Chaos Theory: An Introduction

Sardar Patel Institute of Technology A Report On S.E. Information Technology Group C2 (Roll Numbers: 54-60) A report submitted in partial fulfillment of the requirement of Communication and Presentation Techniques syllabus: Report Writing October 2012

CERTIFICATEThis to certify that the work on the project titled Chaos Theory has been carried out by thefollowing students, who are bonafide students of Sardar Patel Institute of Technology,Mumbai, in partial fulfilment of the syllabus requirement in the subject “Communication andPresentation Techniques” in the academic year 2012-2013: 1. Gajanan Shewale 2. Nayana Shinde 3. Aditya Shirode 4. Suntej Singh 5. Jayesh Solanki 6. Madhuri Tajane 7. Gaurav TripathiProject Guide: ____________________ (Madhavi Gokhale)Principal: _________________________ (Dr.Prachi Gharpure)

PREFACEThe trouble with weather forecasting is that its right too often for us to ignore it and wrongtoo often for us to rely on it.- Patrick YoungThe quote perfectly fits. Whenever our meteorological department predicts some kind ofweather, we know that nature is going to do exactly opposite. But why does this happen? Ifwe can exactly predict at what time any astronomical body is going to enter earth’satmosphere and where it is going to strike, why can’t weather? Does God really play dice?Not quite. The unpredictability of weather can be explained by Chaos Theory.The word “Chaos” comes from the Greek word “Khaos”, meaning "gaping void". Chaos inother words means a state of utter confusion or the inherent unpredictability in the behaviorof a complex natural system.Chaos theory is a field of study in mathematics, with applications in several disciplinesincluding physics, engineering, economics, biology, and philosophy which primarily statesthat small differences in initial conditions (such as those due to rounding errors in numericalcomputation) can yield widely diverging outcomes for chaotic systems, rendering long-termprediction impossible in general.Acknowledging the fact that chaos theory is widely used in many spheres of life and isknown to very few people inspired us to prepare a report on this topic to make the massesaware regarding this phenomenon. We hope that this project serves as a useful tool foranyone who is interested in understanding this topic.

ACKNOWLEDGEMENTSWe, as a group, have taken efforts in this project. However it would not have been possiblewithout the kind support and help of many individuals and organization, who helped us indeveloping the project and willingly helped out with their abilities. We would like to extendour sincere thanks to all of them.We are grateful to Prof. Madhavi Gokhale, Lecturer in ‘Communication and PresentationTechniques’, for her guidance and support. Her valuable comments and discussions on earlierversions of this report lead us to improvements and propelled us in right direction. Her aptsuggestions and constant encouragement have enabled us to focus our efforts.

ABSTRACTChaos theory is a mathematical field of study which states that non-linear dynamical systemsthat are seemingly random are actually deterministic from much simpler equations. Thephenomenon of Chaos theory was introduced to the modern world by Edward Lorenz in 1972with conceptualization of ‘Butterfly Effect’. As chaos theory was developed by inputs ofvarious mathematicians and scientists, it found applications in a large number of scientificfields.The purpose of the project is the interpretation of chaos theory which is not as familiar asother theories. Everything in the universe is in some way or the other under control of Chaosor product of Chaos. Every motion, behavior or tendency can be explained by Chaos Theory.The prime objective of it is the illustration of Chaos Theory and Chaotic behavior.This project includes origin, history, fields of application, real life application and limitationsof Chaos Theory. It explores understanding complexity and dynamics of Chaos.

TABLE OF CONTENTSPreface ……………………………………………………………………………… iAcknowledgement ……………………………………………………………….. iiAbstract ……………………………………………………………………………… iii1. Introduction ………………………………………………………………… 1 1.1 What is chaos theory? 1.2 Chaos Theory: Before and After Lorenz2. Concepts of Chaos Theory ………………………………………………… 5 2.1 Key Terms 2.1.1 Chaotic Systems 2.1.2 Attractors 2.1.3 Fractals 2.2 The Butterfly effect3. Aspects of Chaos Theory …………………………………………………………. 12 3.1 Predictability 3.2 Control of chaos 3.3 Synchronization of chaos4. Applications of Chaos theory …………………………………………………. 16 4.1 Stock market 4.2 Population dynamics 4.3 Biology (human being as a chaotic system) 4.3.1 Predicting heart attacks 4.4 Real time applications 4.4.1 Chaos to produce music 4.4.2 Climbing with chaos 4.5 Random Number Generation5. Limitations of chaos theory …………………………………………………. 236. Conclusion …………………………………………………………………. 24Appendix …………………………………………………………………………. 25List of References …………………………………………………………………. 31Bibliography …………………………………………………………………………. 32

1. INTRODUCTION1.1 WHAT IS CHAOS THEORY?Changes in the weather. Cardiac arrhythmias. Traffic flow patterns. Urban development anddecay. Epidemics. The behavior of people in groups. Any idea what holds all of these ideastogether?Richard Feynman, a well known physicist, quoted that, “Physicists like to think that all you haveto do is say, these are the conditions, now what happens next?”The world of science has been confined to the linear world for centuries. That is to say,mathematicians and physicists have overlooked dynamical systems as random and unpredictable.The only systems that could be understood in the past were those that were believed to be linear,that is to say, systems that follow predictable patterns and arrangements. Linear equations, linearfunctions, linear algebra, linear programming, and linear accelerators are all areas that have beenunderstood and mastered by the human race. But there were some areas that just could not beexplained, like weather patterns, ocean currents, or the actions of cells. There were too manythings going on to keep track of with linear equations.Answer to both questions posed above is Chaos, a theory related to which was given by Lorenz,‘during a coffee break’.What is Chaos exactly? Mathematicians say it is tough to define chaos, but is easy to “recognizeit when you see it.” Chaos and order are two sides of the same coin. Chaos is interrelated withorder. It keeps stagnation from setting into any one system.Systems considered to be chaotic are not really chaotic at all – they are just not as predictable asthe cause-and-effect kind of ideas associated with linear dynamics. Mythology and early sciencehave presented ideas of chaos to explore. In many mythologies the creation of the universe issymbolized by the gods of order conquering chaos. “While the universe, including the gods, mayoriginate from chaos, order seems to emerge also. Order banishes chaos but never really destroysit.”1 Despite its etymological Greek origin ‘ ’ (zhaos), the notion of chaos appears in manydifferent ancient narrations about the origins of the World. Chaos is abundant, but also regimedependent.What is Chaos Theory then?To state as a definition, Chaos theory is the study of complex, nonlinear, dynamic systems.It is a branch of mathematics that deals with systems that appear to be orderly (deterministic)but, in fact, harbor chaotic behaviors. It also deals with systems that appear to be chaotic, but, infact, have underlying order. Chaos theory studies the behavior of dynamical systems that arehighly sensitive to initial conditions, an effect which is popularly referred to as the butterfly

