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Why pretend the world is not chaotic ?
For the past several hundred years Newtonian's paradigm has dominated all branches of science. Newton worked all of his life to find the universal and simplistic laws that would explain all phenomena in the entire universe with extreme precision. The laws had to be valid at the macro level of planets and galaxies and at the micro level of atomic structures. The only complication of his final work is that Einstein found small inaccuracies in the predictability of Newton's laws once they were applied at the atomic micro level. Yet the two greatest scientists assured the world that everything measurable could be accurately predicted at any particular point in the future. That scientific method of looking at all natural phenomena is famous for its deterministic and linear characteristic. It simply means that a change in the original conditions of an experiment always result in proportionate outcomes. In that way science perceived to look at the world as a balanced and strictly ordered system.
Chaos theory on the other hand challenges this linear approach of observing and predicting natural phenomena. It does it as it observes the nonlinear phenomena in our universe and proposes chaotic organization of all matter that behaves in aperiodic way. The theory does not challenge Newton's law F=ma or Einstein's conclusion on relativity but it questions the validity of the linear scientific methods used to build up a universal paradigm. Chaos rejects the determinism or the boasting predictability methods because there are million things we cannot explain with the existing linear methods.
Chaos is around us every day. One morning you get up five minutes later. As a result of that you have to rush to work and since everything is hectic you forget your tie. By the time you get it you have missed the bus for work. Then you are late for your meeting at ten and your boss gets mad and fires you. So, from those insignificant five minutes you have a horrible life for then next twenty years. It happens and it is all chaos as small changes get amplified and result in more complex and unpredictable outcomes. The story sounds very realistic but not many people would look at it and say "this sounds very much like chaos theory."
Chaos presents the chaotic world as a mix of nonlinear dynamical systems full of unpredictable changes. The Newtonian and Einstein view of the world see the world in an equilibrium due to its ordered and deterministic behavior. The most interesting part of chaos theory is that the theory proves the existence of particular aperiodic patterns in chaotic or nonlinear systems proving the existence of universal governing laws.
Many scientists have not worked on chaos since chaos involves the solution of numerous nonlinear equations which is possible with the arrival of the modern computer. Scientists have also avoided considering chaos because if they encountered it, they did not pursue to investigate their experimental findings thoroughly. Instead they assumed there was just noise in the experiment or that there was some calculation error. Since chaos theory presents serious challenges to Newton's laws, it could not find legitimate existence.
The trouble with introducing the idea of nonlinearity in science is that scientists have been in love with linear systems for nearly 300 years where the sum is always equal to the sum of its parts. The alterations in the initial state of linear systems would necessarily result in proportional alterations in any final state. If a system was to be exactly equal to the sum of its parts, then each component is free to do anything on its own regardless of the other components. Of course, that model makes the mathematics relatively easy for complete analysis since outcomes always deviate proportionately to changes in the components.
People were preoccupied with breaking down structures and systems to their infinitesimal atomic levels but never explained their complex behavior. Many people need to agree that science has certainly discovered the atom and quarks for us but how relevant are the rules at that structural level to the upper levels of matter? Scientists have hoped that by breaking up the structure and by finding the governing rules, they can construct that magical universal law. But it has not happened yet due to the deterministic (linear) methods used.
The most significant breakthrough of the theory is that it certainly has some universal validity. The theory has succeeded in uniting all branches of social and natural sciences and offers vast opportunities for collaborative work. Such integration of all branches of science could be best exemplified with the foundation of the Santa Fe Institute where notable scientists such as Nobel Laureates Murray Gell-Man and Keneth Arrow try to give a stronger start to the nonlinear scientific methods. In the Santa Fe Institute social and natural scientists try to observe and explain the chaos in their science field work. Then they collaborate and test its relation in another field and results have been astounding. For example, the study of artificial bird life by super computers has its similarities to the explanation of chaotic but organized aperiodic organization of hydrogen and oxygen atoms into water. The way the birds organize themselves into flock on the screen of a computer is precisely the way the atoms form molecules and then water with the complex characteristics of three different equilibrium states - evaporated (steam), liquid, ice.
Chaos theory has ultimately started the rivalry between Newton's order and chaos, linearity and nonlinearity, prediction and unpredictability. Recent developments of the theory are due to computer simulations and mathematical proofs of dynamic systems behavior. The behavior of such dynamics systems changes constantly due to their nonlinear character. It is also characterized as the unstable relationship between the variables in the system. The best example drawn from our daily lives would be the weather or the stock market. Stock markets behave very much in a chaotic way if we observe the movement of stock and bond prices. There is no certain rule that determines the prices at a certain point in time.
