle, which states that there is always a very small error involved approximately equal to ħ/2 ≈ 10-35 when measuring both the position and momentum of an object. That is, objects when quantum mechanics is taken into account, always have a slight uncertainty of where they’re located or how fast they’re moving. As discussed in the quantum mechanics chapter, this uncertainty in measurement arises as a result of the dual particle and wave-like nature of matter. In the past, this uncertainty, when dealing with objects much larger in size than atoms, has been disregarded as insignificant since it is far from detectable in day-to-day experimental procedures. However, chaos theory asserts that this uncertainty is important, as even slight variations in initial conditions will result in divergent future behavior of the system.
Quantum chaos does certainly pose a dilemma to the scientific world. On one hand, tiny differences in initial conditions significantly influence the evolution of a system, while on the other hand it is known that it is impossible to know the exact initial conditions due to Heinsenberg’s uncertainty principle. So how can we ever make predictions? Unfortunately, there is no easy way around this problem. Some experts believe that it is misguided to apply the principles of quantum mechanics and chaos theory at the same time, as the mathematics that govern the two theories are quite different. However, if the two theories are reliable, they should be applicable to all physical processes. For instance, a “correspondence principle” is used as a check in which quantum mechanical formulas are applied to non-quantum mechanical or “large” systems. That is, as masses become much greater than the mass of an atom, and velocities become much slower than the speed of light, quantum mechanical formulas give the same results as classical formulas. However, on the flipside, as masses become very small and speeds approach the speed of light, formulas used in classical mechanics do not adequately explain things, and therefore their validity must be questioned.
The question is then, is chaos theory applicable on all levels, especially on the quantum or very small scale? It should be. There is no reason, neither conceptually nor mathematically, to think that its principles aught to break down on a quantum scale. Unfortunately, this results in an inherent indeterminism in nature since precise initial conditions cannot be known with exact accuracy. With this assertion, the deterministic universe seems to have been transformed into a conglomeration of indeterministic events. However, all is not lost. Chaotician, Wesley Salmon argues that the pursuit of scientific investigation is still worthwhile even in this possibly indeterministic world, “If indeterminism is true, it does not follow that there are events that are incapable of being explained.” What must be done is change somewhat slightly the direction of science, away from a completely deterministic and identifiable solution, and toward somewhat of a probabilistic and dynamical understanding of systems. This sort of adaptation is currently the subject of much debate as theoretical scientists ponder the best course of action for the future.
What then does all this determinism/indeterminism talk tell us about the possibility for free will? Opinions vary, but most who possess a command of the principles involved agree that chaos theory does not provide a full proof justification for the existence of free will, even though it does imply an indeterministic universe. Quantum mechanics holds that it is impossible to know the exact position and velocity of a particle at the same time, but this does not necessarily mean that the particle does not have a position and velocity at a given time. Instead, there is something else that chooses exactly where a particle is at a given time (and the location is constantly “hovering” anywhere within this allowed box of uncertainty). The free will problem is quite a hefty problem, but in order to fully address the problem one must consider the notion of time. Einstein showed the world that time is nothing at all like we’ve thought – it’s strange – it moves at different paces depending on what’s going on around it. Even if it is granted that events within the universe cannot be traced to a definite cause (which quantum chaos does not necessarily imply), it is still possible that events could be predetermined even though they are not necessarily caused by outside forces. For, it may be possible to think of time as existing much like a mountain range exists, with the human race marching along the top of the mountain range facing backwards. More specifically, it’s possible to see what’s behind us (history, or the back of the mountain range), but it is impossible to see what is in front of us (the future, or the front of the mountain range). However, the reason that humans are able to see the past but not the future is because they are positioned the wrong way; humans can only look backwards. Perhaps someone else, existing outside of time, sitting on a bench looking at the mountain range could see time in its entirety. Even with the puzzling implications of quantum chaos, there is no reason to believe that the future is not just as fixed as the past until a better conception of time is understood. To offer some conciliation however, even if all events in the universe (both “past” and “future”) are already known by an observer outside of time, this still does not imply that they are caused. Just think about the doctor who advises his smoking patient. The doctor knows that with the patient’s weak lungs, if he does not quite smoking he will soon develop lung cancer. This does not mean that the doctor causes the cancer however. The doctor can foresee a future event, but he does not cause that future event. Similarly, it is not entirely accurate to say that just because the future may be already known by an observer that exists outside of time that free will must be abandoned.
The existence of free will for humans, at this time, cannot be outright proven using chaos theory. While chaos theory does suggest some obvious shortcomings in the area of determinism, it cannot touch the much more complicated subject of free will. Ambitious philosophers have attempted to draw conclusions on the subject, but unfortunately the problem is still more complicated than may be addressed with the current knowledge available. However, it is likely that in the future, chaos theory might have something strong to say about the nature of free will. It is important to keep the problem fresh in our minds, and in doing so, it’s very likely that the existence of free will just might be proven in the not too distant future.

Conclusion

For a discipline that did not really get any attention until the 1970s, chaos theory has emerged onto the recent scientific scene with a bang. Although many experts doubt the strength of the theory, more and more research institutions are developing departments to study chaos theory and nonlinear dynamics and the results being generated from these departments speak for themselves.
The primary reason that chaos theory has not received the amount of attention that it may deserve is because it was introduced at a time roughly contemporary with the introduction of two other incredibly important branches of physics, namely those of relativity and quantum mechanics. Furthermore, relativity and quantum mechanics, thanks to a good deal of research, are understood at a more encompassing level than is chaos theory. Though the principles underlying chaos theory were hinted at by previous thinkers, it was really not until the technology revolution and the introduction of the computer that chaotic phenomena could be readily studied. It is quite likely that if the discovery of chaos theory came approximately 100 years earlier (along with the discovery of the computer), its importance might be presently more greatly appreciated. It’s like going to a banquet where you meet William Shakespeare, Sir Isaac Newton, and Bill Gates. All three, historically, will go down as great individuals as they all have significantly changed the way in which the world operates. It is likely that any one of these individuals would create quite a hubbub if they were the only one at the banquet, but collectively they tend to take away from each other. Shakespeare and Newton would probably dominate any given banquet over Gates, as they’re older, and perhaps better respected. There will be time for Gates later and he would likely stand somewhat in the shadow of the other two. This is more or less the attitude toward chaos theory. It has the potential to be great, but it doesn’t quite get the attention of its predecessors yet.
However, like any good theory, chaos theory must first be questioned. The Copernican theory of a heliocentric solar system took quite a long time before it was accepted. In fact, it was only shortly before the turn of the twenty-first century that the church officially accepted the theory, and apologized to Galileo for executing him for supporting the theory (a bit too late, however). Quantum mechanics too had to withstand a great deal of criticism, most notably from Einstein who never cared much for a universe that was governed by probability and bell curves. In all, a group of minds thinking alike will never generate new conclusions, for breakthroughs involve the abandonment of traditional thought and the embracing of something new and unexplored. Einstein himself held the belief that “Great spirits have always encountered violent opposition from mediocre minds.”
Chaos theory, with quantum mechanics and relativity is shaping the future of science. Traditional scientific techniques are being shown inadequate, and, if science is to advance to an all-encompassing theory, it’s going to have to do so by incorporating, at one level or another, chaos theory. This sort of transition will not be easy, especially to scientists who have been taught to think in traditional terms and to hold classical viewpoints. The good news is that the transition to this new world is being made, and as more and more people begin to accept its importance, the transition for the remaining skeptics will become additionally forced. I believe that chaos theory must be thought of more as a paradigm, governing the direction of science and knowledge, than of a scientific theory in the traditional sense. The acceptance of new knowledge and new thinking is of paramount importance in a functioning society – but it is more than this – for it is excitement and hope that gives the pursuit of knowledge meaning. In fact, a complete theory of the universe may not be all that good after all, for do we really want to live in world where nothing is mysterious? Looking to the future, a great deal remains mysterious, but the uncovering of mysteries is being shaped mainly by developments that occurred during the lifetime of today’s adults. Recent theories reiterate Virginia Woolf’s view of interconnectedness in A Sketch of the Past where she believes, “It is a constant idea of mine, that behind the cotton wool [of reality] is hidden a pattern; that we – I mean all human beings – are connected with this; that the whole world is a work of art; that we are parts of the work of art.”
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Link: http://www.duke.edu/~mjd/chaos/Ch3.htm