effect. Small differences in initial conditions (such as those due to rounding errors in numericalcomputation) yield widely diverging outcomes for chaotic systems, rendering long-termprediction impossible in general. This happens even though these systems are deterministic,meaning that their future behavior is fully determined by their initial conditions, withno random elements involved. In other words, the deterministic nature of these systems does notmake them predictable. This behavior is known as deterministic chaos, or simply chaos.Nature is highly complex, and the only prediction you can make is that she is unpredictable. Theamazing unpredictability of nature is what Chaos Theory looks at. Why? Because instead ofbeing boring and translucent, nature is marvelous and mysterious. And Chaos Theory hasmanaged to somewhat capture the beauty of the unpredictable and display it in the mostawesome patterns. Nature, when looked upon with the right kind of eyes, presents her as one ofthe most fabulous works of art ever.Chaos Theory holds to the axiom that reality itself subsists in a state of ontological anarchy.Chaos theory is most commonly attributed to the work of Edward Lorenz. His 1963 paper,Deterministic Nonperiodic Flow, is credited for laying the foundation for Chaos Theory. Lorenzwas a meteorologist who developed a mathematical model used to model the way the air movesin the atmosphere. He discovered by chance that when he entered a starting value at threedecimal .506 instead of entering the full .506127. It caused vast differences in the outcome of themodel. In this way he discovered the principle of Sensitive Dependence on Initial Conditions(SDIC), which is now viewed as a key component in any chaotic system. This idea was thenimmortalized when Lorenz gave a talk at the 139th meeting of the American Association for theAdvancement of Science in 1972 entitled “Predictability: Does the Flap of a Butterfly’s Wings inBrazil Set Off a Tornado in Texas”. With this speech the idea of the Butterfly Effect was bornand has been used when talking about chaos theory ever since. The basic principle is that even inan entirely deterministic system the slightest change in the initial data can cause abrupt andseemingly random changes in the outcome.

1.2 CHAOS THEORY: BEFORE AND AFTER LORENZWhile Lorenz’s discovery has achieved widespread attention that it did not initially receive uponits initial publication is well-deserved outcome, largely being read initially by meteorologists andclimatologists who did not fully appreciate its broader mathematical implications. Nevertheless,the possibility of SDIC had been realized by others much earlier, including Maxwell (1876),Hadamard (1898), and Poincaré (1890). The idea of fractality also had a long history precedinghim going back at least to Cantor (1883), with the idea of endogenously erratic dynamics thatcome close to following periodicity arguably going back as far as the pre-Socratic Greekphilosopher, Anaxagoras [according to Rössler (1998)], although first clearly shown by Cayley(1879).Playing a foundational role, Anaxagoras created the qualitative mathematical notions used sosuccessfully later by the Poincaré school: deterministic flow, cross-section through a flow, i.e.and – most important – the notions of mixing and unmixing.Yet another figure who has been seen to foreshadow modern chaos theory is the Renaissancepolymath, Leonardo da Vinci. Some of his drawings involving wind depict spectacularturbulence, which in fluid dynamics has long been a central area of study associated with chaostheory. Leibniz (1695) posited the possibility of fractional derivatives. “Big whorls have little whorls, Which feed on their velocity; And little whorls have lesser whorls, And so on to viscosity”Another mathematician, Richardson, stated the above, in the molecular sense, in 1922.Professor of Political Economy in Britain, Thomas Robert Malthus, in the first edition of hisfamous Essay on the Principle of Population (1798), mentioned that history would tend greatlyto elucidate the manner in which the constant check upon population acts.Alan Turing conducted work on the process of Morphogenesis in the early 1950s that argued fora mathematical explanation of the process. Morphogenesis is the biological process that causesan organism to develop its shape.Now seeing the work done by his predecessors, one might infer that Lorenz did not do anythingof any particularly great importance. But Lorenz can be claimed to have “discovered chaos” bothbecause he put all the elements together, sensitive dependence on initial conditions, with strange(fractal) attractors, and the resulting erratic dynamics, and because he saw them all together andin the context of a computer simulation situation that could be replicated. He was the one whoderived the mathematical understanding.2

What Lorentz did in weather, Robert May did in ecology. His work, in the early 1970s, helpedpin down the concepts of bifurcation (a split or division by two) and period doubling (one way inwhich order breaks down into chaos). A period is the time required for any cyclic system toreturn to its original state. In the early 1970s, May was working on a model that addressed howinsect birthrate varied with food supply. He found that at certain critical values, his equationrequired twice the time to return to its original state--the period having doubled in value. Afterseveral period-doubling cycles, his model became unpredictable, rather like actual insectpopulations tend to be unpredictable. Since May’s discovery with insects, mathematicians havefound that this period-doubling is a natural route to chaos for many different systems.Even though Lorenz coined the term Butterfly Effect, the name chaos was coined by Jim Yorkein 1974, an applied mathematician at the University of Maryland.In 1971, David Ruelle and Floris Takens described a phenomena they called a strangeattractor (a special type of attractor today called achaotic attractor). This strange phenomenonwas said to reside in what they called phase space (a geometric depiction of the state space of asystem). Strange attractor is recognized as the general pattern followed by chaotic systems.Another pioneer of the new science was Mitchell Feigenbaum. His work, in the late 1970s, wasso revolutionary that several of his first manuscripts were rejected for publication because theywere so novel they were considered irreverent. He discovered order in disorder. He lookeddeeply into turbulence, the home of strange attractors, and saw universality. Feigenbaum showedthat period doubling is the normal way that order breaks down into chaos.How come mathematicians have not studied chaos theory earlier? The answer can be given inone word: computers. The calculations involved in studying chaos are repetitive, boring andnumber in the millions. A multidisciplinary interest in chaos, complexity and self-organizingsystems started in 1970’s with the invention of computers.Benoît Mandlebrot found the piece of the chaos puzzle that put all things together. Mandelbrotpublished a book, The Fractal Geometry of Nature (1982), which looked into a mathematicalbasis of pattern formation in nature, much like the earlier work of Turing. In this he outlined hisprinciple of self similarity, which describes anything in which the same shape is repeated overand over again at smaller and smaller scales. His fractals (the geometry of fractional dimensions)helped describe or picture the actions of chaos, rather than explain it. Chaos and its workingscould now be seen in color on a computer. As a graphical representation of the mathematical rulebehind self similarity Mandelbrot created an image that has come to be known as the MandelbrotSet, or the Thumbprint of God.3 The Mandelbrot Set’s properties of being similar at all scalesmirrors a fundamental ordering principle, repeated in nature.

2. CONCEPTS2.1 KEY TERMSThe typical features of chaos include:• Nonlinearity. If it is linear, it cannot be chaotic.• Determinism. It has deterministic (rather than probabilistic) underlying rules every future stateof the system must follow.• Sensitivity to initial conditions. Small changes in its initial state can lead to radically differentbehavior in its final state. This “butterfly effect” allows the possibility that even the slightperturbation of a butterfly flapping its wings can dramatically affect whether sunny or cloudyskies will predominate days later.• Sustained irregularity in the behavior of the system. Hidden order including a large or infinitenumber of unstable periodic patterns (or motions). This hidden order forms the infrastructure ofirregular chaotic systems---order in disorder for short.• Long-term prediction (but not control!) is mostly impossible due to sensitivity to initialconditions, which can be known only to a finite degree of precision.42.1.1 CHAOTIC SYSTEMSChaos is qualitative in that it seeks to know the general character of a systems long-termbehavior, rather than seeking numerical predictions about a future state.Chaotic systems are unstable since they tend not to resist any outside disturbances but insteadreact in significant ways. In other words, they do not shrug off external influences but are partlynavigated by them. These systems are deterministic because they are made up of few, simpledifferential equations, and make no references to implicit chance mechanisms. A dynamicsystem is a simplified model for the time-varying behavior of an actual system. These systemsare described using differential equations specifying the rates of change for each variable.A deterministic system is a system in which no randomness is involved in the development offuture states of the system. 5It is said to be chaotic whenever its evolution sensitively depends onthe initial conditions. This property implies that two trajectories emerging from two differentclose-by initial conditions separate exponentially in the course of time.

