Articles on Chaos Theory

The Man From CHAOS By William Green first appeared: Dick Morley, inventor of the floppy disk, wants to reinvent your factory. His message: to gain control, you've got to lose control. Factories are complex organisms," preaches Richard Morley, unofficial leader of a movement to overthrow the old notions of how to run a manufacturing plant. Morley and his band of fellow visionaries want to replace the control-freak mind-set of the Industrial Age with an approach derived from chaos theory. "You don't have a prayer in hell of ever understanding factories. You really don't have control. By striving to get control, you only make it worse." Dick Morley is an outspoken thinker who readily compares his extraordinary views on technology and business to those of a Martian. (“Let us assume we are a Martian slowly descending upon the city of Boston," he wrote in an article that uses chaos theory to explain how taxi systems work.) At 63, the acclaimed inventor rides a Harley, keeps 11 pairs of skis by the front door of the New Hampshire barn where he lives and works, has sponsored more than two dozen foster children, and has started roughly 50 companies. Morley was one of the technicians who created the floppy disk and is also the father of the programmable logic controller - an invention that transformed the factory, spawned a $4 billion industry, and pushed manufacturing toward mass customization. Since the mid-1980s, Morley has devoted much of his time to designing a computer system that draws on the principles of chaos theory to run factories. By studying everything from the weather to the swirls of smoke emitted by cigarettes, chaos theorists have concluded that there are complex patterns underlying what once seemed to be erratic behavior. A school of thought most closely associated with the fashionable Santa Fe Institute, chaos theory is now influencing everything from Wall Street trading strategies to software design. And now manufacturing. According to Morley, factories are havens of erratic behavior, places where, he says, "Everything is going wrong all the time." Morley's solution: abandon the illusion that you can predict these technological headaches, or that you can avert them with forward planning. Instead he designs his computer systems with what he describes as "the ability to solve problems, to deal with stuff that you cannot explain." In 1992, Ernest Vahala began testing Morley's chaos computer system in a General Motors assembly plant in Fort Wayne, Indiana. At the time Vahala was director of manufacturing engineering for GM's worldwide truck group, so he had the clout to try an experiment that even he regarded as a trifle bizarre. For the next three years, Morley's computer system ran the paint shop in a GM factory in which robots paint 60 trucks an hour. The sophisticated software needed to paint trucks must enable the robots at each paint booth to perform a series of complex tasks. Among other things, the robots treat the surface of the truck with phosphate, apply base coats and clear coats, and set the paint in an oven at a temperature of up to 300 degrees Fahrenheit. Vahala says Morley's system handled these tasks with ease. More important, the chaos-based approach "reduced the software we required by nearly 100 times," says Vahala - no small matter given the enormous cost of software engineering. For example, in the past, a single paint booth might have been required to paint a red truck, then a blue truck, and then a black truck. As a result, paints would have to be changed and machinery cleaned before each truck could be painted. Morley's system completely changed the game: it enabled the paint booths themselves to "bid" for the right to paint certain trucks. If a particular booth had been painting black trucks all day, it would bid to paint any subsequent trucks that were to be painted black. Says Morley, "The booth is empowered to decide what it does." "It ran beautifully," says Vahala. "It saved $1 million a year in paint alone. You don't have to be a rocket scientist to say, `Holy damn!' I became a complete believer." Despite this triumph, General Motors recently notified Morley that it has dismantled his computer system. Vahala explains that the company is replacing its hydraulic paint robots with electric ones, an overhaul that requires the installation of new software. He says GM is likely to re-explore the chaos approach later but admits, "There continues to be resistance to chaos because people don't understand it. It defies good logic." Morley is sufficiently self-confident not to take this setback personally. I'm disappointed," he says, "for them. It's like seeing your child decide to be a garbage man instead of a doctor." Morley's computer system is gaining greater acceptance in Japan where Yaskawa Electric Corp., a licensee, has sold about 90 chaos computers (average price: $20,000). One current application is to simulate the operation of the Central Japan Railway's famous bullet trains. Other customers are using chaos computers to run steel plant and to schedule the loading and unloading of boats in dock operations. Morley believes the Japanese will continue to embrace the chaos approach, but he's given up on selling his computers in the United States. "People here don't want this stuff," he says. "Americans like reorganization. They don't like technology. When Japan cleans our clock," he says, "then we'll rush to catch up. Again." William Green wgreen1995@aol.com, who has written for the "New Yorker" and the London Independent", lives his life according to chaos theory in Brookline, Massachusetts. CIM Perspectives - Chaos Theory Applied To Manufacturing By Golden E. Herrin, Product Manager Vickers, Inc., Lebanon, OH The Chaos theory and manufacturing at first glance appear to be at best very strange partners. The very word "chaos" infers negative things that no one wants to associate with an efficiently run shop. For example, the dictionary definition of chaos is great confusion, complete disorder, a confused mass or mixture, the infinite space or formless matter before the universe existed. And because of this negative connotation, there has been a gradual shift to the term Complexity Science to describe Chaos projects. But, ever since the Chaos theory was developed in the 1980s, scientists, researchers, and engineers are reaching a growing awareness that it is offering them new tools for seeing order and pattern where only unpredictable, random, or erratic behavior had been previously observed. Industry is all too familiar with the complexity of manufacturing