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Sugestão de leitura de fim de semana ou do final deste serão

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Chaos Theory: The Mergence of Science and Philosophy

Section I: Introduction
Section II: The Beginnings of a New Theory
Section III: The Fractal World
Section IV: Applications of Chaos Theory
Section V: Chaos Theory and Free Will
Section VI: Conclusion



We tend to think science has explained everything when it has explained how the moon goes around the earth. But this idea of a clocklike universe has nothing to do with the real world.

-Jim Yorke, University of Maryland Physicist

As any beachcomber will excitedly admit, there’s a surfeit of great stuff lying along the shores of the world - just waiting to be discovered. In fact, anyone who has ever taken a trip to the ocean (and walked the seashore at low tide) will likely share very nearly the same feeling. From unique rocks and fossils, to fishing lures, to washed-up sea creatures, there’s always something that catches the eye of even the most arid wanderer.
Out of the corner of the eye, the meandering beachcomber detects the shiny pebble – that one rock that stands out like a beacon amidst the surrounding clutter of the more ordinary rubble of the coastline. So the beachcomber picks up that shiny rock and places it in his pocket, keeping it (for it is the best rock) as a memento of his scavenging experience. The beachcomber returns home, and, proud of his new souvenir, places the rock on a shelf for display. However, after a good night's sleep and awakening the next morning, the beachcomber admires the rock on the shelf and sees somewhat of a different rock. Something has changed: the rock on the shelf is in some way not the same rock as the one that was found on the beach, although it is quite definitely the same material rock (all of the atoms are still there). In the new environment of the shelf, the rock has somehow lost its charm.
The reason is that the rock’s identity did not belong exclusively to the rock itself, but was bound-up within the neighboring, rather ordinary beach clutter as well. The individual details of the beachcomber's finding of the rock: the smell of the sea, the wind against the beachcomber’s back, the rock’s location in a tide pool of gray, white and black pebbles - all these tiny and rather seemingly insignificant intricacies contributed to the identity of the rock. Take away these details, and the rock loses something. Without these crucial, although often de-emphasized details, the rock has changed in some way – it has lost its magic.
The rock scenario just presented illustrates one of the most central principles of a breaking new branch of science and philosophy known as chaos theory. Tiny details are often neglected in studying many phenomena in nature, but these details prove to be quite important in the big picture of things. Introductory physics instructors (the brave ones anyway) often attempt to exemplify the equations of physics by performing classroom demonstrations. For instance, while covering projectile motion, an instructor might calculate on the blackboard exactly how far a ping-pong ball will travel after being shot out of a catapult at an angle of a certain degree with a certain initial velocity of definite amount. Any introductory textbook will assert that Newton’s laws of classical mechanics may be used to calculate exactly where the ball will land. So then, attempting to liven up the classroom (as quite a few of the students have already zoned-out by the time projective motion is covered in class), the teacher brings out an actual catapult, sets it to exactly the angle and initial velocity calculated on the blackboard, and shoots the ball across the room.
The instructor repeats the demonstration ten times and each time the ball lands in a slightly different location, no location corresponding to where Newton predicts the ball to land. But how can this be? Classical physics asserts that the ball should land in exactly the same spot every time. But of course it doesn’t. One time it’s a little to the left of the projected landing spot, while another time it’s a little to the right. For a number of the runs the ball does not travel quite as far as the calculation predicted, while for some runs it travels a little further. What has happened?
Well, the reason for the differing results of the experiment has to do with the fact that the world is not perfect. The spring constant on the catapult was only 99.9 % accurate and the blackboard problem assumed a 100% accuracy, thus, the actual initial velocity of the ping-pong ball was not quite as large as expected. Furthermore, the air conditioner was on in the back of the room, thus creating an air current that sent the ball on a slightly off-center trajectory (and the air conditioner had switched on and off a number of times over the course of the ten shootings). The ping-pong ball itself had a mass that was slightly greater than the mass used in the calculation, as it was dirty and had acquired a very small layer of dust around its surface. In bulk, the demonstration was not conducted in a vacuum with perfect conditions, so the ball did not hit the ground exactly where it was "supposed" to hit the ground. All these tiny factors, the spring in the catapult, the air conditioner, the dirty ball, though seemingly insignificant in themselves, when taken together contributed to the ball landing really nowhere near where it was predicted to land (and thus making the professor look quite foolish – no doubt reinforced by a flippant remark from one of the students sitting in the back row who shouted, “It’s a good thing we know these principles in theory, because we sure as hell can’t demonstrate them”).
The reason that the ball didn’t land where it was supposed to land was not because the equations being used were themselves wrong, but because the world in which the experiment was taking place was not perfect. There are billions and billions of tiny factors that affect the world and it’s impossible to just study one or two of these factors when exact predictions are being sought. This is the idea behind chaos theory. It is the idea that the real world is not a perfect place, as it is in the textbook; there are tiny details everywhere, even on the other side of the globe, that affect what’s going on in every corner of the universe because tiny factors turn into huge factors when taken together or allowed to grow over time. Chaos theory asserts that the equations of physics cannot be completely trusted because they only work in ideal situations and make far too many approximations. In the words of the 1960s IBM researcher and inventor of fractal geometry, Benoit Mandelbrot, “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”