Lorenzs experiment: Difference between starting values of these curves is only .000127 Source: http://www.imho.com/sites/default/files/images/fig1_0.gifThe fact that some dynamical model systems showing the above necessary conditions possesssuch a critical dependence on the initial conditions was known since the end of the nineteenthcentury. However, only in the last thirty years of twentieth century, experimental observationshave pointed out that, in fact, chaotic systems are common in nature. They can be found, forexample, in Chemistry (Belouzov-Zhabotinski reaction), in Nonlinear Optics (lasers), inElectronics (Chua-Matsumoto circuit), in Fluid Dynamics (Rayleig-BeHnard convection), etc.Many natural phenomena can also be characterized as being chaotic. They can be found inmeteorology, solar system, heart and brain of living organisms and so on.The peculiarities of a chaotic system can be listed as follows:61. Strong dependence of the behavior on initial conditions2. The sensitivity to the changes of system parameters3. Presence of strong harmonics in the signals4. Fractional dimension of space state trajectories5. Presence of a stretch direction, represented by a positive Lyapunov exponentThe last can be considered as an “index” that quantifies a chaotic behavior.Solutions of chaotic systems can be complex and typically they cannot be easily extrapolatedfrom current trends. The game of Roulette is an interesting example that might illustrate thedistinction between random and chaotic systems: If we study the statistics of the outcome ofrepeated games, then we can see that the sequence of numbers is completely random. That ledEinstein to remark: “The only way to win money in Roulette is to steal it from the bank. On theother hand we know the mechanics of the ball and the wheel very well and if we could somehowmeasure the initial conditions for the ball/wheel system, we might be able to make a short termprediction of the outcome.7

A double pendulum is a classic chaotic system. It consists of one pendulum attached to another. Double Pendulum Source: http://web.mat.bham.ac.uk/C.J.Sangwin/Teaching/pendulum/dp1.jpg2.1.2 ATTRACTORSAn attractor is a set of states (points in the phase space), invariant under the dynamics, towardswhich neighboring states in a given basin of attraction asymptotically approach in the course ofdynamic evolution. An attractor is defined as the smallest unit which cannot itself bedecomposed into two or more attractors with distinct basins of attractionIn chaos theory, systems evolve towards states called attractors. The evolution towards aspecific state is governed by a set of initial conditions. An attractor is generated within thesystem itself.8There are several types of attractors9. The first is a point attractor, where there is only oneoutcome for the system. Death is a point attractor for human beings. No matter who we are, howwe lived our life or whatever, we die at the end of our life.The second type of attractor is called a limit cycle or periodic attractor. Instead of moving to asingle state as in a point attractor, the system settles into a cycle. While we can then not predictthe exact state of the system at any time, we know it will be somewhere in the cycle.The third type of attractor is called a strange attractor or a chaotic attractor. It is a double spiralwhich never repeats itself (or it would be periodic attractor), but the values always move towardsa certain range of values. There are certain states in which the system can exist and others itcannot. If the system were to somehow move out from the acceptable range of states it would be“attracted” back into the attractor. It is a geometrical shape which represents the limited region ofphase space ultimately occupied by all trajectories of a dynamical system.

The giant red spot on Jupiter, also called ‘Soliton’, is a good example of a strange attractor. Weknow it has been stable since first viewed around 1660. The surface of Jupiter consists of belts ofhighly volatile gasses rotating around the planet at extremely high speeds. The friction betweenthe layers and the turbulence created resulted in the formation of the spot. Lorenzian waterwheelis also experimental example of a strange attractor. A.Point attractor B.Limit Cycle C.Limit Tori D. Strange attractor Source: http://psycnet.apa.org/journals/amp/49/1/images/amp_49_1_5_fig2a.gifTry this: • Pick a number, any number • Plug it in for x in .8x + 1 • Take the result and plug that back in for x again in .8x + 1 • Repeat this processAfter some number of iterations, you should notice the list of numbers (the "orbit") convergingon a particular numberStrange attractors are shapes with fractional dimension; they are fractals.2.1.3 FRACTALSFractals are objects that have fractional dimension.A fractal is a mathematical object that is self-similar and chaotic. Fractals are infinitely complex:the closer you look the more detail you see. Fractals are pictures that result from iterations ofnonlinear equations, usually in a feedback loop. Using the output value for the next input value, aset of points is produced. Graphing these points produces images.

Again, by creating a vast number of points using computers to generate those points,mathematicians discovered these wonderfully complex images which were called fractals, a termcoined by Benoit Mandelbrot, one of the first to discover and examine these images.Two important properties of fractals are: self-similarity and fractional dimensions.Self-similarity means that at every level, the fractal image repeats itself. Fractals are shapes orbehaviors that have similar properties at all levels of magnification. Just as the sphere is aconcept that unites raindrops, basketballs and Mars, so fractal is a concept that unites clouds,coastlines, plants and strange attractors. (Look Appendix I for images of various fractals.)Fractals are quite real and incredible; however, what do these have to do with real life? Is there apurpose behind these fascinating images? Fractals make up a large part of the biological world.Clouds, arteries, veins, nerves, parotid gland ducts, and the bronchial tree all show some type offractal organization. In addition, fractals can be found in regional distribution of pulmonaryblood flow, pulmonary alveolar structure, regional myocardial blood flow heterogeneity,surfaces of proteins, mammographic parenchymal pattern as a risk for breast cancer, and in thedistribution of arthropod body lengths.There is a strong link between chaos and fractals. Fractal geometry is the geometry that describesthe chaotic systems we find in nature. Fractals are a language, a way to describe this geometry.Euclidean geometry is a description of lines, circles, triangles, and so on. Fractal geometry isdescribed in algorithms- a set of instructions on how to create the fractal. Computers translate theinstructions into the magnificent patterns, we see as fractal images.Somewhere after Fractals:A scientist by the name of Feigenbaum was looking at the bifurcation diagram again. He waslooking at how fast the bifurcations come. He discovered that they come at a constant rate. Hecalculated it as 4.669. In other words, he discovered the exact scale at which it was self-similar.Make the diagram 4.669 times smaller, and it looks like the next region of bifurcations. Hedecided to look at other equations to see if it was possible to determine a scaling factor for themas well. Much to his surprise, the scaling factor was exactly the same. Not only was thiscomplicated equation displaying regularity, the regularity was exactly the same as a muchsimpler equation. He tried many other functions, and they all produced the same scaling factor,4.669. This was a revolutionary discovery. He had found that a whole class of mathematicalfunctions behaved in the same, predictable way. This universality would help other scientistseasily analyze chaotic equations. Universality gave scientists the first tools to analyze a chaoticsystem. Now they could use a simple equation to predict the outcome of a more complexequation.