systems. In an effort to improve production and quality, meet the market challenges of global competition, build to order, and build it quicker than the competition, manufacturing systems are being pushed to new limits. Older existing sub systems that have been integrated into newer systems have created enormous and very complex software configurations which are often not very robust. This has caused many to take a look at solving the problem of complexity using the Chaos theory. The Chaos theory refers to processes that can be defined by nonlinear equations. Also, the theory of Chaos indicates that success can never be achieved by a top-down, command and control approach. Such systems become so intertwined that they are nearly impossible to change. Chaos instead advocates simpler rules applied to components of a process, which in turn come together to create a robust, agile system. These components or "autonomous agents" have behaviors that emerge as the system runs rather than being defined in advance and then rapidly executed as in the top-down system. When the shop floor is viewed as networks of individual agents, each responding autonomously to local conditions, the factory is viewed not as a single machine but as a community of loosely coupled processors. Each of these processors or entities has its own agenda to allow agents to respond locally to changing conditions. Changing to this structure requires management to change from the traditional role of specifying the top level schedule activities to specifying the local decision rules that agents use to schedule their own operation. Chaos was first applied to manufacturing in 1991 by R. Morley, Inc. (RMI) to seven truck body painting booths at General Motors. Each booth became an autonomous control agent that individually selected its next task by bid from a conveyor line of mixed body styles and color specifications. The results minimized paint changes in several booths saving time and materials. The simple system replaced a simulation that ran more than 500 pages of computer code by reducing it to just a few pages of computer code. Researchers have used similar concepts of Chaos as tools to optimize simple computer code and engineering designs, and to control complex industrial systems so it is only natural that they should be extended to the shop floor. There are at least four organizations with programs working to bring about the integration of Chaos and Complexity Science into industry:    The Santa Fe Institute, a not-for-profit research organization that is considered by many to be the nation's premier resource for the study of Complex systems. The National Center for Manufacturing Sciences, a consortium of American manufacturers that facilitates precompetitive R&D to help its members make better, more competitive use of advanced technologies. The Industrial Technology Institute which is a not-for-profit contract R&D organization dedicated to improving the competitiveness of North American manufacturing through the appropriate use of advanced technologies. R. Morley Inc. (RMI), which organizes and conducts conferences on Chaos applied to manufacturing.  Early expectations of Chaos and Complexity Science are high. This is a field of research that merits watching closely in the next few years to see if it can provide the necessary tools needed for shops to meet the manufacturing challenges of the 21st century. In application, chaos becomes much less theory and more reality Chaos Theory Abstract Chaos Theory, though among the youngest of the sciences, has emerged from its obscure roots in the sixties and seventies to become one of the most fascinating fields in existence. At the forefront of much research on physical systems and seemingly dynamic behavior, chaos science promises to be influential in the shaping of scientific inquiry in the future. The following paper attempts to give a broad view of the evolving field then focuses on some of the areas it is being applied to today including those related to business and logistics management. Introduction/General Concept In a large office half occupied by a primitive Royal McBee computer system on the Massachusetts Institute of Technology campus, meteorologist Edward Lorenz programmed twelve simple equations into the computer that would, he hoped, model and predict weather patterns. Though it didn't accurately predict the weather in any particular location, the computer program was able to predict what the weather might be. A few months after the initial experiment Lorenz wanted to reproduce a particular weather pattern that the program had previously modeled. In order to save time, Lorenz decided to enter a number from the middle of the sequence rather than starting from the beginning. When the machine began its process, Edward left the room to take a quick break. Upon returning he found the outcome of the second sequence to be far different from that of the first. After examining the printouts from both models, Lorenz came to the fundamental conclusion that is the basis of Chaos Theory. When entering the starting point for the second sequence he rounded the number to the nearest thousand thinking such a small difference would have no effect. Now he realized the importance of this error. In non-linear, dynamic events such as weather patterns, small changes in initial circumstances result in amplified differences in the eventual outcome. "Lorenz had entered the shorter, rounded-off numbers, assuming that the difference - one part in a thousand - was inconsequential. Yet in Lorenz's particular system of equations, small errors proved catastrophic" (Gleick, 1987, p. 16). This effect, now known as the butterfly effect, has become the basic principle describing Chaos Theory. As the theory suggests, the flapping of a butterfly's wings today produces a small change in the state of the atmosphere. Over a period of time this small atmospheric change can amplify itself and theoretically cause a devastating tornado halfway across the world or by the same argument, prevent one. This phenomenon, common to Chaos Theory, is also known as sensitive dependence on initial conditions. Just a small change in the initial conditions can drastically change the long-term behavior of a system. Though Lorenz is commonly credited with the modern introduction of Chaos Theory to everyday occurrences rather than just scientific theorization, several other people were investigating the possibility of latent trends in chaotic behavior. Benoit Mendelbrot was working in the research division at IBM (International Business Machines) investigating