Before jumping into the realm of chaos theory, I believe that a brief prelude to the validity of the theory is necessary. Relativity and quantum mechanics, subjects that have come to the forefront to physics in the course of the last century, are highly scientific, mathematical disciplines whose theorems, though somewhat paradoxically at times, have been observed via numerous physical experiments and observations. That is to say, these subjects are accepted as real and reliable branches of physics by the majority of contemporary scientists. Both are taught in just about every college undergraduate physics curriculum, and the principles of these two theories, at least the most fundamental principles, are generally accepted as providing the best explanations for certain behaviors in the universe. However, the branches of relativity and quantum mechanics have strong philosophical undertones as well. That is, by crunching through the math behind problems of relativity or quantum mechanics, interesting conclusions that have philosophical repercussions emerge. Generally however, these philosophical complications are secondary to the mathematics of the two theories, and, while these philosophical questions are worth discussion, they are not generally regarded as the bulk of the theory. Otherwise stated, quantum mechanics seeks a (more or less) purely scientific understanding of the atom, and, if certain paradoxes relating to parallel universes or wave superposition emerge, then so be it. However the pursuit of these philosophical questions is not the chief aim of the majority of particle physicists – the aim remains in science, or physics, as, as unfortunate as it might be, particle physics is much more fundable than is debating the possibility of parallel universes. Societies are much more inclined to grant money for Magnetic Resonance Imaging (MRI) studies that have applications in medical clinical trials than in granting equivalent amounts of money to figuring out the meaning of life. To make a long story shore, relativity and quantum mechanics work from physics to philosophy. They are philosophical disciplines in the sense that they raise questions that are not traditionally thought of as scientific questions, but these disciplines seek scientific explanations to these questions at a highly specific level. This is how science in general works; a scientist may devote his entire life to the understanding of one piece of a 1000 piece puzzle, which in itself may seem insignificant, but will hopefully one day be combined with the work of 999 others (and their graduate students) to form a coherent solution to a big problem.
I believe that chaos theory does not follow this pattern of philosophical questions arising from scientific investigation. Instead, chaos theory, though it is certainly debatable and can in many ways be applied in this way, is a science derived from philosophy. The foundations of chaos theory have resulted from people’s everyday experiences and feelings on certain seemingly non-mathematical or non-scientific phenomena. That is to say, chaos theory has its origins not in scientific laboratories, but instead in ordinary observations by ordinary people. Meteorologists, artists, computer engineers and philosophers have played key roles in the discovering of this new branch of knowledge. It is from the natural world and non-scientific disciplines (or at least disciplines that are not exclusively scientific) that the equations and patterns governing chaos theory have been drawn. This occurrence has two important consequences. First, chaos theory is presently not regarded as a core subject within the physics community. That is to say, it lives somewhat in the shadow of relativity and quantum mechanics, each of which are presently much more scientifically dependable (i.e. they are able to be used to generate definite results for scientific papers, publications, etc.). The primary reason that chaos theory is not as highly regarded as relativity and quantum mechanics in the current physics community is that the conclusions that chaos theory points to are often not backed by mathematical dependability. These new conclusions rely more on patterns and somewhat hazy prediction than on numbers and single quantities, in many cases. Because of this novelty, it is difficult to understand exactly what these conclusions mean. This is not to say that the professional scientific community abandons chaos theory, for many universities across the world have now developed departments in studying chaos theory, complex systems, and nonlinear phenomenon, but it is to say that across the board, the developments resulting from studies in chaos theory are not as central in twenty-first century physics as those of relativity and quantum mechanics. The second point pertaining to chaos theory worth mentioning is that the scope of the theory is quite controversial. Some experts believe that natural processes are so complex that it is simply best to ignore the minor eccentricities of nature and stick with the equations Newton has provided, and we ought to go quietly about our business. The others, of whom I am one, believe that chaos theory has potentially more to offer to the human race than any other field to date. The reason being that chaos theory is not just a mathematical or scientific discipline, but is instead the beginnings of an all-encompassing discipline. Chaos theory models how the world works, from weather patterns, to stock market shifts, to art, to brain patterns, to social structures, to something as seemingly non-scientific as interpersonal relationships themselves. In all of these areas, and many more, chaos theory is beginning to provide powerful explanatory power.
Chaos theory, at least in terms of most questions, does not provide definite answers, as scientific experiments have come to seemingly provide. Instead, at least in this stage of the theory, chaos theory shows that there is much more to the world than can be modeled in a few equations. The world is unimaginably complex, thereby implying that there is little hope for it to be understood or accurately modeled by humans. However, as will soon be shown, the remarkable aspect of chaos theory is that incredibly complex systems show definite patterns. Though the world is infinitely complicated, there appear to be spooky patterns that exist within the world. By studying these patterns within complicated system (instead of attempting to unrealistically isolate just a few variables within the system), a new science begins to emerge. This new discipline is scientific as well as philosophical, and it is quickly changing the direction of a great many fields of study.