In order to demonstrate the significance of chaos theory we need to go over the basics of the theory with some examples and explanations. That will contrast the essential differences between the present linear science paradigm established by Newton and the new revolutionary challenge from chaos theory.
First, let us take an orange juice production company. If the weather is bad in Florida, the stocks of this company will jump up since people will face higher prices for the orange juice. At the same time, if the FDA announces that oranges from Florida are diseased with Hepatitis A, every broker will sell the shares in that company. There are many variables which determine the unpredictability or nonlinear behavior of the market. Many would agree the weather, on the other hand, is purely unpredictable. We all know how small winds in the Mexican Gulf could turn out to be deadly tornadoes on the south coast of the States. Also, the problem of global warming might affect weather dynamics. Therefore, when all variables are included in the observation of a system, no specific outcome can be predicted.
The first significant characteristic of the theory is the initial sensitivity of conditions. It implies that any change in the initial relationship between variables in a system would result in the so-called positive effect. The change of such relationships can destroy the existing structure and behavior and establish new equilibrium in the form of a new complex system. This part of the theory strongly rejects the dominant Newtonian view of a predictable universe. The change of weather in Florida could end all speculative moves by all brokers as well as news from the FDA that it made an inaccurate report. Another case would be the total ban of orange consumption for some valid reason. Nobody expects a constant relationship to hold for long in a dynamic system such as the market. Weather is another system that is very sensitive to any change in temperature or other variables (humidity, chill factor, wind speed and direction). With a small change in the direction of a tornado, a whole city could be completely destroyed. These examples are just part of the picture for proving the importance of sensitivity of initial conditions.
It is important to note that such small changes can also lead to remodeling of the systems into another type of complex system with a whole new set of variable relationships. The small change of speed of some winds can intercept the tornado and form just heavy winds instead of a change in direction. A similar analogy could be described with human feelings and emotions, probably one of the most complicated variables to analyze as an outcome to some combination of linear sounds. In order to see this, imagine that you are listening to one of Mozart's symphonies, which you may agree is not as simplistic as just playing some sounds of a violin or piano. If you listen to the same symphony but each musical instrument performed separately, it will most certainly evoke different feelings in you. You would be observing chaos: simplistic systems (the separate musical sounds) mixed together producing a complex system. This phenomenon confirms that in complex or nonlinear systems the sum is not equal to its parts.
Another part of the theory is the explanation of nonlinear behavior. In chaotic or so-called nonlinear systems, simple equations or relationships appear to reconfigure their system around a point called the attractor. The attractor is the final state of aperiodic nonlinear behavior. All it means is that even though everything is chaotic the system achieves some kind of pattern but not in a determined periodic sequence. The winter of 1996 was full of record high snow precipitation but that certainly did not mean no snow in the winter of 1997. Weather pressure levels and components of the wind above Singapore from 1965 to 1985 show the two-year periodic behavior. Since the repetition of the cycles are not identical and vary with other variables in the weather system, we witness a perfect example of aperiodic nonlinear behavior.
The stock market has its own system pattern. The general profit trend for bonds and stocks has been substantially positive in the long run. In daily prices the market indexes vary enormously, but when we get to the weekly, monthly, and finally annual graphs, we can see the beauty of nonlinear patterns.
(source: Complexification, p.104)
The general trend has been positive for most of the time. Then if we look at the subparts of the big trends, we see the finger-looking patterns of ups and downs but still governed by the attractor in a positive direction.
In principal, chaos is deterministic. It may sound like a paradox, but as mentioned above, nonlinear behavior amplifies small changes into greater outcomes and those outcomes tend to follow some pattern approaching the attractor of the system. There is a certain difficulty of showing this assertion of the theorem about nonlinear systems like the weather, for example, because totally accurate measurements are nearly impossible. In other words, any small change of the initial conditions for a dynamic system result in a completely different structure than what should have been expected. Chaos just attempts to explain the formation of the final outcomes and their approximate characteristic contrary to the strong predictability of linear models that forecast exact numbers and values, which unsurprisingly do not turn out to be quite as first predicted.