historical fluctuations in different data sets for some years prior to Lorenz's discovery (www.ems.psu.edu, 1999). Also, a less known French astronomer, Michel Henon, discovered chaos in his studies of planetary motion and the movement of stars around the galactic center about the same time that Lorenz stumbled upon the now famous butterfly effect. "But the first deliberate, coordinated attempt to understand how global behavior might differ from local behavior came from mathematicians. Among them was Stephen Smale of the University of California at Berkeley, already famous for unraveling the most esoteric problems of many-dimensional topology" (Gleick, 1987, p. 45). Smale was exploring non-linear oscillators - pendulums and springs, and came to the conclusion that most dynamic systems tend to settle into behavior that is not truly dynamic at all but rather recurring. So despite the majority of the credit being given to Edward Lorenz, Chaos Theory was actually a concept of interest to many scientists and mathematicians at the time. An idea that must be discussed when exploring Chaos Theory is that of fractals. Fractals, which are geometric shapes that are similar to themselves at different scales, are said to illustrate the graphic representation of chaotic systems. Fractals exhibit self-similar behavior no matter how large they are magnified or how small they are condensed. Therefore, chaotic systems, according to Chaos Theory, may seem random and dynamic if looked at on small scales but if magnified, should show relatively constant patterns similar to geometric fractals. Business Applications In recent years Chaos Theory has gained acceptance and popularity in several areas related to business. Probably the most widespread and profitable area it has been applied is in the modeling and predicting of the world's stock markets. Through massive compilations of historical total stock market data and data related to individual stocks, it is possible to find continuous and predictable recurring patterns. It has been discovered that the trends found in months, years and even decades worth of data, occur to a smaller extent in hours, days and weeks in the fractal manner. That is to say that in market price fluctuations, as you look at monthly, weekly, daily and intra daily movement graphs, the structure tends to be similar. This has allowed those people who can correctly apply the trends to current markets to make very large amounts of money relatively quickly. One such case is found in a small Santa Fe, New Mexico based company, Prediction Co. Doyne Farmer and Norman Packard, the same two men responsible for winning millions of dollars using Chaos Theory to beat Las Vegas casinos at roulette, founded this company and have proved it is a leader in this new field of investment strategy. In fact the company has gained so much recognition that it recently became a trading advisor for Swiss Bank Corp. (now UBS), the world's third largest bank. Prediction Co. has perfected the theory to such an extent that they are able to make millions of dollars in a single day in times considered to be a market collapse. For example, "on the infamous October 27, 1997, Dow crash, the predictors' portfolio made a profit - of the multimillion-dollar kind" (www.abqjournal.com, 1999). They did so by determining which individual stocks were at the point in their trend where a sharp rise was inevitable and putting a large portion of their holdings in those after retrieving the investments from areas that showed the possibility of a major decline. Prediction Co. is not alone in achieving this accuracy. A man by the name of John Mathews has been able to predict the JSE (Johannesburg Stock Exchange) with an amazing 95% accuracy (http://tqd.advanced.org, 1998). Results like this are sure to attract attention and broaden the base of investors applying Chaos and Complexity Theory to the business world. If an event as seemingly random as stock market performance, which is based on the independent purchase and sale of securities by millions of individuals, can be predicted with this degree of accuracy, why can't any event? It is only a matter of how quickly we can collect and compile the historical data concerning these occurrences and find the latent patterns. Chaos theory is also gaining popularity in fields related to logistics. Though these techniques and applications are relatively new, they seem to be promising to the industry especially in production management and the forecasting of demand. In March of this year AMR Research, a market research company, held the second annual Chaos Conference allowing participants to apply the principles of Chaos Theory to practical problems of manufacturing and process control. "Using recent breakthroughs in the understanding of non-linear natural systems such as ecologies and economies, the conference introduced concepts that manufacturing managers can use to create organizations that are robust, flexible, simple to model, and relatively inexpensive to operate" (www.amrresearch.com, 1999). The importance of Chaos Theory to manufacturing managers is that it makes it possible to create self-governing systems that optimize performance without specifying the activities of any individual element within the system. The role of the manager then changes from that of a supervisor to that of a facilitator, creating an environment in which the desired behavior of the system (the manufacturing workers and machinery) emerges. This results in a system that is immune to the failure of any single event since the system is self-configuring and self-adjusting to problems it encounters. This is an alternative to current forecasting and planning techniques, which are too simple for today's dynamic manufacturing environment. A system guided by the concept of "chaos" with the ability to adjust to unseen events, makes it possible to adapt very quickly to changes, rather than trying to foresee them. These systems are controlled by so called "emergent behaviors" which act like the decision rules commonly found in computer systems. These behaviors optimize the resources of the manufacturing plant to handle all necessary production requirements in order to meet a specified demand. An example of this is AGV's, automated guided vehicles, which are increasingly used by production plants to haul products around within the facility. When an order is ready to be moved to a different location, the closest AGV is directed to the location using decision rules performed within the vehicles computer, thereby optimizing the distance any resource must