The Beginnings of a New Theory

The name chaos carries with it a connotation of disorder and confusion, so what exactly is the idea behind a chaos theory? Such a theory seems to be a contradiction in terms. Can there be a theory of disorder, a theory of chaos? It depends on the definition of “disorder,” for as it turns out, there is indeed a good deal of order in even the most seemingly disordered processes.
Unfortunately, there is not a universally agreed upon definition for the theory, as the scope of the theory and its proper use is still disputed by many. Philosopher and author, Stephen H. Kellert, in his book In the Wake of Chaos, defines chaos theory as “The qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems.” Put more simply, chaos theory asserts that it is possible to observe complex, unpredictable and seemingly random behavior from relatively simple (or at least seemingly simple) systems in nature. However, chaos theory also deals with the reverse scenario. That is, chaos theory takes seemingly random systems and shows there to be order within the apparent randomness. In the words of Shakespeare’s Polonius, “Though this be madness, yet there is method in’t.”
In understanding a broad definition of chaos theory (for that is all that is really available at the present time), it is necessary to acknowledge the uniqueness of the theory. Chaos theory, unlike many other branches of mathematics and physics, is not exclusively dependent on numerical calculations in developing predictions. The reason for this is that the type of problems chaos theory deals with are nonlinear problems. Nonlinear problems are generally more difficult to study because the behavior of nonlinear systems cannot be predicted in a "straight" manner. Furthermore, due the nonlinearity implies in the majority of cases of nonlinear and chaotic systems, the systems cannot be solved exactly in the traditional mathematical sense; thus, patterns and models must be sought out. What is being dealt with in nonlinear investigations are systems that are not straight and predictable, but are instead twisted and distorted. Just think about a simple linear equation, say in the area of population dynamics. In a simple scenario, the predicted annual population of a given group is equal to the rate of growth of the group multiplied by the number of years the group has existed. This is a linear relationship because when this equation is plotted (the predicted population against the years of existence), a straight line with a slope equal to the rate of growth is obtained. This sort of linear relationship makes it really easy to predict future populations; one can either plug-in different years to the actual equation, or one can just look at the line graph - the equation has a definite solution that is easy to calculate. Scientists, mathematicians, economists, and sociologists really like linear relationships because they allow for future predictions to be generated quickly and easily.
Unfortunately, many relationships in nature are not linear. With the population growth equation just presented, it is clear that this is a very poor model for future group populations. What about death rates? What about droughts, epidemics, and relative predatory populations? All of these factors contribute to the future population of the group. As all of these slightly more subtle variables are accounted for, it becomes clear that there is no longer a linear relationship between population level and time. That is, maybe the predicted population is proportional to the square (second power) of the year multiplied by the growth rate, minus the cube (third power) of the year, divided by another variable that accounts for droughts and yet another that accounts for epidemics. What happens when these powers of 2 and 3 to the equation are considered is that the plot is no longer a straight line, but is instead now something that wiggles. Instead of the plot simply growing linearly, it curves up and down, depending on the year, the predator population, the available resources, the effects of global warming and the myriad of other factors that are relevant. The more nonlinear factors that are introduced, the more complex the equation becomes. At some point the equation becomes so elaborate and complex that it is impossible to solve by hand.
Until about 20 years ago, this inability to solve complex nonlinear equations by hand posed a huge problem to researchers. If equation couldn’t be solved, then how could they be useful? This problem was generally overcome by just ignoring a great many factors that complicated the equation. If enough of these nonlinear factors are ignored, then the equation is simplified into something that can be solved using traditional mathematical techniques. The problem with this approach is that it results in a pretty shoddy model for the system being studied.
However, recently scientists have been blessed with a great new tool in studying these incredibly complex equations: the computer. Computers can generate plots of nonlinear equations in fractions of a second, thereby allowing researchers to observe the behavior of nonlinear systems that were previously overlooked as unsolvable. The computer is to chaos theory what the wheel was to transportation: It provided people with the means to begin exploring a whole new world. If the computer and its number crunching abilities had been available at the turn of the twentieth century, then it’s quite likely that the study of chaos theory would have originated much earlier. However, because the computer did not really become available until somewhere between the 1970s and 1980s, the emergence of chaos theory had to be postponed.
So then, the equations that govern the workings of chaos theory are highly complex, so complex that they cannot be solved by ordinary pencil and paper techniques. However, the computer can be used as a tool to model the behavior of these equations and produce previously unobtainable data. What sort of physical systems are governed by such complex, nonlinear equations? As it turns out, a whole bunch. In fact, the applications reach far beyond just systems in physics. Current research on chaos theory is being carried out in just about every branch of western thought. Biology, chemistry, physics, mathematics, music theory, sociology, psychology, economics and philosophy all have begun to place considerable emphasis on the possibility of using the principles of chaos theory to model certain previously unexplainable phenomena. In fact, in my own work, I have provided some of the background material on chaos and economics for a company called Tetrahex that attempts to model the ups and downs of the stock market with a new kind of “Fractal Finance” software. Dr. Steven Strogatz has outlined numerous interesting applications of nonlinear dynamics in his textbook Nonlinear Dynamics and Chaos, in which he presents examples ranging from electrical circuit behavior, to the encoding of secret messages, to the modeling of love affairs.
Like many great discoveries, chaos theory was first discovered by accident. The unlikely hero was meteorologist, Edward Lorenz who was seeking to discover the secrets to accurate long-term weather prediction. Lorenz, as a student, had studied under the mathematical physicist, David Birkhoff and during this time in his life developed a considerable appreciation for the mathematics governing a variety of physical systems. Lorenz, like all meteorologists, was unable to make accurate long-term weather predictions, but he believed that if all of the present weather conditions were known, then future weather patterns (what the temperature would be in Helena, Montana in five years) would be able to predicted with absolute accuracy. This idea is quite central to scientific investigation. If the initial state of a system is known, then by using laws, it is possible to predict the future state of the system.
In 1960, Lorenz was making use of his digital computer to solve a set of 12 mathematical equations that roughly modeled certain weather patterns. The weather on earth, Lorenz realized, was more complex than could be incorporated into 12 equations, but he hoped that by looking at the 12 equations on his computer, certain patterns pertaining to weather cycles would emerge. After all, Lorenz acknowledged that though the weather was incredibly unpredictable, the weather conformed to definite patterns. For instance, there are always cycles of low and high rainfall, heat waves tend to come in patterns, as well as do cold fronts. Lorenz believed that though it was very difficult to predict what exactly would be the weather on a future date, by looking at patterns in the weather certain educated guesses could be made (e.g. it’s going to be warmer in July of 2025 than it’s going to be in July of 2035). Lorenz ran his program consisting of his 12 mathematical equations and looked at the results. Indeed, there were certain patterns that developed in the resulting graph.
Like any good researcher, Lorenz sought to acquire more data than was obtained from just the single run. For, the results from this particular experiment were a fluke; the results could not be reliable unless they could be reproduced again and again. However, a problem with computers in the 1960s was that they were unfathomably slow by today’s standards. A computer bought as recently as 1995 has less than 1/10th of the speed of a computer bought just five or six years later, and computers of the 1960s were nowhere near as powerful as the computers of the 90s. This reason, combined with a bit of fortunate impatience as it would turn out, led Lorenz to rerun the simulation not from the initial starting point, but from a different, later starting point. That is, instead of running the whole simulation again, he only ran the second half or so. By doing this, he cut the amount of time it took to run the simulation in half (which allowed for him to collect more data by conducting more runs).
The results of the new runs were quite startling. Lorenz expected the second simulation to correspond to the second half of the first run, but, after the first couple time steps, the pattern began to drastically diverge. The second run was nothing at all like the second half of the first run! The reason for this was that in the second run, Lorenz had entered the starting coordinates for the simulation by hand, which he simply read from the printout from the first simulation. The computer, being the accurate machine that it is, stored the numbers to six decimal places, while Lorenz only entered the numbers to three decimal places. Instead of entering the number 0.500354 for example, Lorenz would only input 0.500. Accuracy to three decimal places is very, very good and accuracy to six decimal places is almost unheard of due to experimental and laboratory errors. Thus, Lorenz gave little thought to rounding the input numbers down to three decimal places.
However, because the numbers entered were not exactly the same as the original numbers, the plot outputted by the computer quickly diverged from its original pattern. This occurrence led Lorenz to the conclusion that complex systems such as the weather are incredibly sensitive on the initial conditions of the system. In Lorenz’s own words,

It implies that two states differing by imperceptible amounts may eventually evolve into two considerably different states. If, then, there is any error whatever in observing the present state – and in any real system such errors seem inevitable – and acceptable prediction of an instantaneous state in the distant future may well be impossible. An alteration so small that it only affected the one-millionth place value of a decimal point, comparable to a butterfly flapping its wings perhaps, could throw off the whole prediction.

