Where Chaos begins, classical science ends.Ever since physicists haveinquired into the laws of nature, the have not begun to explore irregular sideof nature, the erratic and discontinuous side, that have always puzzledscientists.They did not attempt to understand disorder in the atmosphere, theturbulent sea, the oscillations of the heart and brain, and the fluctuations ofwildlife populations.All of these things were taken for granted until in the1970’s some American and European scientists began to investigate the randomnessof nature.

They were physicists, biologists, chemists and mathematicians but theywere all seeking one thing: connections between different kinds of irregularity.

“Physiologists found a surprising order in the chaos that develops in the humanheart, the prime cause of a sudden, unexplained death.Ecologists explored therise and fall of gypsy moth populations.Economists dug out old stock pricedata and tried a new kind of analysis.The insights that emerged led directlyinto the natural world- the shapes of clouds, the paths of lightning, themicroscopic intertwining of blood vessels, the galactic clustering of stars.”(Gleick, 1987)The man most responsible for coming up with the Chaos Theory wasMitchell Feigenbaum, who was one of a handful of scientists at Los Alamos, NewMexico when he first started thinking about Chaos.Feigenbaum was a littleknown scientist from New York, with only one published work to his name.Hewas working on nothing very important, like quasi periodicity, in which he andonly he had 26 hour days instead of the usual 24.He gave that up because hecould not bear to wake up to setting sun, which happened periodically.Hespent most of time watching clouds from the hiking trails above the laboratory.

To him could represented a side of nature that the mainstream of physics hadpassed by, a side that was fuzzy and detailed, and structured yet unpredictable.

He thought about these things quietly, without producing any work.

After he started looking, chaos seemed to be everywhere.A flag snapsback and forth in the wind.A dripping faucet changes from a steady pattern toa random one.A rising column of smoke disappears into random swirls.”Chaosbreaks across the lines that separate scientific disciplines.Because it is ascience of the global nature of systems, it has brought together thinkers fromfields that have been widely separated…Chaos poses problems that defy acceptedways of working in science.It makes strong claims about the universalbehavior of complexity.The first Chaos theorists, the scientists who set thediscipline in motion, shared certain sensibilities.They had an eye forpattern, especially pattern that appeared on different scales at the same time.

They had a taste for randomness and complexity, for jagged edges and suddenleaps.Believers in chaos– and they sometimes call themselves believers, orconverts, or evangelists–speculate about determinism and free will, aboutevolution, about the nature of conscious intelligence.They feel theat theyare turning back a trend in science towards reductionism, the analysis ofsystems in terms of their constituent parts: quarks, chromosomes, or neutrons.

They believe that they are looking for the whole.”(Gleick, 1987)The Chaos Theory is also called Nonlinear Dynamics, or the Complexitytheory.They all mean the same thing though- a scientific discipline which isbased on the study of nonlinear systems. To understand the Complexity theorypeople must understand the two words, nonlinear and system, to appreciate thenature of the science.A system can best be defined as the understanding ofthe relationship between things which interact.For example, a pile of stonesis a system which interacts based upon how they are piled. If they are piled outof balance, the interaction results in their movement until they find acondition under which they are in balance.A group of stones which do nottouch one another are not a system, because there is no interaction.A systemcan be modeled. Which means another system which supposedly replicates thebehavior ofthe original system can be created. Theoretically, one cantake a second group of stones which are the same weight, shape, and density ofthe first group, pile them in the same way as the first group, and predict thatthey will fall into a new configuration that is the same as the first group. Ora mathematical representation can be made of the stones through application ofNewton’s law of gravity, to predict how future piles of the same type – and ofdifferent types of stones – will interact.Mathematical modeling is the key,but not the only modeling process used for systems.

The word nonlinear has to do with understanding mathematical models usedto describe systems. Before the growth of interest in nonlinear systems, mostmodels were analyzed as though they were linear systems meaning that when themathematical formulas representing the behavior of the systems were put into agraph form, the results looked like a straight line.Newton used calculus as amathematical method for showing change in systems within the context of straightlines. And statistics is a process of converting what is usually nonlinear datainto a linear format for analysis.

Linear systems are the classic scientific system and have been used forhundreds of years, they are not complex, and they are easy to work with becausethey are very predictable. For example, you would consider a factory a linearsystem. If more inventory is added to the factory, or more employees are hired,it would stand to reason that more pieces produced by the factory by asignificant amount. By changing what goes into a system we should be able totell what comes out of it.But as any factory manager knows, factories don’tactually work that way.If the amount of people, the inventory, or whateverother variable is changed in the factory you would get widely differing resultson a day to day basis from what was predicted.That is because a factory is acomplex nonlinear system, like most systems found in nature.

When most natural systems are modeled, their mathematicalrepresentations do not produce straight lines on graphs, and that the systemoutputs are extremely difficult to predict. Before the chaos theory wasdeveloped, most scientists studied nature and other random things using linearsystems. Starting with the work of Sir Isaac Newton, physics has provided aprocess for modeling nature, and the mathematical equations associated with ithave all been linear. When a study resulted in strange answers, when aprediction usually came true but not this one time, the failure was blamed onexperimental error or noise.

Now, with the advent of the Chaos theory and research into complexsystems theory, we know that the “noise” actually was important informationabout the experiment.When noise is added to the graph results, the resultsare no longer a straight line, and are not predictable.This noise is what wasoriginally referred to as the chaos in the experiment.Since studying thisnoise, this chaos, was one of the first concerns of those studying complexsystems theory, Glieck originally named the discipline Chaos Theory.