travel. Using these emerging technologies a manufacturing control system can adjust materials and manufacturing resources to compensate for dynamic events that occur on the manufacturing floor and by doing so allow for the prediction of the overall system (so that it will meet demand) though what happens at a specific job or operation may vary greatly (www.amrresearch.com, 1999). Perhaps the most promising area for Chaos Theory in the future is in predicting or forecasting non-deterministic events. If the theory that all non-linear systems exhibit underlying patterns is conclusively confirmed, it is only a matter of time until those patterns are found in data concerning every event imaginable. In the field of logistics management this will prove exceedingly useful in allowing companies to accurately forecast product demand and allocate their resources to meet that level. It is only a matter of time, one could argue, until we amass enough data to determine where the trends occur. Once forecasting techniques become a science rather than an art, the modern day troubles regarding the production of goods will be eliminated and the ability to meet demand will no longer be a major focus of logistics since demand will not be considered a dynamic variable. No current technique shows any possibility of competing with the abilities that Chaos Theory may possess in the area of predicting future events. An interesting use of Chaos Theory in the real world deals with the design of a washing machine manufactured by Goldstar Co. created back in 1993. Using the concepts evolving from the exploration of chaotic events, this machine supposedly produces cleaner and less tangled clothes. The key to this machine is in the fact that there are identifiable and predictable movements in non-linear systems. By using a small pulsator that stirs the water with a chaotic motion randomly rising and dropping as the main pulsator rotates, the clothes inside are actually cleaned chaotically rather than with a linear motion, which would not produce the volume of air and water flushing through the clothes that is created by Goldstar's washer. When introduced to the washing machine market in 1993, this was the first consumer product that exploited the random yet predictable motions of Chaos Theory. Though it seems a strong marketing pitch if nothing else, it was eventually outperformed by a newly introduced "bubble machine" from Daewoo that combined the concepts of Chaos Theory with those of Fuzzy Logic to create an even more scientific sounding method for cleaning clothes. Though it is not clear whether these theories really add to the washing capabilities of either machine, it is clear that Chaos Theory has not gone unnoticed by today's consumer market. Conclusion Chaos Theory is a quickly developing concept and as more scientists begin to delve into its vast possibilities, it will continue its rapid ascent in the scientific world. But it is not only an area of interest and importance to those trained in mathematics and physical sciences. Compared to the discoveries of the likes of Einstein and Newton, Chaos Theory should be ignored by no one and especially not by companies interested in being competitive in the future. At the least it should be recognized as an up-and-coming theory and those who begin to take advantage of its immense capabilities will have a growing advantage over those who discard it as just another scientific hypothesis. Works Cited 1. "Chaos: Manufacturing and Science Explore Complex System Control." http://www.amrresearch.com 12 April, 1994. 2. "Chaos Theory." http://www.ems.psu.edu/info/explore/chaos 14, September, 1999. 3. "Complexity Theory Attracts Practical Applications." http://www.amrresearch.com 16, April, 1999. 4. Gleick, James. Chaos: Making a New Science. New York: Penguin Books, 1987. 5. "History of Chaos." http://www,tqd.advanced.org 12, November, 1998. 6. Pentz, Michelle. "Chaos Theory Takes on the Stock Market." ABQ Journal 6 December, 1999. Part 1: History of Chaos Theory Be sure to check out Author's Homepage links in Footnotes Introduction Real life never goes as smoothly as this music. Click icon. The word "chaos" might have first appeared in Hesiod's Theogeny (@700 B.C.E.) in Part I: "At the beginning there was chaos, nothing but void, formless matter, infinite space." Later in Milton's Paradise Lost: "In the beginning, how the heav'ns and earth rose out of chaos". Both Shakespeare (Othello) and Henry Miller (Black Spring) refer to chaos. In these instances one inferred that chaos was an undesirable disordered quality. Historically our vernacular incorporated this idea of disorder into chaos; dictionaries defined chaos as turmoil, turbulence, primordial abyss, and biblical references to Tohu and Bohu had the same referential character of undesired randomness. Scientifically, Chaos implied the existence of the undesirable randomness, but the self-organization concept at the edge of chaos denoted the order we get out of chaos. The American essayist and historian Henry Adams (1858-1918) expressed the scientific meaning of "chaos" succinctly: "Chaos often breeds life, when order breeds habit".(1) Li and Yorke(2) coined the word chaos to refer to the mathematical problem in chaos theory that described a time evolution with sensitive dependence on initial conditions. Robert May, a mathematician-biologist whose research was well read, used the word and the theory from Li and Yorke's paper, thus making them and the word famous. Chaos theory came in the back door, so to speak, of the researcher's world. It was not a law like thermodynamics or quantum physics, but it did enable the researcher to analyze events or areas with many problematic intricacies. Cambel reported that it had even been proposed that we call chaos "divinamics"(3) after the ancient Roman divinatio described by Cicero. (Because of the ubiquity of chaos found in nature, and because my research is in the area of religion, I would certainly go along with that name.) Ilya Prigogine, the 1977 Nobel Prize winner in chemistry, pioneered the work in entropy of open systems; this was the inflow and outflow of matter, energy, or information between the system and its environment. Prigogine used dissipative systems to show that more complex structures can evolve from simpler ones, or order coming out of chaos. What is Chaos/Complexity theory? Daniel Stein, in the Preface to the first volume of lectures given at the 1988 Complex Systems Summer School for the Sante Fe Institute in New Mexico, compares Chaos/Complexity to a "theological concept", because lots of people talked about it but no one knew what it really was.