This incredible dependence on initial conditions was labeled by Lorenz as the “Butterfly Effect.” A butterfly flapping its wings in Colombia and thus changing the initial conditions within the atmosphere (though by only a fractionally small amount) could cause rain in Texas. This clearly led to tough problems for meteorologists. For, the weather was so complex that its future behavior could be affected by something as small as a butterfly flapping its wings. How could all of these tiny factors be accounted for? Even the meteorologist’s very efforts to account for these effects would have to be taken into account if an accurate weather prediction was to be made.
Lorenz became increasingly interested in his newfound observations, and he continued to perform experiments relating to sensitivity in initial conditions. Lorenz began taking equations that he was familiar with, such as equations that modeled convection currents in the earth’s atmosphere, and simplified them considerably so that they could be readily studied. In fact, Lorenz simplified the equations to such an extent that they really did not model any physical systems whatsoever, as his interest was now not in meteorology but in this strange new phenomenon that he was uncovering. He was using a system of three differential equations, given by

x’ = a (y - x)
y’ = x (b - z) - y
t’ = xy - c z

What is important to note about these equations is that they are differential equations. Differential equations are a special type of equations that utilize a branch of mathematics called calculus and are often very helpful in modeling many physical systems. It is easy to tell that the equations are differential equations because of the prime (‘) that is located over some of the terms in the equations. Solving differential equations is often a tricky process; many times it is impossible to go about solving differential equations like ordinary algebraic equations such as x + 3 = 5 because of the intricacies that the calculus introduces to the equations. Solving differential equations is quite empirical in fact, what one generally has to do is try and match the differential equation of consideration with an equation that has a known solution, and then just apply a similar solution to that problem. Physicists do this quite often. In fact, I believe that few physicists would dispute that there’s only a handful of real, honest-to-goodness equations that one knows how to solve easily. Instead, the idea is to take an equation that one doesn’t know how to solve, rearrange different terms and expressions, and get it to look like something that can be dealt with.
Differential equations have different solutions, but what they all have is a dependence on an initial condition. That is, since differential equations model physical systems, then to predict the future state of they system, it is absolutely necessary to know the current state of the system. If a car is traveling at 65 miles per hour, how far will it be from Boston in 3 hours? This question doesn’t make any sense unless you know where the car started (and in what direction it was moving). The number of initial conditions needed to solve the equation depends on the order of the differential equation (the equations above are first order since they are expressed as x’, y’, or t’, however if x’ were to be replaced by x’’ or x’’’ then the differential equation would grow to order two or three respectively). First order differential equations such as the ones that Lorenz used require only one initial condition. However, second order differential equations require two types of initial conditions, third order differential equations require three initial conditions, and so forth.
Lorenz substituted very nearly the same initial conditions for his system of equations but found that regardless of how slight the initial variation, the numbers that the equations were spitting out always diverged drastically after a relatively short period of time.
To illustrate what Lorenz’s equations are actually showing, it is helpful to think of a waterwheel-like structure. The waterwheel looks much like a Ferris wheel, except instead of having carriages for people, it has containers for water. At the bottom of each of the water containers, a small hole is punched to allow for water to drain out. The waterwheel is placed beneath a water source, such as a large faucet. When the faucet is turned on, water begins to accumulate in the top cup of the waterwheel. If the stream of water coming from the faucet is slow enough, then all of the water will ultimately drip right through the hole in the bottom of the cup, and the wheel will not move. However, if the stream of incoming water is too fast, the cup will not be able to dispose of all of the water through the hole and water will begin to accumulate in the cup. Once the cup has become heavy enough, gravity will force the cup downward and the waterwheel will begin to move. As the wheel rotates, other cups will fill with water and the wheel will continue to rotate. If the stream of water is not too fast, then the wheel will exhibit relatively uniform behavior. However, if the velocity of the water is increased greatly, the waterwheel will begin to exhibit chaotic, or unpredictable, behavior. The wheel might move left for a few seconds, and then to the right for five or six seconds, left again for maybe only one second and on and on (but with no discernible pattern). From studying the wheel it is clear that the system exhibits chaotic behavior. Once the incoming water reaches a certain velocity, there’s no way to predict how the waterwheel will behave. Furthermore, the wheel will never fall into a set pattern or rotation – it will always behave unpredictably.
The equations that Lorenz was using were actually meant to model convection in the earth atmosphere, but as it turns out the waterwheel is governed by a similar set of equations. Two points are worthwhile to note when observing either the convection equations or the waterwheel. First, the systems are both dissipative. If the energy source is removed, such as the water faucet in the waterwheel example, then the system becomes motionless. The equivalent energy source for the convection cycles in earth’s atmosphere is heat; if the heat is removed, then the convection stops. When a system obeys such dissipative behavior it is said to be an attractor since all trajectories “attract” to a stable solution under the right circumstances. However, what Lorenz had discovered was not just any attractor, but it had a further distinguishing feature as well. That is, there was an extreme and sensitive dependence on initial conditions, meaning that two nearby but not identical trajectories on the attractor must quickly diverge. These two features of the system seem to be geometrically contradictory – how can nearby trajectories both converge onto the attractor and also diverge? Stephen Kellert explains how the seemingly contradictory behavior may be reconciled by appealing to a “stretching” and “folding” analogy,

This apparent contradiction is reconciled by one of the main geometric features of strange attractors: a combination of stretching and folding. The action of a chaotic system will take nearby points and stretch them apart in a certain direction, thus creating the local divergence responsible for unpredictability. But the system also acts to “fold” together points that are at some distance, causing a convergence of trajectories in a different direction.

The shape that Lorenz discovered, today known as the Lorenz attractor, looks much like a butterfly or two neighboring spirals, with the two spirals rotating in opposite directions. In a folding and stretching pattern, trajectories are constantly switching back and forth between the two rather distinct geometrical configurations of the right and left spiral. Two nearby points may quickly move to occupy positions on completely opposite sides of the attractor, but the trajectories are always confined to a set, overlapping spiral shape – they do not have absolute freedom to roam wherever they like. This unique shape, stretching and folding yet maintaining a definite geometrical outline is known as a fractal. More formally, systems that possess sensitive dependence on initial conditions demand that, when the equations governing their behavior are plotted, all points on the surface of the plot must have neighboring points that diverge very steeply. The problem with two-dimensional objects is that they cannot accommodate for this behavior, as these exponentially diverging trajectories would at some point have to cross in a two-dimensional plot. The problem with trajectories crossing is that if they cross, then they must have an option of paths to follow at these points of intersection, thereby contradicting the deterministic nature of the equations. The alternative seems to be that the structure must not exist in two dimensions, but instead three. However, this poses problems as well. What Lorenz argued was that due to the nature of the asymptotic behavior of the initial conditions chosen for the attractor equations, if a three dimensional space were chosen, then the attractor would have to be without volume, which too cannot be possible. Grasping the details of the reasoning requires a good bit of imagination as well as geometrical aspiration, but the end result is that fractal structures, such as the Lorenz attractor, must exist in a special dimension which is greater than 2 and less than 3. Fractals are chiefly distinguished from other geometrical structures, such as circles and cubes, because they must exist in nonintegral dimensions.
The definitions governing chaos and fractal geometry are often quite mathematically finicky, but an understanding of the underlying principles involved can certainly be gained without delving into the specifics. A fractal is something new, a structure that cannot be represented in traditional two or three-dimensional geometry, but follows its own set of rules and patterns. Some of these patterns are an extreme and sensitive dependence on slight variations in initial configurations, exhibiting behavior that quickly diverges on even the slightest of variations. However, as was illustrated with the Lorenz attractor, there is a general geometrical pattern to the behavior, a “method to the madness” as Shakespeare would contend. The method, as will soon be shown, yields terrific and almost spooky results, as many notions that are often accepted as unexplainable, notions such as luck and chance, are not quite as mysterious as they appear to be. The fractal world goes well beyond waterwheels and convection currents. It transcends science and philosophy alike, encompassing the shadowy mysteries surrounding everything from free will and determinism to beauty and art.