Another word that is vital to understanding the Complexity theory iscomplex.What makes us determine which system is more complex then another?There are many discussions of this question. In Exploring Complexity, NobelLaureate Ilya Prigogine explains that the complexity of the system is defined bythe complexity of the model necessary to effectively predict the behavior of thesystem. The more the model must look like the actual system to predict systemresults, the more complex the system is considered to be. The most complexsystem example is the weather, which, as demonstrated by Edward Lorenz, can onlybe effectively modeled with an exact duplicate of itself.One example of asimple system to model is to calculate the time it takes for a train to go fromcity A to city B if it travels at a given speed.To predict the time we needonly to know the speed that the train is traveling (in mph) and the distance (inmiles). The simple formula would be mph/m, which is a simple system.

But the pile of stones, which appears to be a simple system, is actuallyvery complex.If we want to predict which stone will end up at which place inthe pile then you would have to know very detailed information about the stones,including their weights, shapes, and starting location of each stone to make anaccurate prediction.If there is a minor difference between the shape of onestone in the model and the shape of the original stone, the modeled results willbe very different.The system is very complex, thus making prediction verydifficult..

The generator of unpredictability in complex systems is what Lorenzcalls “sensitivity to initial conditions” or “the butterfly effect.” The conceptmeans that with a complex, nonlinear system, a tiny difference in startingposition can lead to greatly varied results.For example, in a difficult poolshot a tiny error in aim causes a slight change in the balls path.However,with each ball it collides with, the ball strays farther and farther from theintended path.Lorenz once said that “if a butterfly is flapping its wings inArgentina and we cannot take that action into account in our weather prediction,then we will fail to predict a thunderstorm over our home town two weeks fromnow because of this dynamic.”(Lorenz, 1987)The general rule for complex systems is that one cannot create a modelthat will accurately predict outcomes but one can create models that simulatethe processes that the system will go through to create the models.Thisrealization is impacting many activities in business and other industries.Forinstance, it raises considerable questions relating to the real value ofcreating organizational visions and mission statements as currently practices.

Like physics, the Chaos theory provides a foundation for the study ofall other scientific disciplines.It is a variety of methods for incorporatingnonlinear dynamics into the study of science.Attempts to change thediscipline and make it a separate form of science have been strongly resisted.

The work represents a reunification of the sciences for many in the scientificcommunity.

One of Lorenz’s best accomplishments supporting the Chaos Theory was theLorenz Attractor.The Lorenz Attractor is based on three differentialequations, three constants, and three initial conditions. The attractorrepresents the behavior of gas at any given time, and its condition at any giventime depends upon its condition at a previous time. If the initial conditionsare changed by even a tiny amount, checking the attractor at a later time willshow numbers totally different. This is because small differences will reproducethemselves recursively until numbers are entirely unlike the original systemwith the original initial conditions.But, the plot of the attractor, or theoverall behavior of the system will be the same.

A very small cause which escapes our notice determines a considerableeffect that we cannot fail to see, and then we say that the effect is due tochance.If we knew exactly the laws of nature and the situation of theuniverse at the initial moment, we could predict exactly the situation of thatsame universe at a succeeding moment.But even if it were the case that thenatural laws had any secret for us, we could still know the situationapproximately.If that enabled us to predict the succeeding situation with thesame approximation, that is all we require, and we should say that thephenomenon has been predicted, that it is governed by the laws.But it is notalways so; it may happen that small differences in the initial conditionsproduce very great ones in the final phenomena.A small error in the formerwill produce an enormous error in the latter.Prediction becomesimpossible…” (Poincare, 1973)The Complexity theory has developed from mathematics, biology, andchemistry, but mostly from physics and particularly thermodynamics, the study ofturbulence leading to the understanding of self-organizing systems and systemstates (equilibrium, near equilibrium, the edge of chaos, and chaos).”Theconcept of entropy is actually the physicists application of the concept ofevolution to physical systems. The greater the entropy of a system, the morehighly evolved is the system.”( Prigogine, 1974)The Complexity theory is alsohaving a major impact on quantum physics and attempts to reconcile the chaos ofquantum physics with the predictability of Newton’s universe.

With complexity theory, the distinctions between the differentdisciplines of sciences are disappearing.For example, fractal research is nowused for biological studies.But there is a question as to whether the currentresearch and academic funding will support this move to interdisciplinaryresearch.

Complexity is already affecting many aspects of our lives and has agreat impacts on all sciences.It is answering previously unsolvable problemsin cosmology and quantum mechanics.The understanding of heart arrhythmias andbrain functioning has been revolutionized by complexity research.There havebeen a number of other things developed from complexity research, such a theSimLife, SimAnt, etc. which are a series of computer programs.Fractalmathematics are critical to improved information compression and encryptionschemes needed for computer networking and telecommunications. Geneticalgorithms are being applied to economic research and stock predictions.

Engineering applications range from factory scheduling to product design, withpioneering work being done at places like DuPont and Deere ; Co.

Another element of the nonlinear dynamics, Fractals, have appearedeverywhere, most recently in graphic applications like the successful FractalDesign Painter series of products. Fractal image compression techniques arestill being researched, but promise such amazing results as 600:1 graphiccompression ratios.The movie special effects industry would have much lessrealistic clouds, rocks, and shadows without fractal graphic technology.

Though it is one of the youngest sciences, the Chaos Theory holds greatpromise in the fields of meteorology, physics, mathematics, and just aboutanything else you can think of.

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