(4) Several explanations for Chaos theory called for the words synthesis, cross-discipline, edge of chaos, dynamical, cellular automata, or neural networks, but all carry with them the concept of complex systems. The implications of Chaos are profound, for who could know the absolute conditions of any system for a complete prediction to be made of the behavior of that system? HOW IT ALL STARTED For thousands of years humans have noted that small causes could have large effects and that it was hard to predict anything for certain. What had caused a stir among scientists was that in some systems small changes of initial conditions could lead to predictions so different that prediction itself becomes useless. At the end of the 19th century, French mathematician, Jacques Hadamard proved a theorem on the sensitive dependence on initial conditions about the frictionless motion of a point on a surface or the geodesic flow on a surface of negative curvature. All this was about billiard balls and why you can't predict what three of them will do when they careened off each other on the table. French physicist Pierre Duhem understood the significance of Hadamard’s theorem. He published a paper in 1906 that made it quite plain that prediction was "forever unusable" because of the necessarily present uncertain initial conditions in Hadamard's theorem. These papers went unnoticed or rather unnoted by the man who was recognized as the Father of Chaos theory, Henri Poincare (1854-1912). In 1908 he published SCIENCE ET METHODE (5), that contained one sentence concerning the idea of chance being the determining factor in dynamic systems because of some factor in the beginning that we didn't know about. All three of these men and their ideas went unnoted because quantum mechanics had disrupted the whole physics world of ideas; and because there were no tools such as ergodic theorems(6) about the mathematics of measure; and because there were no computers to simulate what these theorems prove. In 1846, the planet Neptune was discovered, causing quite a celebration in the classical Newtonian mechanical world, this revelation had been predicted from the observation of small deviations in the orbit of Uranus. Something unexpected happened in 1889, though, when King Oscar II of Norway offered a prize for the solution to the problem of whether the solar system was stable. Henri Poincaré submitted his solution and won the prize, but a colleague happened to discover an error in the calculations. Poincaré was given six months to rectify the matter in order to keep his prize. In consternation, Poincaré found there was no solution.(7) Poincaré had found results that upset the accepted view of a purely deterministic universe that had reigned since Sir Isaac Newton lined out linear mathematics. In his 1890 paper, he showed that Newton's laws did not provide a solution to the "three-body problem", in other words, how one deals with predictions about the earth, moon and sun. He had found that small differences in the initial conditions produce very great ones in the final phenomena, and the situation defied prediction. Poincaré's discoveries were dismissed in lieu of Newton's linear model, one was to just ignore the small changes that cropped up. The three-body problem was what Poincare had to interpret with a two-body system of mathematics. Why was it a problem? He was trying to discover order in a system where none could be discerned. Poincaré’s negative answer caused positive consequences in the creation of chaos theory. About eighty years later, as early as 1963, Edward Lorenz8, using Poincaré’s mathematics, described a simple mathematical model of a weather system that was made up of three linked nonlinear differential equations that showed rates of change in temperature and wind speed. Some surprising results showed complex behavior from supposedly simple equations; also, the behavior of the system of equations was sensitively dependent on the initial conditions of the mathematical model. He spelled out the implications of his discovery, saying it implied that if there were any errors in observing the initial state of the system, and this is inevitable in any real system, prediction as to a future state of the system was impossible.(9) Lorenz labeled these systems that exhibited sensitive dependence on initial conditions as having the "butterfly effect": this unique name came from the proposition that a butterfly flapping its wings in Hong Kong can effect the course of a tornado in Texas. During 1970-71, interest in turbulence, strange attractors and sensitive dependence on initial conditions arose in the world of physics.(10) E. N. Lorenz published a paper, called "Deterministic nonperiodic flow" in 1963 that proved that meteorologists could not predict the weather. Jim Yorke, an applied mathematician from the University of Maryland was the first to use the name Chaos, but actually it was not even a chaos situation, but the name caught on.(11) A chaotic system is sensitive to initial conditions and causes the system to become unstable. Cambel identifies chaos as inherent in both the complexity in nature and the complexity in knowledge. The nature side of chaos entails all the physical sciences. The knowledge side of chaos deals with the human sciences. Chaos may manifest itself in either form or function or in both. Chaos studies the interdependence of things in a far-from-equilibrium state. Every open nonlinear dissipative(12) system has some relationship to another open system and their operations will intersect, overlap and converge. If the systems are sensitive to the initial conditions, in other words, you don’t know exactly in detail every little piece of information, and then you have a potentially chaotic system. Not all systems will be chaotic, but those where a lack of infinite detail is unknown, then these systems have an indeterminate quality about them. You can’t tell what’s going to happen next. They are unpredictable. If these systems are perturbed either internally or externally, they will display chaotic behavior and this behavior will be amplified microscopically and macroscopically. Further research in non-linear dynamical systems(13) that displayed a sensitive dependence on initial conditions came from Ilya Prigogine, a Nobel-prize winning chemist, who first began work with far-from-equilibrium systems in thermodynamic research.