The Fractal World

The IBM researcher Benoit Mandelbrot is credited with coining the name, “fractal geometry.” In his own words,

I coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means “to break”: to create irregular fragments. It is therefore sensible - and how appropriate for our needs! - that, in addition to “fragmented” (as fractional or refraction), fractus should also mean “irregular,” both meanings being preserved in fragment.

Fractals describe the rough details of the world. Many people probably have a preconceived notion of what a fractal is – most likely a brightly colored design of swirls and clusters that seem to be presently filling the pages of many popular calendars and postcards. However, fractal images are not restricted to simply these computer-generated images. In fact, the natural world is filled with fractals. Mountains, trees, snowflakes, coastlines, the scattering of leaves and even the human body itself are all prime examples of fractals and fractal patterns.
Put simply, fractals are the patterns and tracks left behind as a result of dynamical (changing) behavior. While chaos theory attempts to explain how dynamical systems change over time (and why they change over time), fractal geometry deals with the actual images that these dynamical systems produce. Anything that is complex, which includes the majority of natural processes, tends to leave behind fractal images. A pile of autumn leaves, the path of a winding seashore, or even a single snowflake are all great examples of fractal patterns generated by nature. Think about all of the factors that are involved in a pile of leaves looking exactly as it does. The process may begin perhaps with a squirrel, forgetting where he stashed his supply of acorns. The acorns, assuming that they’ve been left in an environment conducive to their growth, begin to grow and blossom into trees over time. During a tremendous thunderstorm, one of the trees is struck by lightning, which in turn significantly stunts the tree’s growth. Two neighboring trees are affected by an odd disease, that, among other things, discolors the leaves and bark of the tree. Years later, an October storm with powerfully strong winds forces the leaves off the group of trees and scatters them into piles of varying sizes in the surrounding vicinity. The next day, a man walking his dog comes across one of these piles of leaves, and notices the incredible variation in the stack. A good 1/3 of the leaves are considerably darker than the other leaves, while a few of the leaves are quite small – others quite large. The lightning storm, the disease, the storm, not to mention the absentminded squirrel (who is now long gone) all played significant roles in getting that pile of leaves to look just as it does. Furthermore, this doesn’t even begin to mention the factors that caused these factors. Perhaps the trees became infested with the disease that discolored the leaves as a result of a farmer disposing of his own infected tree in the middle of the field holding the squirrel’s forgotten acorns, which he in fact did not forget about, but found himself unable to retrieve the acorns after he was hit by a car while a hurried teenager was driving to school. The factors go on and on, and perhaps someone with a good imagination could tell a better story than I have told here, but what is clear is that the natural world is incredibly interwoven, and sensitive to even the most seemingly trivial things.
What makes fractals so interesting is not that they are just the images left behind by dynamical processes, but that they have haunting features that reveal an order to these seemingly unfathomably complicated processes. Self-similarity in fractal scaling is one of the most intriguing features of fractal geometry. Self-similarity asserts that the shape of the fractal image shows a similar shape when it is viewed at different magnifications. Consider the fractal image of the winding seashore, jutting in and out as it twists its way down the coast. The seashore is formed as a result of dynamical processes, formed by a variety of factors some of which being erosion and ocean turbulence. Viewed from the air, the seashore looks like nothing more than a winding coastline. But if one walks along the seashore, viewing it on a more personal and closer level, it is found that the shoreline is not just one big thing, but a conglomeration of little things, such as coves, bays, inlets, peninsulas and pools. Now suppose just one of these coves on the shoreline is examined even more closely. That is, zoom in from the entire seashore to just this one part of the seashore. What will be found in just this cove is a self-similarity to the entire coastline, it’s just that the new twists and turns evident in the cove exist on a much smaller scale. Zoom in yet again to a small tide pool within the cove, and a similar pattern emerges yet again. Why should systems that are so dynamically complex and chaotic possess this self-similarity?
This self-similarity is often explained in terms of holism, an interpretation of the geometry in terms of parts. In his book Fractals: The Patterns of Chaos, science writer, John Briggs explains the holistic phenomenon in terms of the weather,

Obviously, the weather at its different scales displays a self-similarity, a fractal structure. One way to explain this is to say that the weather is holistic, which means that between its “parts” (its fronts, patches of rain or snow, high-pressure and low-pressure zones) are other “parts of parts,” and “parts of parts of parts” (right down to the shimmers of heat rising from the sweating body of one of the hikers, or the chemical heat generated inside her straining muscle tissue). The result is that when all these “parts and “parts of parts” start feeding into each other, they can generate images (such as weather maps) whose patterns have scaling detail. These patterns illustrate the fact that the system’s whole movement takes place continuously at every scale.

Mathematically, this holism can be explained by appealing to the dimensional peculiarity of fractal images. In traditional two-dimensional or three-dimensional objects, such as lines and spheres, magnification of the image does not reveal any new information about the figure itself. Looking at a line either two feet away or 200 feet away doesn’t make it do anything fancy. However, as mentioned earlier, fractal images have dimensions somewhere between 2 and 3, and because of this difference, information about fractal images may be gained by looking at the image on different scales. The dimensions within the fractal may be thought to be tangled up or folded together, unable to reveal their true nature unless they are examined with the utmost scrutiny.
Fractals possess the property of showing self-similarity on different scales, and, as has been shown, fractals pop up all over nature. However, there are also a great deal of mathematically generated fractal images. These images are often quite colorful and flamboyant, which, due to their aesthetic appeal, are reproduced on calendars, postcards, computer screensavers and posters. So then, how are these fractal images created? A large majority of these computer images belong to a set of fractals called the Mandelbrot set, named after the father of fractal geometry, Benoit Mandelbrot himself. Mandelbrot was using his computer to plot iterative equations on the complex plane. In other words, Mandelbrot was using a simple equation, something that looked like:

Changing Number + Set Number = Result

and illustrating the behavior of the equation on his computer.

Initially, pick a number to put in the “Set Number” slot, and start with 0 in the “Changing Number” slot. Calculate the “Result” and then plug this value back into the “Changing Number” slot and do the calculation again. Continuing to do this type of calculation, where the result is taken and plugged back in to another parameter in the equation is called iterating the equation. Iteration is a central process in dealing with fractals and chaotic systems. However, the numbers that Mandelbrot uses are not real numbers such as 1, 2, 45 or 77.9, but imaginary numbers on the complex plane. The reason for this is that when real numbers are added, subtracted, multiplied or divided, despite what your high school Algebra teacher told you, nothing particularly exciting happens. However, imaginary numbers behave quite differently. Imaginary numbers are represented in terms of the letter i, which is equal to . The value of i is imaginary, because working with conventional mathematical common sense, there’s no way to ever get a value of . Whenever the square of a number is taken, the resulting number is always positive. Thus, it’s impossible to have a value of , or the square root of anything negative for that matter. Imaginary numbers possess a variety of interesting properties however, four of which are,