2.2 THE BUTTERFLY EFFECTButterfly effect is a way of describing how, unless all factors can be accounted for, large systemslike the weather remain impossible to predict with total accuracy because there are too manyunknown variables to track. It is also called as sensitive dependence on initial condition.Lorenz coined the term and put forward the idea of ‘Butterfly Effect’. His paper was titled as-‘Predictability: Does the Flap of a Butterfly’s wings in Brazil Set Off a Tornado in Texas?’10Intrigued by results he obtained as a mistake in inputs during his computer simulation, Lorenzbegan creating a mathematical explanation that would show the sensitive dependence of large,complex systems like the weather. To simplify his findings, Lorenz coinedthe butterfly explanation.What he meant broadly was the butterfly effect is a way of describing how, unless all factors canbe accounted for, large systems like the weather remain impossible to predict with total accuracybecause there are too many unknown variables to track.The butterfly does not cause the tornado. The flap of the wings is a part of the initial conditions;one set of conditions leads to a typhoon while the other set of conditions does not. The flappingwing represents a small change in the initial condition of the system, which causes a chain ofevents leading to large-scale alterations of events.A better analogy for butterfly effect is an avalanche. It can be provoked with a small input (aloud noise, some burst of wind), its mostly unpredictable, and the resulting energy is huge.As a literary device used to entertain, the concept of Butterfly Effect suggests that our presentconditions can be dramatically altered by the most insignificant change in the past.Chaos theory and butterfly effect are not the same. The Butterfly effect is a symbol of chaos. It isa simple and entertaining way of describing one component of the greater Chaos Theory, namely"Sensitive dependence on initial conditions." Essentially, the Butterfly Effect describes howsmall changes at one point of a nonlinear system can result in larger differences to a later state.Chaos Theory itself is a much larger system of theorems and formulas for predicting andunderstanding the behaviors of complex, nonlinear systems. Saying that the Butterfly Effect isthe same as Chaos Theory is a bit like saying that the cheese and the cheese burger are the samething.Because of the "Butterfly Effect", it is now accepted that weather forecasts can be accurate onlyin the short-term, and that long-term forecasts, even made with the most sophisticated computermethods imaginable, will always be no better than guesses.

The presence of chaotic systems in nature seems to place a limit on our ability to applydeterministic physical laws to predict motions with any degree of certainty. The discovery ofchaos seems to imply that randomness lurks at the core of any deterministic model of theuniverse.11Because of this fact, some scientists have begun to question whether or not it is meaningful at allto say that the universe is deterministic in its behavior. This is an open question which may bepartially answered as science learns more about how chaotic systems operate.

3. ASPECTS OF CHAOS THOERY3.1 PREDICTABILITY OF CHAOSNotion of chaos can be made more precise by seeing its sign, mapping them and observing theconditions under which a system drifts into a chaotic state.Characteristics of a Chaotic System: • No periodic behavior • Sensitivity to initial conditions • Chaotic motion is difficult or impossible to forecast • The motion looks random • Non-linearBecause of the various factors involved in chaotic systems, they are hard to predict. A lot ofcomplicated computations and mathematical equations are involved. But still the prediction mayvary from observed output.Chaos in the atmosphere:The atmosphere is a chaotic system, and as a result, small errors in our estimate of the currentstate can grow to have a major impact on the subsequent forecast. Because of the limited numberof observations available and the uneven spread of these around the globe, there is always someuncertainty in our estimate of the current state of the atmosphere. In practice this limits detailedweather prediction to about a week or so ahead.Accepting the findings from chaos theory about the sensitivity of the prediction to uncertaintiesin the initial conditions, it is becoming common now to run in parallel a set, or ensemble, ofpredictions from different but similar initial conditions. The Ensemble Prediction System (EPS)provides a practical tool for estimating how these small differences could affect the forecast.This weather prediction model is run 51 times from slightly different initial conditions. To takeinto account the effect of uncertainties in the model formulation, each forecast is made usingslightly different model equations. The 51 scenarios can be combined into an average forecast(the ensemble-mean) or into a small number of alternative forecasts (the clusters), or they can beused to compute probabilities of possible future weather events.

3.2 CONTROL OF CHAOS12 The idea of controlling chaos is that when a trajectory approaches ergodically a desiredperiodic orbit embedded in the attractor, one applies small perturbations to stabilize such anorbit. If one switches on the stabilizing perturbations, the trajectory moves to the neighbourhoodof the desired periodic orbit that can now be stabilized. This fact has suggested the idea that thecritical sensitivity of a chaotic system to changes (perturbations) in its initial conditions may be,in fact, very desirable in practical experimental situations. (This is known as Ott, Grebogi, andYorke (OGY) approach of controlling chaos.)There are three ways to control chaos13: 1. Alter organizational parameters so that the range of fluctuations is limited. 2. Apply small perturbations to the chaotic system to try and cause it to organize. 3. Change the relationship between the organization and the environment.Due to the critical dependence on the initial conditions of chaotic systems and due to the factthat, in general, experimental initial conditions are never known perfectly, these systems areintrinsically unpredictable. Indeed, the prediction trajectory emerging from a bonafide initialcondition and the real trajectory emerging from the real initial condition diverge exponentiallyin course of time, so that the error in the prediction (the distance between prediction and realtrajectories) grows exponentially in time, until making the systems real trajectory completelydifferent from the predicted one at long times.For many years, this feature made chaos undesirable, and most experimentalists considered suchcharacteristic as something to be strongly avoided. Besides their critical sensitivity to initialconditions, chaotic systems exhibit two other important properties: • Firstly, there are an infinite number of unstable periodic orbits embedded in the underlying chaotic set. In other words, the skeleton of a chaotic attractor is a collection of an infinite number of periodic orbits, each one being unstable. • Secondly, the dynamics in the chaotic attractor is ergodic, which implies that during its temporal evolution the system ergodically visits small neighbourhood of every point in each one of the unstable periodic orbits embedded within the chaotic attractor.A relevant consequence of these properties is that a chaotic dynamics can be seen as shadowingsome periodic behavior at a given time, and erratically jumping from one to another periodicorbit. Indeed, if it is true that a small perturbation can give rise to a very large response in thecourse of time, it is also true that a judicious choice of such a perturbation can direct thetrajectory to wherever one wants in the attractor, and to produce a series of desired dynamicalstates. This is the idea of control of chaos.