(14) Ilya Prigogines' research in non-linear dissipative structures led to the concept of equilibrium and far-from equilibrium to categorize the state of a system. In the physical studies of thermodynamics, Prigogines' research revealed far-from-equilibrium conditions that led to systemic behavior different from what was expected by the customary interpretation of the Second Law of Thermodynamics. Phenomena of bifurcation and self-organization emerged from systems in equilibrium if there was disruption or interference. This disruption or interference became the next step to Chaos Theory; it became Chaos/Complexity Theory. Prigogine talked about his theory as if he were Aristotle: a far-from-equilibrium system can go ‘from being to becoming’.(15) These ‘becoming’ phenomena showed order coming out of chaos in heat systems, chemical systems, and living systems. From Lorenz simulation, René Thom, mathematician, proposed ‘catastrophe theory’, or a mathematical description of how a chaos system bifurcates or branches. Out of these bifurcations came pattern, coherence, stable dynamic structures, networks, coupling, synchronization and synergy. From the study of complex adaptive systems used by Poincaré, Lorenz and Prigogine, Norman Packard and Chris Langton developed theories about the ‘edge of chaos’ in their research with cellular automata.(16) The energy flowing through the system, and the fluctuations, cause endless change which may either dampen or amplify the effects. In a phase transition of chaotic flux, (when a system changes from one state to another), it may completely reorganize the whole system in an unpredictable manner.(17) Two scientists, physicist Mitchell Feigenbaum(18) and computer scientist Oscar Lanford(19) came up with a picture of chaos in hydrodynamics using Renormalization ideas. They were studying non-linear systems and their transformations.(20) Since then, chaos theory or Nonlinear Science has taken the scientific world by a storm, with papers coming in from all fields of science and the humanities. Strange attractors were showing up in biology, statistics, psychology and economics and in every field of endeavor. Properties of complexity Complexity or the edge of chaos yielded self-organizing, self-maintaining dynamic structure that occurred spontaneously in a far-from-equilibrium system. Complexity had no agreed upon definition, but it could manifest itself in our everyday lives. Intense work is being done on the implications of complexity at the Santa Fe Institute in New Mexico. Here Ph. D.’s from many fields use cross-disciplinary methods to show how complexity in one area might link to another. Erwin Laszlo, from the Vienna International Academy, has the most interesting statement about Complexity: In fact, of all the terms that form the lingua franca of chaos theory and the general theory of systems, bifurcation may turn out to be the most important, first because it aptly describes the single most important kind of experience shared by nearly all people in today’s world, and second because it accurately describes the single most decisive event shaping the future of contemporary societies.(21) Bifurcation once meant splitting into two or more forks. In chaos theory it means: When a complex dynamical chaotic system becomes unstable in its environment because of perturbations, disturbances or ‘stress’, an attractor draws the trajectories of the stress, and at the point of phase transition, the system bifurcates and it is propelled either to a new order through self-organization or to disintegration. The phase transition of a system at the edge of chaos began with the studies of John Von Neumann(22) and Steve Wolfram(23) in their research on cellular automata.(24) Their research revealed the edge of chaos was the place where the parallel processing of the whole system was maximized. The system performed at its greatest potential and was able to carry out the most complex computations. At the bifurcation stage, the system was in a virtual area(25) where choices are made--the system could choose whatever attractor was most compelling, could jump from one attractor to another--but it was here at this stage that forward futuristic choices were made: this was deep chaos. The system self-organized itself to a higher level of complexity or it disintegrated. The phase transition stage may be called the transeunt stage, the place where transitory events happen. Transeunt is a philosophical term meaning that there is an effect on the system as a whole produced from the inside of the system having a transitory effect; and, a scientific term in that it is a nonperiodic signal of sudden pulse or impulse. After the bifurcation, the system may settle into a new dynamic regime of a set of more complex and chaotic attractors, thus becoming an even more complex system that it was initially. Three kinds of bifurcations happen: 1. Subtle, the transition is smooth. 2. Catastrophic, the transition is abrupt and the result of excessive perturbation. 3. Explosive, the transition is sudden and has discontinuous factors that wrench the system out of one order and into another.(26) Per Bak(27), with his co-researchers Chao Tang and Kurt Wiesenfeld reckons nature abiding on the edge of chaos or what they call ‘self-organized criticality’. Our daily encounter with Chaos/Complexity is seen in traffic flow, weather changes, population dynamics, organizational behavior, shifts in public opinion, urban development and decay, cardiological arrhythmias, epidemics. It might be found in the operation of the communications and computer technologies on which we rely, the combustion processes in our automobiles, cell differentiation, immunology, decision making, the fracture structures, and turbulence. Here are a few of the statements that Cambel makes about the ubiquity of chaos: 1. Complexity can occur in natural and man-made systems, as well as in social structures and human beings. 2. Complex dynamical systems may be very large or very small, indeed, in some complex systems, large and small components live cooperatively. 3. The system is neither completely deterministic nor completely random, and exhibits both characteristics. 4. The causes and effects of the events that the system experiences are not proportional. 5. The different parts of complex systems are linked and affect one another in a synergistic manner. 