All powers of imaginary numbers may be simplified and calculated in terms of these identities. For instance, i5 may be thought of as i * i4 or i * 1 = i. Remembering the details of these identities or even being able to apply them is not necessarily important in grasping the concept behind the Mandelbrot set, what is important is appreciating that imaginary numbers behave somewhat differently than real numbers. It is this different behavior that allows for iterative equations to yield interesting and unconventional results. Namely, when different imaginary numbers are plugged in to the simple iterative equation just shown, and the results are plotted on the complex plane (a plot of imaginary numbers as opposed to real numbers), funny things happen. For certain “Set numbers,” the result will hover around a fixed value, while for others it might blow-up quickly to infinity. Computers analyze the simple equations by using huge amounts of different “Set numbers” (which are imaginary numbers) and looking at what happens to the “Result.” What the computer does is iterate the equation a few thousand times for different “Set numbers” and then observes whether the result is stable or unstable. If the result is stable, then perhaps the pixel on the computer screen corresponding to the used “Set number” is colored black, while if the result goes to infinity, the pixel is given a different color (and perhaps different colors depending on how quickly the result goes to infinity).
By giving each pixel on a computer screen a corresponding value in the iterative equation, a whole computer screen may be used to illustrate the behavior of the equation. The result is a beautiful fractal image belonging to the Mandelbrot set, which shows both the self-similarity inherent in fractal images and paints a very aesthetically pleasing picture. Non-mathematicians as well as mathematicians have come to appreciate the beauty of these fractal images that such simple equations produce. The images create a bond between aesthetics and mathematics; mathematics might just be beautiful after all!
Chaoticians, or those who study chaos theory, have used their imagination to construct quite a few different variations of fractal images, in addition to those of the Mandelbrot set. A discussion of chaos theory would not be complete without mentioning at least two of these other variations, as they are both often alluded to in discussing chaotic systems and their properties. In addition to using equations to iterate results, as Mandelbrot did, it is also possible to make “classical” fractals, or fractals made by adding or taking away elements of a geometrical structure in an iterative process. The German mathematician Helge von Koch used an iterative process to create his famous Koch island, or Koch curve. To construct this structure, take a line and to the middle of it add an equilateral triangle (with the base of the equilateral triangle being the original line). Now take this figure, and to the middle of each line (for now there are four) add another equilateral triangle. Continue this iterative process, and an interesting looking figure begins to emerge. The resulting, spiraling figure is a fractal. It shows self-similarity, however, the structure also illustrates the non-integral dimension that fractals occupy. Each time the iteration is conducted, more lines to the figure are introduced, and thus the total perimeter of the figure is lengthened. However, the area of the Koch curve is never greater than a semi-circle drawn around the curve itself. Thus, what results is a figure with an infinite perimeter that has a bounded area. By thinking in terms of two or three dimensions, this poses a clear problem. How can something have an infinite perimeter yet a finite area? On the other hand, by accepting the non-integral dimensions that represent the space in which fractal images dwell, the paradox is quickly eliminated. For, geometrical structures that have non-integral dimensions can possess infinite perimeters and finite areas.
A second classical fractal that is often mentioned in discussing chaos theory is the Cantor dust, first discovered by the Russian-born, German mathematician, Georg Cantor. Instead of adding an equilateral triangle to a line as Koch did, Cantor took a line and subtracted the middle third of it. From the remaining two lines, he subtracted the middle third of each of them as well, resulting in now four lines. Continuing the iteration process, one gets a “dust” of infinitesimally thin points, distinctively spaced. The Cantor dust has been used in a variety of applications, including modeling the way stars are distributed in clusters and the formation of bubbles trapped in sandstone. The Koch curve has been used to simulate realistic shoreline patterns as well as the way in which other dynamical systems, such as forest fires, spread and grow.
Fractals tend to pop-up all over both the natural and mathematical world. Weather patterns, coastlines, cave formations, waterfalls and even brainwaves all display fractal rendering. Mathematicians and computer scientists have made use of the computer to generate fractal images iteratively, with speeds unimaginable as recently as the 1950s. Even something as seemingly unscientific as music has been captured within the realm of fractal geometry. Massachusetts Institute of Technology graduate student, Diana S. Dabby has used the Lorenz attractor to generate creative and quite appealing variations of different musical themes. More than any single advancement however, fractal geometry has shown that the language of mathematics is not restricted to an exclusive group of quirky mathematicians and eccentric physicists. For, fractal geometry goes far beyond science, as it sheds insight on just about all dynamical processes, whether they be in nature, art, sociology or economics. The idea that there is a way to model seemingly unpredictable processes, and that this process holds both beautiful and definite structures, points to the conclusion that there might be a greater "force" governing the behavior of the universe.

Applications of Chaos Theory

Experts and novices alike agree that chaos theory is fun to think about. It has a great name and is able to generate pretty pictures – but is there anything that this theory can really contribute to science? Science traditionally seeks to provide an explanation for why certain events happen, and, in doing so, it provides a mechanism for man to control the outcomes and behavior of certain natural processes. Just think about something as simple as a baseball flying out of a ballpark. Physicists, using fundamental equations derived from Newton’s laws of motion, have been able to calculate with exact accuracy where a baseball will land if its initial velocity and trajectory are known. Plug the values into the equations, which have been confirmed to be accurate through experimentation, and it’s possible to know how far the ball will travel. Can chaos theory provide this kind of problem solving ability? That is, can the fundamental principles of chaos theory be applied to the natural world – and can these types of traditional results be obtained? If so, what experiments use this sort of problem solving technique? These are the sorts of questions that any good scientist would likely ask, and they are certainly worthwhile questions to be asking.
In short, the answer is no: Chaos theory, as it is understood today, may not be used in the traditional sense to provide scientific explanatory power. If anything, chaos theory asserts that traditional scientific methods cannot be used in modeling natural processes because the isolation and control techniques they employ by definition compromise the validity of the experiments themselves. The New Nork Institute of Technology computer graphics researcher, Peter Oppenheimer commented on the use of chaos theory in the scientific world,

Science likes to think its goal is to make objective representations of nature, but it seems to me that all such representations, visualizations, or models merely isolate a few select parameters, a few aspects of the object and say, what happens if we just look at these? Each different approach gives you a slightly different result…A lot of knowledge we’re gaining from computer pictures is very intuitive and must not be seen to be objective…I don’t think we’ve figured out just what kind of knowledge it is. One reaction to all this is dismay at the limits of our ability to figure things out, but maybe we have to take some sort of leap of faith. Wow, we can’t figure it all out, isn’t that wonderful? Let’s accept these pictures, but let’s accept them as something else than the kind of knowledge we’re used to. Maybe it becomes art rather than science. It’s still knowledge, but a different kind…Just seeing how sensitive things are to their initial conditions has changed my notion of our place in the universe and our ability to make things happen.