The important point here is that, because of chaos, one is able to produce an infinite number ofdesired dynamical behaviors (either periodic or not periodic) using the same chaotic system, withthe only help of tiny perturbations chosen properly. We stress that this is not the case for a non-chaotic dynamics, wherein the perturbations to be done for producing a desired behavior must, ingeneral, be of the same order of magnitude as the unperturbed evolution of the dynamicalvariables.There are two methods used in Control of Chaos, namely: - Ott-Grebogi-Yorke Method Calculated swift, tiny perturbations are applied to the system once every cycle. - Pyragas Method A continuous controlling signal is injected into the system which approaches zero as the system reaches the desired orbitThe applications of controlling chaos are enormous, ranging from the control of turbulent flows,to the parallel signal transmission and computation to the parallel coding-decoding procedure, tothe control of cardiac fibrillation, and so forth.3.3 SYNCHRONIZATION OF CHAOSChaos has long-term unpredictable behavior. This is usually couched mathematically assensitivity to initial conditions—where the system’s dynamics takes it is hard to predict from thestarting point. Although a chaotic system can have a pattern (an attractor) in state space,determining where on the attractor the system is at a distant, future time given its position in thepast is a problem that becomes exponentially harder as time passes.Can we force two chaotic systems to follow the same path on the attractor? Perhaps we could‘lock’ one to the other and thereby cause their synchronization? The answer is, yes.Why would we want to do this? The noise-like behavior of chaotic systems suggested early onthat such behavior might be useful in some type of private communications. One glance at theFourier spectrum from a chaotic system will suggest the same. There are typically no dominantpeaks, no special frequencies. The spectrum is broadband. There have been suggestions to usechaos in robotics or biological implants. If we have several parts that we would like to acttogether, although chaotically, we are again led to the synchronization of chaos.There is identical synchronization in any system, chaotic or not, if the motion is continuallyconfined to a hyperplane in phase space. The most general and minimal condition for stability, isto have the Lyapunov exponents associated with be negative for the transverse subsystem. TwoLorenz systems can be synchronized together using CR (Complete Replacement) technique.14

Systems with more than one positive Lyapunov exponent, called hyperchaotic systems, can besynchronized using one drive signal.Pecora & Carroll method of synchronization:15 Synchronization of chaos can be achieved by introducing a coupling term between two chaoticsystems and providing a controlling feedback in one that will eventually cause its trajectory toconverge to that of the other and then remain synchronized with it.The method of Pecora and Carroll is to split an autonomous dynamical systemu = f(u) into a drive system v = g(v; w) and a response subsystem = h(v; w). One then makes acopy of the response subsystem ’= h(v; w) and asks under which conditions we can expect= ’– 0 as tLasers that behave chaotically could be synchronized. Two solid state lasers can couple throughoverlapping electromagnetic lasing fields. The coupling is similar to mutual coupling except it isnegative. This causes the lasers to actually be in oppositely signed states. Such lasersynchronization opens the way for potential uses in fiber optics.1617 Synchronization of chaos may be used to build a means of secure communication over a publicchannel.

4. APPLICATIONS OF CHAOS THEORY4.1 CHAOS THEORY IN STOCK MARKETThe science of chaos supplies us with a new and provocative paradigm to view the markets andprovides a more accurate and predictable way to analyze the current and future action of acommodity or stock. It gives us a better map with which to trade. It does not depend onconstructing a template from the past and applying it to the future. But it concentrates on thecurrent market behavior, which is simply a composite of (and is quite similar to) the individualFractal behavior of the mass of traders. Chaos analysis has determined that market prices arehighly random, but with a trend. The amount of the trend varies from market to market and fromtime frame to time frame.The price movements that take place over the period of several minutes will resemble pricemovements that take place over the period of several years. In theory, big market crashes shouldnever happen. But Mandelbrot predicts that a market crash should occur about once a decade.Given the fact that weve had major crashes in 1987, 1998 and 2008 - roughly once a decade - itsclear that Mandelbrot made a pretty good prediction.The new Fractal Market Hypothesis, based on Chaos Theory explains the phenomena in financialbranch, which the Efficient Market Hypothesis could not deal with. In the hypothesis, Hurstexponent determines the rate of chaos and distinguished fractal from random time series.Lyapunov exponent determines the rate of predictability. A positive Lyapunov exponentindicates chaos and it sets the time scale which makes the state of prediction possible. Plottingstock market variations and matching them with chaotic analyses of above exponents, one mightpredict future behavior of market.

4.2 POPULATION DYNAMICSAnother concept where chaos theory can be applied is the prediction of biological populations.The simplest equation that takes this into account is the following:Next years population = R * this years population * (1 - this years population)In this equation, the population is a number between 0 and 1, where 1 represents the maximumpossible population and 0 represents extinction. R is the growth rate determined by two factors:the rate of reproduction, and the rate of death from old age. The question is, how does thisparameter affect the equation?One biologist, Robert May, decided to see what would happen to the equation as the growth ratevalue changes18. At low values of the growth rate, the population would settle down to a singlenumber. For instance, if the growth rate value is 2.7, the population will settle down to .6292. Asthe growth rate increases, the final population would increase as well. Then, something weirdhappens.As soon as the growth rate passes 3, the line breaks in two. Instead of settling down to a singlepopulation, it would jump between two different populations. It would be one value for one year,go to another value the next year, then repeat the cycle forever. Raising the growth rate a littlemore would cause it to jump between four different values. As the parameter rose further, theline bifurcates (doubles) again. The bifurcations came faster and faster until suddenly, chaosappears. Past a certain growth rate, it becomes impossible to predict the behavior of the equation.However, upon closer inspection, it is possible to see white strips. Looking closer at these stripsreveals little windows of order, where the equation goes through the bifurcations again beforereturning to chaos.Population biology illustrates the deep structure that underlies the apparent confusion in thesurface behavior of chaotic systems. Some animal populations exhibit a boom-and-bust patternin their numbers over a period of years. This boom-and-bust pattern has been seen elsewhere,including disease epidemics.

Bifurcation graph showing chaotic behavior of population Source: http://mathewpeet.org/science/chaos/chaos.jpg4.3 BIOLOGYChaos theory can also be applied to human biological rhythms. The human body is governed bythe rhythmical movements of many dynamical systems: the beating heart, the regular cycle ofinhaling and exhaling air that makes up breathing, the circadian rhythm of waking and sleeping,the saccadic (jumping) movements of the eye that allow us to focus and process images in thevisual field, the regularities and irregularities in the brain waves of mentally healthy andmentally impaired people as represented on electroencephalograms. None of these dynamicsystems are perfect all the time, and when a period of chaotic behavior occurs, it is notnecessarily bad. Healthy hearts often exhibit brief chaotic fluctuations, and sick hearts can haveregular rhythms. Applying chaos theory to these human dynamic systems provides informationabout how to reduce sleep disorders, heart disease, and mental disease.