There is a simple mathematical example that could be observed with a stable attractor, converging oscillation and chaotic behavior that proves the implications of linear models. The behavior of the Logistic Map ( x(t+1)=rx(1-x) ) could be observed with the use of a simple spreadsheet. We need an initial condition of some Xt and parameter with constant value K. We put Xt in cell A1 and K into B1 and then write the formula ($B$1*$A1)*(1-A1). The cell A2 represents the value Xt+1. Then we just copy cell A2 down. The graph bellow shows the sensitivity of simple systems just like the logistic equation. For both series r = 3.9, and for the first series, X=.965, and for the second one, X=.97. The small change of initial measurements for any simple-looking system could form a new form of chaos under absolutely new rules. The tiny change of our experiment proves the positive effect of simple linear systems. You can imagine what happens with changes in nonlinear systems if such a major reconfiguration is observed for linear ones.
(Note: X must be between 0 and 1, and r values are to be between 0 and 4)
The blue series has value X=0.965 and the red X=0.97. Both series have r = 3.9. As the two series oscillate they become more stable as we expect them to obtain certain pattern around the attractor. It is obvious that the behavior of the two series is chaotic. The change of 0.005 in X produced such huge pattern differences in the outcome of the series that is astounding. The graph depicts the prediction of Chaos Theory. The bigger the initial changes of a system, the bigger the final outcomes of the system. According to linear approaches the logistic equation must have responded proportionately to the initial changes of the system but as the graph shows that does not happen.
The deterministic and simple equation seen above has very much complex behavior following the prediction of the theory - periodic behavior, chaos, and attractor achieved (periodic oscillations). The time series turns out to unfold the basics of dynamic systems, quite widely used by the mathematician biologist Robert May at Princeton University to study different populations of species. He relied on the fact that the present population sometimes predicts the population size for next year. If the population is small, it will grow freely and then when food is exhausted by the larger population the following year, reproduction will be down again. That could be one explanation for the dynamics of species' populations.
The reason why we observe chaotic behavior is the fact that each value is not independently calculated. It is a time series equation, quite realistic and analogous to real life systems, and each value depends on the previous one and so forth. Therefore, the initial change determines the value we are looking for and it is not just a simple plug-in value somewhere in the middle of the cycle.
Many scientists have rejected the nonlinear behavior because when an experiment did not work out as predicted, they assumed there was noise or just computational error. Randomness is a chaotic-like behavior but the difference is that it has no attractor in any future state - no patterns or oscillation takes place - and since the attractor is missing, the system would not be deterministic. The computational errors are easy to spot but randomness or the noise factor in a experiment is harder to find.
Epidemiologists in New York tried to build a deterministic model of fixed oscillations for the growth of flu cases but their "recent research suggests the culprit may be chaos, a strange type of mathematical order that appears to be random but actually [following] very precise rules."(1) Randomness is just a sequence of events in which anything can happen next out of all possibilities but without following some specific pattern. Since the researchers found specific patterns over periods of time they could blame the variation of high and low flue seasons with randomness but only with chaos.
One of the biggest discoveries of nonlinearity and its chaos theory governing rules was accidentally discovered by Edward Lorenz. In the early 1960s, Lorenz, a meteorologist at MIT, tried to develop models of atmospheric convection using a large number of differential equations. The complicated equations varied with time and produced solutions resembling similar behavior of weather variation over long periods of time. Once he restarted his computer using numbers rounded to three significant digits rather than six in the form of a shortcut, he envisioned the beauty of initial sensitivity in a dynamic system. The solutions preceded the old ones for a certain time but soon they started to depart from the original path and appeared to carry no relation to one another. He has also started from a different point in his weather model. Then he thought of noise because he rounded the numbers to three digits. Unfortunately, after changing them to six digits again and starting from the midway point he picked before the shortcut, Lorenz still got the same chaotic behavior of the weather model.
The Lorenz attractor in the three-dimensional phase space spanned by the Lorenz model variables x, y, z. (the equations are dx/dt=10(y-x); dy/dt=-xz+28x-y; dz/dt=xy-(8/3)z )
(source: Exploring Chaos, p.73)
Lorenz looked for even simpler ways of describing the complex behavior and that is how he came to the three equations above. The relationship between them is not proportional but nonlinear. The equations are the simplest examples of chaos arising from a system of ordinary differential equations. The Lorenz attractor resembled the wings of a butterfly and therefore acquired the name "the butterfly effect." The circular wings of the three dimensional wings showed how weather could almost repeat itself but never exactly overlap identically and that small changes cause large effects of magnitude. The picture of the wings behaved within specific boundaries because chaotic models might be crazy looking but they have limits due to their organized nature. The model is deterministic, it does not just blow out of proportion with no limits for us to observe.