The analogy Oppenheimer uses, where he links chaos theory with “knowledge” but not necessarily with traditional scientific knowledge is right on track. Chaos theory is a new discipline altogether and it is misguided to try and bend it to fit traditional expectations. It is this alternative view of science and progress that has been emphasized in this chapter, but this way of looking at chaos theory is certainly not the end of the story. As any university faculty member will admit, universities tend to discourage new and creative enterprises if they are not fundable, unlikely to yield productive research and scientific publications. However, many universities do now have chaos theory and nonlinear dynamics departments, which generate scientific knowledge by using principles of chaos theory and nonlinear mathematics. Programs in Chaos and Complex Systems at Duke University, the University of Maryland, Princeton and the University of Michigan (all institutions that have well-respected Physics programs), among many others, have emerged over the past half-century. This means that there must be some traditional applications of the theory – it must be good for something.
Understanding the stress/strain patterns and strength of different materials is often a central topic of experimental as well as theoretical nonlinear dynamics research. In addition, the spatial structure of populations and how these populations evolve and grow over time is a prime example of a complex system studied using chaos theory. Dr. Joshua Socolar and Dr. William Wilson at the Duke University Nonlinear Dynamics and Complex Systems department work on such a problem; they argue,

To understand the population dynamics of biological systems it is sometimes necessary to take into account the spatial structure of the population. That is, different types of organisms subject to the same external environmental pressures may thrive or not, depending on how the individuals tend to be arranged in space. For example, a species that tends to form dense clusters may be more susceptible to extinction due to a disease that has only a minor effect on a species that tends to be more sparsely distributed.

In addition, major universities across the world are using chaos theory and feedback techniques to understand complex fluid flow and contact lines. Chaos theory has biological applications as well, as many researchers are currently using feedback and small perturbation methods in understanding and controlling cardiac dynamics in the heart.
However, as many economists will say, the applications of chaos theory reach far beyond the realm of science. As misguided as it is, the structure of society places a greater degree of praiseworthiness and acclaim to those with financial success than those who discover new knowledge. For this reason, many naïve money seekers looking for quick financial success, have turned their attention to applying chaos theory to predicting the fluctuations in the stock market. In fact, Mandelbrot, the inventor of fractal geometry, did not become interested in the subject for the sake of knowledge exclusively, but he was instead attempting to use his computer to locate trends in long-term cotton-price fluctuations. Thus, in a way, fractal geometry was discovered through economics. So then, can chaos theory be used to predict the stock market? The value of stocks are certainly influenced by a large number of factors, and they do exhibit long term trends as both historians and economists will likely agree. It would be nice, at least from a financial standpoint, to be able to look at the long-term trends of the stock market and apply some sort of self-similarity principle for day-to-day trading. On October 19, 1987, the Dow Jones average dropped some 508 points, about 23 % of it’s value – it would be nice if this sort of drastic change could be foreseen in some way. An ability to foresee such drastic changes would save stock traders a great deal of money.
In order to be successful at predicting short-term stock market variations, chaos theory would have to locate the variables that affect the variations in a given stock market. The fewer number of these variables, the easier it would be to make accurate predictions about the system. The problem is delineating between the variables that do contribute to the dynamics of the system (e.g. speeches by the President of the United States) and variables that do not contribute to the system (e.g. the temperature in Scotland – maybe). By separating variables into these two categories, it may be possible to understand which “state variables” actually contribute to the system directly, and which variables, like the temperature in Scotland, merely change the environment of the system but do not actually affect the system itself. Assuming that one was able to identify a few, central state variables for the system, then prediction using chaotic self-similarity principles would be possible. The problem is that nobody has really been able to identify a set of core variables that govern the stock market, and experts have been looking with close inspection for quite some time. What’s likely is that there are a whole bunch of variables that contribute to the dynamics of the system, thereby drastically increasing the dimensionality of the fractal system and thus making precise, short-term prediction near impossible. Despite what some ambitious economic chaoticians may argue, there is simply not enough knowledge at the present to understand the seemingly unpredictable day-to-day fluctuations in the stock market. This is not to say that long term predictions may not be made, as these have been shown to be relatively effective, it is simply the case that presently there is no way to accurately use chaos theory to predict short term fluctuations in the stock market.

Chaos Theory and Free Will

Like relativity and quantum mechanics, chaos theory sparks a good deal of debate on a number of philosophical issues. The most notable of these philosophical issues deals with determinism and the possibility of human free will. Since the dawn of introspective thought, most of mankind has held the belief that individual actions are not predetermined or predestined, but are direct results of the individual mind. The great English poet, John Milton, in his epic poem Paradise Lost, argues that the preservation of the freedom of the human will is the most important feature of man’s existence,


The mind is its own place, and in itself
Can make a heav’n of hell, a hell of heav’n.
What matter where, if I be still the same,
And what I should be, all but less than he
Whom thunder hath made greater? Here at least
We shall be free; th’ Almighty hath not built
Here for his envy, will not drive us hence:
Here we may reign secure, and in my choice
To reign is worth ambition though in hell:
Better to reign in hell, than serve in heav’n.


Milton is proposing that it is better to compromise everything, even the possibility for salvation and goodness, in order to preserve the vital and unparalleled power of the freedom of the human will. Many agree with Milton that life would hold little meaning if it was somehow proven that free will does not exist. Although, a great many scholars have argued that the existence of completely free actions is overly hopeful. American author, Kurt Vonnegut, though ironically, admits of the deterministic nature of the universe in his book, Slaughterhouse Five. Billy Pilgrim, the protagonist of the tale, is kidnapped by the Tralfamadorians, an alien race whose knowledge of the workings of the universe far surpasses the knowledge of the earthlings. The Tralfamadorians admit that their kidnapping of Pilgrim was not at all a matter of choice, but arose as a result of a mandate brought about by necessitation in the universe. Vonnegut suggests that the kidnapping was no more optional than a fly that was caught in amber to remain caught. In his own words, “Well, here we are, Mr. Pilgrim, trapped in the amber of this moment. There is no why."
The subject of free will transcends beyond just the realm of philosophy, as it is pondered by poets, authors, scientists and nearly all members of the human race. Most members of the human race assert that there is at least some freedom of the will for humans, but this assertion is based more on a matter of faith and hope than on any real factual conviction. It would arguably be quite difficult to go through life with the knowledge that the decisions you make and conclusions you reach are not results of your doings, but are instead mandated by some external power. For this reason, chaos theory has been employed to attempt to provide some form of factual justification for the existence of free will in an ever-increasing scientific and deterministic world.
Before delving into how chaos theory relates to free will in human actions, it is first necessary to look at the philosophic concept of determinism to get an understanding on the subject under discussion. Philosophers define determinism in different ways (generally depending on what they’re out to prove), but the idea behind determinism is that the things that occur in nature have causes, and if these causes are removed, then the given event does not occur. Determinism is the underlying principle that governs all scientific investigation. There’s no reason to attempt to understand the principles, or causes, that govern the universe if they are not linked with the actual workings of the universe. Early in the twentieth century, it was believed that if the current state of the universe were known exactly, that is, if the positions and momentums of all objects in the universe were known, then it would be possible to predict the state of the universe at a future time with exact accuracy.
Chaos theory asserts that future behavior of complex and dynamical systems are incredibly sensitive to tiny variations in initial conditions. Initially, chaos theory does not seem to threaten the deterministic nature of science. However, now combine chaos theory with quantum mechanics and the new branch of “quantum chaos” arises. When you “turn on” quantum m