6. There is positive and negative feedback. The level of complexity depends on the character of the system, its environment, and the nature of the interactions between them.28 WHERE ITS ALL GOING If we lived in a completely deterministic world there would be no surprises and no decision making because an event would be caused by certain conditions that could lead to no other outcome. Nor could we consider living in a completely random world for there would be, as Cambel says, "no rational way of reaching a well-reasoned decision". (29) What kind of answers do we get when we recognize that a system is indeed unstable and that it is indeed an example of chaos at work. The American Association for the Advancement of Science published nineteen papers presented at their 1989 meeting that was devoted entirely to chaos theory usage on such ideas as chaos in dynamical systems, biological systems, turbulence, quantized systems, global affairs, economics, the arms race, and celestial systems. Stambler(30) reported that the Electric Power Research Institute was considering the applications of chaos control in voltage collapses, electromechanical oscillations, and unpredictable behavior in electric grids. Peng, Petrov and Showalter(31) were studying the usefulness of chaos control in chemical processing and combustion. Ott, Grebogi, and Yorke cited the many purposes of chaos and said it might even be necessary in higher life forms for brain functioning. Freeman studied just such brain functions related to the olfactory system and concluded that indeed chaos "affords an opportunity to exploit further these manifestations of brain activities"(32). Not only are research papers prolific, but an array of books are being published monthly on chaos applications. Bergé, Pomeau, and Vidal assert that chaos theory has "great predictive power"(33) that allows an understanding of the overall behavior of a system. Kauffman(34) uses the self-organization end of chaos to assert that nature itself is spontaneous; Cramer claimed that by overcoming the objections to mysticism and scientism(35), that the "theory of fundamental complexity is valid" (this will most likely turn into a book--so many researchers refer to it). This perhaps gives some idea as to far reaching applications of chaos theory in the scientific areas. A few last words about the edge of chaos will be added here because they will allow you to see how research has gone from linear science to nonlinear applications. Wentworth d'Arcy Thompson, in his book On Growth and Form(36), used transformations of coordinates to compare species of animals. Comparing one form of a fish, as an example, with another could be shown on a coordinate map and used to show how they differ and how they were alike. The same kind of transformation coordinate map could compare chimpanzee skulls to human skulls. Where Thompson used order to compare the workings of nature, Stuart Kauffman, in his book The Origins of Order: Self-Organization and Selection in Evolution,(37)took the next step in studying nature. He was seeking the origins of order in complex systems that were chaotic. His research is rife with examples of the interconnectedness of selection and self-organization. The essence of his findings are that much of the order seen in organisms stems from spontaneous generation from systems operating at the edge of chaos, or in other words, systems that are unstable purposely. Thompson applied physics to biology, and now Kauffman is applying chaos /complexity theory to biology. Cramer sees the interaction of order and disorder as a necessity in nature. "In nature, then, forms are not independent and arbitrary, they are interrelated in a regular way...And even organs arising to serve new functions develop according to the principle of transformation. At the branch points where something new emerges, disruptions of order are in fact necessary; abrupt phase changes occur. Indeed, the interplay of order and chaos constitutes the creative potential of nature."(38) The great French mathematician Henri Poincaré first noticed the idea that many simple nonlinear deterministic systems can behave in an apparently unpredictable and chaotic manner. Other early pioneering work in the field of chaotic dynamics were found in the mathematical literature by such luminaries as Birkhoff, Cartwright, Littlewood, Levinson, Smale, and Kolmogorov and his students, among others. In spite of this, the importance of chaos was not fully appreciated until the widespread availability of digital computers for numerical simulations and the demonstration of chaos in various physical systems. This realization has had broad implications for many fields of science, and it has been only within the past decade or so that the field has undergone explosive growth. The ideas of chaos have been very fruitful in such diverse disciplines as biology, economics, chemistry, engineering, fluid mechanics, physics, just to name a few. As you can see, Chaos Complexity theory can become a real research tool for many fields. Metaphorically it can be used outside the scientific field. This author plans to apply this theory to religious research. FOOTNOTES 1. Cambel, A. B. Applied Chaos Theory: A Paradigm for Complexity. Academic Press, Inc. San Diego, CA 1993. P. 15. return 2. Li, T. Y. and Yorke, J.A. "Period Three Implies Chaos". American Mathematical Monthly, 82,1975. Pp. 995-992. return 3. Cambel. P. 16. quoting Zeldovich, Y. A., Ruzmaikin, A. A. and Sokoloff, D.D. The Almighty Chance. Singapore: World Scientific. 1990. return 4. Stein, Daniel L. (ed.) Lectures in the Sciences of Complexity. Vol. 1. Redwood City, CA. Addison-Wesley Publishing Co. 1989. P. XIII. return 5. Poincare, Jules Henri. Science and Method. New York: Dover. 1952. English trans. Did you know his cousin Raymond was President of France from 1913-20? return 6. Ergodics are used in statistics, the method says that given a long enough interval a system will return to a similar state it previously had. return 7. Peterson, I. Newton’s Clock: Chaos in the Solar System. New York: MacMillan. 1993. Peterson tells other quite interesting stories about the beginnings of Chaos theory. return 8. Lorenz, Edward N. "Deterministic nonperiodic flow." Journal of the Atmospheric Sciences. 20:130-41. 1963. return 9. Lorenz. P. 133. return 10. Another problem those jumped right into the middle of the chaos theory excitement was complexity. Kurt Godel, logician from Austria, published a paper in 1931 that the mathematician's dream of a complete set of numbers to represent everything is incomplete. ("Within the framework of generally accepted basic assertions concerning integers 1 2,3,...., Godel showed that some assertions can neither be proved on or disproved: these are undecideable assertions. If one increases the number of basic assertions there will nevertheless always remain some undecideable assertions. It was quite earthshaking to mathematics, but now it is accepted to know that the set of all properties of integers and the set of all true assertions about them does not have a finite basis. return 11. Li, T. and J. A. Yorke. Pp. 985-92. return 12. Nonlinear defined without mathematics is an equation that has both negative and positive answers. Dissipative defined without the physics means that it gives off energy and doesn’t get any back. return 13. Kellert, Stephen. In the Wake of Chaos: Unpredictable Order in Synamical Systems. Chicago, Ill.: The University of Chicago Press. 