The effect also attributes to the prediction or rather expectation that flapping butterflies in Brazil would cause a tornado in Arizona. Unfortunately, proving such assertion by mathematical means is impossible since we could not possibly know the exact location of the butterflies, which ones are flapping, and on top of everything, their flapping directions. Measuring the initial conditions is a crucial part of any dynamical system since any inaccuracy would be blown out of proportion and would give an unrealistic result to the actual future state of the system. The graph bellow just shows the positive feedback effect or rather the departure of Lorenz's first experiment with the second one. The second experiment was merely different but it shows how it decreases its similarity to the original data and pattern.
(source: Chaos: The Making of a New Science, p.17)
It is hard to believe that the weather would follow some well defined path in its evolution, influenced by billions of particles. But with the help of 'cluster-analysis' and accumulated historical data, it appears that there are "about 10 different regimes [that] characterize most of the large-scale variability of the atmosphere in the northern hemisphere."(2) Therefore, we prove that chaotic behavior follows strict patterns.
After the discovery of the butterfly effect several phenomena could be modeled by the differential equations. One of the simplest examples that could be approximately described by the Lorenz equations is the thermosiphon. It is a cylindrical tube in a vertical position with fluids inside. Heating one end and cooling the other produces the necessary convection. The warmer fluids rise and cooler ones fall. The convection could begin in either direction. The circulation proceeds in a loop and then suddenly reverses direction, depicting the dynamics of a nonlinear system.
A good example would be the development of regression analysis in economics and mathematics. This method is widely used by many social scientists to predict how the independent variables would predicted outcome. Since a lot of scientific works use this type of analysis, we need to understand its actual usefulness. Regression relates one variable to another in a linear relationship but it also boasts of multiple regression analysis capabilities (relate several variables to one). If an economist is trying to relate inflation, interest rates, and unemployment to GDP growth, he must make the transition from a nonlinear relationships among billions of actors in the world economy and present a single digit result, his "accurate" forecast. He is basically trying to reduce something very complex and irreducible to a single digit. The suggested forecast would be like we can put our feelings on a scale from 1 to 10, it just cannot happen.
The analysis itself presents a poor explanation as far as proving the direction of causation because as Professor Jacobsen from the Economics Department at Wesleyan University says "regressions never prove causality direction, they simply show relationships." If we don't know which variable acts first (referring to the causation direction of the equation) then what good does it do to our economy and to the millions of people that believe in the "accurate" economists' predictions in the future? Practically, this analysis, the heart of economics, fails to account for the nonlinear systems and certainly the economy is one of them. Therefore, predictions about the economy even in the near future might be inaccurate due to the lack of nonlinear methods of analysis.
Calculus is another linear biased approach that scientists use. Calculus takes infinitely small lines to describe the borders of an object and then adds up those parts to get the sum or total surface, length or volume. As the limit of lines approaches infinity, the measurement of a particular object becomes infinitely accurate but at the same time no professor would tell his or her students that that is all calculus can do - just measure. This progress towards simplicity has turned out to be productive but in the wrong direction. We all hear about the atoms and their components but rarely know why and how they behave in a particular way.
If we are trying to get a better understanding of the world, why do we need to tear everything down to relativity and quantum theories? "The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe"(3) which many scientists still fail to understand. If physicists continue to emphasize the importance of elementary particles and their fundamental laws, they would make the evidence more irrelevant to all sciences. Knowing and seeing quarks and electrons does not give us an explanation for the universal law that all scientists have sought for hundreds of years and besides, how does it help us to understand why those micro particles interact to form more and more complex structures?
The main feature of linear problems is that their solutions can be presented in many ways by combining linear equations and obtaining the same final result. In such linear mathematics, the individual solutions can be added together and still get to the total solution of the problems. A nonlinear equation cannot be broken up and its bits reformulated to obtain a solution. It is because even simple looking equations like xy=z does not allow decomposition. Therefore, differential nonlinear equations are intractable for analysts and seen as totality. In other words, anything that could not be presented in a simpler way but only by itself we call irreducible or nonlinear. The unusual stability of linear systems due to their proportional response to outside shocks is the reason why we have highly ordered phenomena like buildings and houses. On the other hand, if we were to increase the humidity of a little part in the atmosphere, there would certainly be an effect in the short or long-term outcome of the nonlinear system. The beauty of dynamics is its ability to be an organized order no matter how bizarre it sounds.