4.3.1 PREDICTING HEART ATTACKS19 Science Daily, in its July 23, 2010 article reported that Chaos models may someday help modelcardiac arrhythmias -- abnormal electrical rhythms of the heart, according to researchers in thejournal Chaos, published by the American Institute of Physics. A space-time plot of the alternans along a cardiac fiber is a solution to the Echebarria-Karma equation. Source: http://images.sciencedaily.com/2010/07/100721145105-large.jpgIn recent years, medical research has drawn more attention to chaos in cardiac dynamics.Although chaos marks the disorder of a dynamical system, locating the origin of chaos andwatching it develop might allow researchers to predict, and maybe even counteract, certainoutcomes.An important example is the chaotic behavior of ventricular fibrillation, a severely abnormalheart rhythm that is often life-threatening. One study found chaos in two and three dimensions inthe breakup of spiral and scroll waves, thought to be precursors of cardiac fibrillation. Anotherstudy found that one type of heartbeat irregularity, a sudden response of the heart to rapidbeating called "spatially discordant alternans," leads to chaotic behavior and thus is a possiblepredictor of a fatal heart attack.Assigning extreme parameter values to the model, the team was able to find chaotic behavior inspace over time. The resulting chaos may have a unique origin, which has not yet beenidentified.20

4.4 REAL TIME APPLICATIONS4.4.1 CHAOS TO PRODUCE MUSIC21 In 1995, Diana S. Dabby, An Associate Professor of Electrical Engineering and Music in OlinCollege, applied Chaotic Mapping to generate musical variations. The goal was to inspirecomposers from the generated ideas.A chaotic mapping provides a technique for generating musical variations of an original work.This technique, based on the sensitivity of chaotic trajectories to initial conditions, produceschanges in the pitch sequence of a piece. A sequence of musical pitches {pi, is paired with thex-components {xi} of a Lorenz chaotic trajectory. In this way, the x axis becomes a pitch axisconfigured according to the notes of the original composition. Then, a second chaotic trajectory,whose initial condition differs from the first, is launched. Its x-components trigger pitches on thepitch axis (via the mapping) that vary in sequence from the original work, thus creating avariation. There are virtually an unlimited number of variations possible, many appealing toexperts and others alike.The technique’s success with a highly context-dependent application such as music22, indicates itmay prove applicable to other sequences of context dependent symbols, e.g., DNA or proteinsequences, pixel sequences from scanned art work, word sequences from prose or poetry,textural sequences requiring some intrinsic variation, and so on.4.4.2 CLIMBIMG WITH CHAOS23 In 2009, Caleb Phillips, a Colorado Graduate and a rock climbing enthusiast, thought thatmathematics could help him use computers to design new climbing routes. Because in indoorrock climbing, the real challenge is to scale the wall using only some of the hand and foot holds,following a specific route.He made a computer program, which he named Strange Beta, which starts with an existingclimbing route, alters some climbing moves, like a left sidepull or a right match, and gives aresulting varied route plan, using a chaotic variation generator and a fourth order Runge-Kuttanumerical integrator. It was discovered that computer aided route setting can produce routeswhich climbers prefer to those set traditionally.This proved that a chaotic systems like this can be described mathematically and computers canproduce some seemingly random results, which can be used to approximate human creativity.

4.5 RANDOM NUMBER GENERATION24 The programming community is majorly interested in design and maintenance of standardpseudo-random number generators that are portable and reusable. The pseudo-random numbergenerators, familiar to all programmers, are derived from deterministic chaotic dynamicalsystems. As we shall see, when pseudo-random number generators are designed properly, thesequences produced are completely predictable.Random number generation has been of great interest from the beginnings of computing.Philosophically and mathematically, the concept of randomness poses many problems.Intuitively, we equate randomness with unpredictability.In the past 25 years, largely due to the separate efforts of Chaitin and Kolmogorov the concept ofrandomness has been made definitive. They accomplished this by developing the concept ofalgorithmic complexity. The complexity of an n-digit sequence is the length in bits of theshortest computer program that can produce the sequence. For a very regular sequence,1111111111, for example, a very small program is needed. As the sequence becomes highlyirregular and as the length of the sequence grows beyond bounds, it can be shown that theshortest program needed to produce the sequence is slightly larger than the sequence itself.Clearly, any algorithmic implementation of the theoretical ideal of randomness on a computerwill be imperfect. From all view points above, a random sequence is a non-computable infinitesequence. A measure of the "goodness" of a pseudo-random number generator is aperiodicity.However, any finite computer algorithm implementation yields only periodic sequences--although of very long period. Thus, random number generators found in computer languages arereferred to as "pseudo-random" number generators.In 1951, Lehmer proposed an algorithm that has become the de facto standard pseudo-randomnumber generator. As it is usually implemented, the algorithm is known as a Prime ModulusMultiplicative Linear Congruential Generator. It is better known as a Lehmer generator. Theform of the Lehmer generator is: f(xn) = xn+1 = a xn mod m.Park and Miller proposed a portable minimal standard Lehmer generator. Their choice of m, theprime modulus, is 231 - 1, and their choice of the multiplier is a = 75. Their minimal standardLehmer generator is given by: f(x) = 16807 x mod 2147483647.Qualitatively the behavior of chaotic systems is non-periodic, greatly disordered, deterministic--yet apparently unpredictable and random. In current usage, a system is chaotic if it has sensitivedependence on initial conditions. Associated with this sensitive dependence is a geometricgrowth in small differences with time.

The simplest chaotic dynamical system is the Bernoulli shift described by Palmore: Xn+1 = D xn mod 1,where D is an integer larger than 1.The Bernoulli shift is a simple algorithm that exhibits complex chaotic behavior. It isdeterministic in that it is a specific algorithm implemented by a definite set of instructions in acomputer language on a computer with finite precision. It is characterized by exponential growthand disorder.By inspection, it is similar in form to the Lehmer generator. With the example of the Bernoullishift above, we observe that prime modulus multiplicative linear congruential generators areimplementations of deterministic chaotic processes. They are Bernoulli shifts on integers. Themultiplier a and the modulus m are chosen with great care to ensure that a full period isproduced. A specific implementation of a full period generator is a single cycle system. Seedingstarts the generator at a given point on the cycle--the entire sequence will be produced if thegenerator is called m times. This algorithm depends sensitively on the choice of multiplier andmodulus.

5. LIMITATIONS OF CHAOS THEORYThe major and most significant limitation of chaos theory is the feature that defines it: sensitivedependence on initial conditions.The limitations of applying chaos theory are in due mostly from choosing the input parameters.The methods chosen to compute these parameters depend on the dynamics underlying the dataand on the kind of analysis intended, which is in most cases highly complex and not alwaysaccurate. Theoretical chaos results are seriously constrained by the need for large amounts ofpreliminary data.Chaos theory in its current form is limited. At its present stage of development, it can be used toask if experimental data were generated by a random or deterministic process, but it is a difficultand frustrating analytical approach to use. It is not clear how much data are required in order toconstruct the phase space set and determine its fractal dimension; the amounts of data are likelyto be extremely large, and biological systems may not remain in a single state long enough togather the required amounts of data. Present technology too, while managing calculations anddata, also falls short at some level.Because this theory is still developing, many ideas continue to emerge, and hence concepts areredefined or complemented continuously. Scientists are trying to link chaos theory with otherscientific disciplines to establish a more general theory.Some systems do not seem to benefit from the results of chaos theory in general. Chaos will notappear in slow systems, i.e., where events are infrequent or where a great deal of frictiondissipates energy and damps out disturbances.One may encounter scenarios and systems with erratic behavior where a source of chaos is notimmediately evident. In this event, it may be necessary to examine different scales of behavior.For example, chaos theory does not help study the flight of a single bird, free to choose whereand when to fly. However, there is evidence of chaos in how groups of birds flock and traveltogether.Even though chaos theory helps us in taking better decisions and designing new strategies forfuture, we cannot completely rely on its outcomes, else this habit of short-term observe-and-manage would put organization’s long term future in danger. Chaos theory is not as simplistic tofind an immediate and direct application in the business environment, but mapping of thebusiness environment using the knowledge of chaos theory definitely is worthwhile.A new statistical hypothesis testing must be designed in near future to analyze the ever growingchaotic patterns and analyze subtle changes in them.