1993. P. 3. A dynamical system is a simplified model for the "time-varying" behavior which gives the recipe for producing the present physical state of a system and for transforming the system to a descriptive of its state in the past or future. By changing the variables one can map the changes that a system goes through to obtain a state from time to time: the use of evolution equations or differential equations is but the necessary process that is sometimes oppressively long. Chaos theory is a part of dynamical systems study but uses nonlinear terms in the equations. These nonlinear terms may be expressions such as x2, sin (x), or 2xy, which makes it impossible to render a single answer. "Chaos theory investigates a system by asking about the general character of its long-term behavior" return 14. Thermodynamics is the study of energy flow. Classical thermodynamic studies closed or near equilibrium systems. Von Bertalanffy actually presented the same idea in his General System’s theory in 1968. Prigogine researched far-from-equilibrium systems of chemical and heat transfer, which displayed self-organizing characteristics. He refined the theory, linked it to living systems and publicized it. return 15. Prigogine, Ilya. and Stengers, I. Order out of Chaos: Man's New Dialogue with Nature. New York: Bantam Books. 1984. return 16. Packard, Norman. "Adaptation Toward the Edge of Chaos." A Technical Report, Center for Complex Systems Research. University of Illinois. CCSR-88-5. 1988. There is an earlier paper by Chris Langton. "Studying Artificial Life with Cellular Automata" in Physica. 22D. 1986. Pp. 120-49. return 17. Waldrop, Mitchell. Complexity: The Emerging Science at the Edge of Order and Chaos. (New York: Simon & Schuster. 1992). Waldrop keeps you interested about the discovery and the implication of these complex systems in this book. return 18. Feigenbaum, M. J. "Quantitative universality for a class of nonlinear transformation," J.STATIST..PHYS. 21 (1979). Pp. 25-52. return 19. Lanford, O. E. "A computer-assisted proof of the Feigenbaum conjectures". BULL .AMER. MATH. SOC.6 (1982). Pp.427-34. return 20. Ruelle, David. CHANCE AND CHAOS. Princeton: University Press. 1991. Pp. 57-79. return 21. Laszlo, Ervin. The Age of Bifurcation: Understanding the Changing World. Philadelphia, Pa.: Gordon and Breach Science Publishers. 1991. P. 4. return 22. von Neumann, John. Theory of Self-Reproducing Automata. Edited by Arthur W. Burks. (Champaign-Urbana: University of Illinois Press. 1966. return 23. Wolfram, S. "Statistical mechanics of cellular automata" in Rev. Mod. Phys. 55:601. 1983. and "Universality and complexity in cellular automata" in Physica 10D:1. 1984. See also: Theory and Applications of Cellular Automata. Singapore: World Scientific. 1986. return 24. Waldrop. P.87. Cellular automata are programs for generating patterns on a computer according to rules specified by the programmer. They are precisely defined and can be analyzed in detail, yet they have a dynamic quality that leads to complexity in the system. Their research tries to find laws that describe when and how such complexities emerge in nature. return 25. Fox, Ronald F. "Quantum Chaos in Two-Level Quantum Systems." The Ubiquity of Chaos. Saul Krasner, ed. Washington, D. C.: American Association for the Advancement of Science. 1990). Pp. 105-113. Fox worked with quantum mechanical models to discover that the periodic modulations were seen to arise from virtual quantum transitions. There had to be the virtual transitions in order for chaos to happen. There was no classical analogue for these findings. return 26. Ashby, W. Ross. Design for a Brain. 2nd. ed. (New York: Wiley Publishers. 1960). Ashby’s work was centered around how a system with many interacting parts adapts to it environment. He was thinking in terms of neural or brain adaptation. See also: Ashby, W. Ross. "Principles of the Self-organizing system." Principles of Self-Organization. Foerster and Zopf, eds. New York: Perganmon Press. 1962. return 27. Bak, Per and P., Tang, C. "Self-Organized Criticality". Physics Review A 38:364. 1988. return 28. Cambel. Pp. 3-4. return 29. Cambel. P. 4. return 30. Stambler, I. "Chaos Creates a Stir in Energy-Related R&D". R&D Magazine. December, p.16. return 31. Peng, B., Petrov, V. and Showalter, K. "Controlling Chemical Chaos". Journal of Physical Chemistry. 95. Pp. 1957-59. 1991. return 32. Freeman, Walter J., "Searching for Signal and Noise in the Chaos of Brain Waves". in The Ubiquity of Chaos. Saul Krasner, ed. Washington, D. C.: American Association for the Advancement of Science. 1990. Pp. 47-55. return 33. Bergé, P., Pomeau, Y., and Vidal, C. Order Within Chaos. Translated by L. Tuckerman. Paris: J. Wiley & Sons. 1984. P.265. The predictive power in this reference is not to predict the exact value of some property of a system, but to allow the researcher to understand the overall behavior of that system, and perhaps even predict what the overall behavior will look like at some future point. It is strictly holistic prediction they refer to. return 34. Kauffman, Stuart A. The Origins of Order: Self-Organization and Selection in Evolution. (New York: Oxford University Press. 1993). return 35. Cramer. Pp. 218-9. Cramer objects that the theory of fundamental complexity is a product of mysticism. He agrees that it might seem mystic because it contains antinaturalistic, nonscientific, and even mystical elements. But, the macroscopic biological realm contains individual molecular events that are subject to feedback coupling operating through amplification mechanisms. The statistical fluctuations can be recognized by nonlinear equations used in chaos theory, thus, these networks become indeterminate under certain conditions. Since chaos theory incorporates these latest scientific findings, it cannot be regarded as mysticism. return 36. Thompson, Wentworth d’Arcy. On Growth and Form. 2ed.. Cambridge: Cambridge University Press. 1966. return 37. Kauffman, Stuart A. The Origins of Order: Self-Organization and Selection in Evolution. return