Statistics is another field of mathematics that we could skillfully attack for a biased understanding of our chaotic world. If we take a square board and an infinitely thin dart we could illustrate the importance of small effects on systems. Since the dart is ultimately of zero thickness, our choices on the board are unlimited. If we aim relatively close to the center, our chances are low; to aim and hit much closer to the center decreases the chances even further. Then if we try to hit the very center of the board, our odds are practically zero, but clearly a probability of zero could not be an impossibility. You have the same probability hitting any other point on the board so your chances are not impossible. So, all statistics does is observe the result rather than predict it. If someone was to hit the arrow while flying towards the board, great theorems in statistics fail to give explanation.
The unfortunate part about linear analysis, heavily exploited in today's universities and high schools, is that statistics pretend to predict outcomes but in reality defeats its purpose. It tells us the chance of obtaining two tails in a row if we use an example of throwing up a dime, but in fact does not predict it to happen or explain any kind of future pattern. Biology and chemistry could not be better examples to give us a portrait of limited scientific methods. It is great that we know all about DNA, atoms, tissues, reactions among proteins and other chemical compounds but their behavior is still impossible for a prediction or detailed explanation. Chaos does not have a superior advantage in explaining why the world is round or why our kidneys consist of tissues, atoms, electrons, etc., but it provides ideas and strong observations of nonlinear systems that has been missing for hundreds of years. People though of chaos as something random just as you would think of our very first example of getting up late and ending up fired as a result of that. The randomness bias or ignorant argument could not be true. There are laws that produce aperiodic behavior with specific patterns.
Coming back to the description of simple and complex systems, we need to look into the application of simplistic structures forming complex patterns through the use of fractal geometry. Chaotic behavior does not retrace previous points during its temporal evolution, and we could see that by using simple two dimensional figures like triangles and squares to measure the coast of England. As we mentioned before, small differences cause amplified results. If we measure the coastline of Britain with 10 meter lines, we would obtain result x. Then if we use 1 meter lines, we get a more accurate estimate but much larger result or outcome of the system. Any further measurements, using shorter units, will change not only the outcome significantly, but also the shape of the coastline which differs every time we measure it.
There are three really major limitations of chaos theory. The first problem with chaos theory is the fact that if we can observe patterns based on some initial sensitive conditions, we cannot make the appropriate measurements accurately. Chaotic outcomes will be drastically different than our predictions since the technology we have today does not allow us to make such sophisticated calculations of tenth or beyond decimal places (for example, air temperatures).
The second complication with the theory is the proof of its significance. Since chaos theory does not define the magic universal law governing our lives, many people wonder why we do need such theory. It is a revolutionary scientific method and it will need some time to be accommodated properly by all scientists. The argument is that we cannot continue to base our lives on the micro level and not lift our heads to look beyond the simplicity of organisms and nature in general. Chaos has been neglected at higher educational institutions and maybe that is the reason our progress of getting a more complex understanding of the world is missing. Our world is clearly nonlinear and in order to study it we cannot analyze it with linear tools and study only the parts of structures and systems. We need to start explaining the behavior of all those micro and macro parts.
The last problem with the theory is that it does not allow us to observe a natural phenomena through the heavy use of models. All scientists take a small part of a complex system and try to explain with a small part all relationships and behaviors within it. Unfortunately, chaos as you recall says that any tiny change of the original conditions would produce variation of outcomes. If you try to isolate a small part of the atmosphere and study that particular part you could not do it. Some things sciences like economics have already struggles with the establishment of models. You will always read or hear that economists first assume the conditions and then predict which means that if any of their assumptions fail the predictability of their models is poor.
When we try to isolate subparts of bigger structures or systems we lose our capability to accurate observation and prediction. The butterfly effect shows exactly what happens when we ignore the small factors that could contribute to something major like a tornado. Therefore, it will take quite a bit of time for technology to allow us to perfectly isolate parts of the dynamic systems and explain their contribution to the more complex and higher in hierarchy systems or structures.
Chaos theory has been used quite widely in social and natural sciences. The ability of the theory to account for many nonlinear phenomena in all branches of science shows the strength and capability of the theory. The nonlinearity of our world could not be narrowed down to atomic structures and their tiny forces as many scientists predict with their limited tools of linearity. Chaos has provided the world with the foundation of a new emerging science paradigm where people from all branches of science can relate and collaborate to investigate natural phenomena just like the Santa Fe Institute. Also, our books should not repeat endlessly the lessons of atomic structures, DNA, Cartesian coordinates, and calculus. It is great to know how to break up structures and systems but we need to know about their behavior of forming the complexity around us.
Bibliography
Book report (on Complexity by Mitchell Waldrop)
Theory Related links to Chaos
Electronic Journal on Complex Systems
Chaos general homepage
Chaos Theory in Psychology and the Life Sciences
Chaos and financial markets
Chaos general page