6. CONCLUSIONIt’s well known that the heart has to be largely regular or you die. But the brain has to be largelyirregular; if not, you have epilepsy. This shows that irregularity, chaos, leads to complexsystems. It’s not all disorder. On the contrary, I would say chaos is what makes life andintelligence possible.- Ilya PrigogineThe natural world has always had a chaotic way about it. We can find chaos theory everywherearound us: in simple pendulum, stock market, solar system, weather forecasting, imageprocessing, biological systems, human body and so on. Chaotic systems are deterministic innature but they may appear to be random. Chaotic systems are very sensitive to the initialconditions which means that a slight change in the starting point can lead to enormously differentoutcomes. This makes the system fairly unpredictable. Chaos systems never repeat but theyalways have some order. Most of the systems we find in the world predicted by classical physicsare the exceptions but in this world of order, chaos rules. Chaos theory is a new way of thinkingabout what we have. It gives us a new concept of measurements and scales. It looks at theuniverse in an entirely different way. Understanding chaos is understanding life as we know it.Because of chaos, it is realized that even simple systems may give rise to and, hence, be used asmodels for complex behavior.Chaos forms a bridge between different fields. Chaos offers a fresh way to proceed withobservational data, especially those data which may be ignored because they proved too erratic.Just as relativity eliminated the Newtonian illusion of absolute space and time, and as quantumtheory eliminated the Newtonian and Einsteinian dream of a controllable measurementprocess, chaos eliminates the Laplacian fantasy of long-time deterministic predictability. Thatis the reason why chaos theory has been seen as potentially “one of the three greatest triumphs ofthe 21st century.” In 1991, James Marti speculated that ‘Chaos might be the new world order.’And it definitely might.

APPENDIX I – Other Applications of Chaos TheoryAPPLICATION OF CHAOS THEORY IN NEGOTIATIONSRichard Halpern1, in an article published in 2008, summarizes impact of Chaos Theory andHeisenberg Uncertainty Principle on case negotiations in law, as follows- (1) Never rely on someone elses measurement to formulate a key component of strategy. A small mistake can cause huge repercussions (SDIC). Hence better do it yourself. (2) Keep trying something new, unexpected; sweep the defence of its feet. Make the system chaotic. (3) If the process is going the way you wanted, simplify it as much as possible. Predictability would increase and chance of blunders is minimized. (4) If the tide is running against you, add new elements: complicate. Nothing to lose, and with a little help from Chaos, everything to gain. You might turn a hopeless case into a winner.CHAOS THEORY IN PHILOSOPHYYou can stay at home and be happily introspective or you can make a choice, step out, and be theButterfly that begins the tempest that changes the world. - John SanfordIn this infinite world, amongst billions of people, we, alone, are just one person. One might thinknobody would care what we think. But our intentions and actions matter, because of the ButterflyEffect. We are profoundly affected by the world around us, but sometimes the reverse is true aswell. If we keep the intention to save all beings from suffering, that influences our actions.Because this universe we find ourselves in is most certainly a complex system, theres no tellingwhat effect those intentions and actions might have. They will, however, most certainly have aneffect on the system... small or large. Everything we think, everything we do, is significant. Onlyby looking backward, can we identify those events that would later prove to have“revolutionary” consequences.2When we realize that every action matters, then there are no wasted moments, no insignificantgestures. Each moment is filled with portent, and possibilities. Maybe the next thing you do willtip the scales, and the world will go one way rather than another. We just dont know.What a responsibility! ! "# $ " %&& (! " ( & & & !) ( *

CHAOS THEORY IN DISASTER MANAGEMENTDisaster response organizations are dynamic systems. A dynamical system consists of two parts:a rule or dynamic, which specifies how a system evolves, and an initial condition or state fromwhich the system starts. Some dynamical systems evolve in exceedingly complex ways, beingirregular and initially appearing to defy any rule. Chaos theory gives a way to analyze suchsystems and develop disaster management techniques.The interaction of the workplace rules of motion with the field of action determines the directionand result of motion in the workplace. By applying the logistic equation (Xn+1 = kXn(1-Xn)) to theappropriate disaster response data, it is possible to determine if a disaster organization orresponse system traces the universal route to chaos. Recent work in evolutionary theory andsimulation studies supports the view that organizations at the edge of chaos tend to be highlyadaptive and hence succeed.The application of chaos theory to disaster management has some preliminary empirical support.The terms make metaphorical sense and seem to be consistent with disaster managersexperiences, hence leading to more effective models.3CHAOS THEORY IN ARTComplexity and self-organization emerge from disorder as the result of a simple process, givingrise to exquisite patterns. + , - " * . / * $ " - 0) 1(2 $ ,

a) Pattern formed by the vibration of sand on a metal plate (Source: http://www.mi.sanu.ac.rs/vismath/jaynew/fig01a.jpg) b) Vibration of a thin film of glycerine, From Cymatics by Hans Jenny (Source: http://www.mi.sanu.ac.rs/vismath/jaynew/fig01b.jpg) Van Goghs painting, "Starry Night", Fractals in art Source: http://www.mi.sanu.ac.rs/vismath/jaynew/fig12.jpgMandelbrot and Julia sets, Lyapunov diagrams also produce beautiful visuals. Computergraphics generated with equations obtained through Chaos Theory also breed brilliant pieces ofmachine-made art.

CHAOS THEORY IN POPULAR CULTUREChaos theory has been mentioned in numerous movies and works of literature. For instance, itwas mentioned extensively in Michael Crichton’s novel Jurassic Park and more briefly in itssequel, The Lost World. Other examples include the film Chaos, The Science of Sleep, TheButterfly Effect, the sitcom The Big Bang Theory and Community, Tom Stoppards play Arcadia,Lawrence Sterne’s novel Tristram Shandy, Pfitz by Andrew Crumey, Ray Bradbury’s short storyA Sound Of Thunder and the video games Tom Clancys Splinter Cell: Chaos Theory andAssassins Creed.The influence of chaos theory in shaping the popular understanding of the world we live in wasthe subject of the BBC documentary High Anxieties — The Mathematics of Chaos directed byDavid Malone. Chaos theory was also the subject of discussion in the BBC documentary "TheSecret Life of Chaos" presented by the physicist Jim Al-Khalil. Indian movie “Dasavathaaram”also used idea of butterfly effect where the 2004 Tsunami in the Indian Ocean is portrayed asbeing caused by the sinking of Lord Vishnu’s idol along with Kamal Hassan in the 12th century.4SOLAR SYSTEM CHAOS Astronomers and cosmologists have known for quite some time that the solar system does notrun with the precision of a Swiss watch. Inabilities occur in the motions of Saturns moonHyperion, gaps in the asteroid belt between Mars and Jupiter, and in the orbit of the planetsthemselves. For centuries astronomers tried to compare the solar system to a gigantic clockaround the sun; however, they found that their equations never actually predicted the real planetsmovement.5Solar system represents a typical 3-body system. The vectors become infinite and the systembecomes chaotic. This prevents a definitive analytical solution to the equations of motion.Extreme sensitivity to initial conditions is quantified by the exponential divergence of nearbyorbits.The recent advances are the beginning of a quest to tease out the critical properties of our solarsystem (and its subsystems) that give it the curious character of being only marginally chaotic ormarginally stable on time spans comparable with its current age. It is but a part of the quest tounderstand what processes of formation (and perhaps initial conditions) led to this remarkablesystem in nature and how common such systems are in our galaxy and the universe.6 " 3 * 0 ) " 4 * !" !# $ % - ! * 5 1 * 6

APPENDIX II – EXAMPLESExamples of Fractals:Sierpinskis Triangle demonstrates this quite well: a triangle within smaller triangles withinsmaller triangles within ever smaller triangles, on and on. Many shapes in nature display thissame quality of self-similarity. Clouds, ferns, coastlines, mountains, etc. all possess this feature. Sierpenski Triangle and its fractal dimensions Source: http://eldar.mathstat.uoguelph.ca/dashlock/ftax/Gallery/Siepinski1D960.gifThe Koch Snowflake is also a well noted, simple fractal image. To construct a Koch Snowflake,begin with a triangle with sides of length 1. At the middle of each side, add a new triangle one-third the size; and repeat this process for an infinite amount of iterations. The length of theboundary is 3 X 4/3 X 4/3 X 4/3...-infinity. However, the area remains less than the area of acircle drawn around the original triangle. What this means is that an infinitely long linesurrounds a finite area.

Koch Snowflake Source: http://www.zeuscat.com/andrew/chaos/vonkoch.htmlLORENZIAN WATER WHEELTo check the equations that he had derived from the weather model, Lorenz created a thoughtexperiment. He used a waterwheel, with a set number of buckets, usually more than seven,spaced equally around its rim. The buckets are mounted on swivels, much like Ferris-wheelseats, so that the buckets will always open upwards. At the bottom of each bucket is a small hole.The entire waterwheel system is than mounted under a waterspout. A slow, constant stream of water is propelled from the waterspout. The waterwheel would beginto spin at a fairly constant rate. Lorenz decided to increase the flow of water, and, as predicted inhis Lorenz Attractor, an interesting phenomenon arose. The increased velocity of the waterresulted in a chaotic motion for the waterwheel. The waterwheel would revolve in one directionas before, but then it would suddenly jerk about and revolve in the opposite direction. The fillingand emptying of the buckets was no longer synchronized; the system was now chaotic. Lorenzobserved his mysterious waterwheel for hours, and, no matter how long he recorded the positionsand contents of the buckets, he could not find a specific relation binding the movements. Thewaterwheel would continue on in chaotic behavior without ever repeating any of its previousconditions. A graph of the waterwheel would resemble the Lorenz Attractor.7 7 ) )! 3 8 ". 8 8 * % !! & ( )* + , %

LIST OF REFERENCES ! - . / / 4( 9 : 4( 9 . 6 *! , )** 5: ;3 ) ) ! ! # !/ " ! . . < % 1 1 . / ! . / ! / .% = ". + " " 1( " 0) " >; (?@ A " /) . " / .!(( " .!(( . (9 B ;) * $ C 6 * )! . 1 " > ?( ( ! / /( 0 / / / /( . ; 8 $ )!* * " 1* 1 1 3 *6 ! 0 ! / 0 6( 9 ( % 1 . / ! . / ! / . .( , )" 2 > ?( - " " * D 0 3 * (6 / %4 9 $ /) * *1* A ) $ ( E6 > ? F 6 0 . : G 1 % ( / !# /$ ! / 1 - . ( 2( 6 !) " 4 ( HH;D * II$ ( 3( . ( > ? < 6 ) **) % ! / / / 2 - -- 23 .! . - ! 9 " $ ) 8 5 3 "0 %6 ! $ 6 ! 6 5 86 G ( 5 ( $ 1 :J / - 3 - 3 -- - % H # $ , I / - - 2 - % , !! , B-) G * - "C > ?( 1 )* 6 6 ; ) ($ ( / / ! ! - B9 % G * C 6 5 8 , ( $ ! 9 ; ! B6 " 9 % G A : *! " ) 6 "C > ? , * * ) 6 # 3 9 ) ( ! / ! / 4 5- 3 . * 5)*! 0 1 0 ! ! / & - - - 3- 2 36 -6 .% >1 : ) " 1)") 6 *! ?(

BIBLIOGRAPHY1. Gleick, James, Chaos – Making a New Science, London: Cardinal, 19872. Stewart, Ian, Does God Play Dice? The Mathematics of Chaos, London: Penguin, 19903. Howe, Vernon, Chaos: A new mathematical paradigm, International Faith and Learning Seminar, June 19944. PhD thesis on Chaos Theory as a Literary Science, indolentdandy.net/phd, 20105. Prigogine, Ilya and Stengers, Isabelle, Order Out of Chaos: Man’s New Dialogue with Nature, London: Flamingo, 19846. Li, Zhong, et al, Integration of Fuzzy Logic and Chaos Theory, 20067. Araújo, V., Deterministic non-periodic flow, XXXVI Semana de la Matemática, PUCV, Chile, 20098. Zeng, Xubin, Chaos Theory and its Application in the Atmosphere, Department of Atmospheric Science, Colorado State University, Paper No.5049. Blesser, Barry, Chaos Theory: Limits of Analysis and Predictions, 200610. Lorenz, Edward, Deterministic Nonperiodic flow, Journal of Atmospheric Sciences, Jan 7,196311. Eckman and Ruelle, Ergodic Theory of Chaos and Strange Attractors, The American Physical Society, 198512. Chaos Thoery: A new Paradigm for Trading, www.traderslibrary.com/chapters/1546201.pdf13. Townsend, James, Chaos Theory: A Brief Tutorial and Discussion, Indiana University, 199214. Kiel and Elliott, Chaos Theory in the Social Sciences: Foundations and Applications, Ann Arbor: The University of Michigan Press, 199615. Kellert, Stephen, In the Wake of Chaos: Unpredictable Order in Dynamical Systems, Chicago: The University of Chicago Press, 199316. Peckham, Morse, Man’s Rage for Chaos: Biology, Behavior and the Arts, New York: Schocken Books, 196717. Nicolis, John, Chaos and Information Processing: A Heuristic Outline, Singapore: World Scientific, 199118. Gulick, Denny, Encounters with Chaos, New York: McGraw-Hill, 199219. Bono, James, Science, Discourse, and Literature: The Role/Rule of Metaphor in Science, 199020. Richard Wright, Art and Science in Chaos: Contesting Readings of Scientific Visualisation, ISEA Conference, 199421. Lewin, Roger, Complexity: Life at the Edge of Chaos, New York: Macmillan, 199222. Lorenz, Edward, The Essence of Chaos, London: UCL Press, 199323. Briggs and Peat, Turbulent Mirror: An Illustrated Guide to Chaos Theory and the Science of Wholeness, New York: Harper and Row, 1989

Chaos Theory: An Introduction

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Chaos Theory: An Introduction Document Transcript