Extended Essay Draft 1

Extended Essay “How have Graphical Interpretations and Formulas in Chaos Theory Have Impacted Science”

Lincoln High School 2996 Elias Mueller Candidate 002996-025

May 2012

Group: 5 Mathematics Word Count: 2,110

How Have Graphical Interpretations and Formulas in Chaos Theory Impacted Science

Elias Mueller 002996-025

Abstract:

This extended essay explores to what extent graphical interpretations, and formulas (mathematics) in Chaos Theory have impacted science. I begin by defining Chaos Theory and what is included within the study, and how the Lorenz Attractor (which is an example named after the discoverer of Chaos Theory, Edward N. Lorenz) brought about this fascinating blend of the branches of Mathematics and Science. Also, I explain Lorenz‟s initial discoveries and what an equation and graph that fits in with Chaos Theory would look like. I further these interpretations by examining the Mandelbrot Set named for its discoverer, Benoît Mandelbrot. After I investigate Butterfly Effect further and its implications on experiments and studies. Furthermore, I am investigating how technology in mathematics and science has affected Chaos Theory and its studiers.

Word Count: 128

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Table of Contents:

Abstract: ................................................. 1 Table of Contents: .................................. 2 Introduction: .......................................... 3 Body: ...................................................... 5 Conclusion ............................................ 12 Work Cited ........................................... 13

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Introduction: Fig. 1 is the “Lorenz Attractor.” In mathematics an attractor is a set which over time develops into a dynamical system. A dynamical system is a concept in mathematics where fixed rules describe the dependency of time for a point in space. The “Lorenz Attractor” specifically is “an attractor that arises in a simplified system of equations describing the two-dimensional flow of

Figure 1

fluid of uniform depth Η , with an imposed temperature difference ΔΤ , under gravity g, with buoyancy α, thermal diffusivity κ, and kinematic viscosity υ” ( Lorenz Attractor, 1999). Edward Lorenz was the meteorologist who discovered this phenomenon and subsequently, his findings were named after him. Meteorology is branch of science, and much like physics, it is heavy in mathematics. As found in the parameters of the attractor the scientific recordings and utilized in a mathematical operation to determine the specific outcome. In 1963 Lorenz released a paper explaining his findings. He went to examine a set which he had viewed before, this time instead of using his usual precision in numbers to start the sequence he skipped to add in the millionth place value, when he returned and the program was finished he was astonished to find the end results were significantly different than before, which he then realized the concept of sensitive dependence on initial conditions (Williams, 1997, p. 18) Sensitive dependence on initial conditions refers to the idea that slight changes in the initial setting of an environment can lead to drastic changes, this is paralleled with the Butterfly Effect. The Butterfly Effect and Lorenz Attractor are the center of modern studies in Chaos theory. Both show that slight changes have 3



significant effects which help with all fields of study because we should be more cautious on the conclusions we make after an experiment unless we have screened the results through slight changes and found a happy medium in the data. This is helpful for life in general because it proves to the person that small changes in the start of their day can significantly impact the end of their day. The mathematics behind the theory is the part that allows examining of how the different initial conditions can affect the outcome through graphically modeling. Chaos theory is the study of the behavior of dynamical systems , and “chaos happens only in deterministic, nonlinear, dynamical systems” (Williams, 1997, p. 9). Essentially chaos is defined by three main elements, being that the system (group of things that function together) is deterministic, nonlinear, and dynamical. A deterministic system is one which “follows a rule” (Williams, 1997, p. 5), “„Nonlinearity‟ refers to something that is „not linear‟” (Beyerchen, 1993, para. 6), and dynamical “is anything that moves, changes, or evolves in time” (Williams, 1997, p . 11). In conclusion, a chaotic system is ordered, and changing at a non-constant rate: The basis of Chaos Theory. Mathematical interpretations have heavily impacted science through Chaos Theory.

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Body: “Where chaos begins, classical science stops.” (Gleick, 1987, p. 3)

Figure 2

Chaos is not typical science. For students science is typically Biology a course where mathematics is used once in awhile to prove the findings. However, overall the view of science to the general public is that science is very qualitative. Chaos Theory is a quantitative study, much like physics, mathematical formulas and interpretations are used in the technique to determine possible results. Therefore, when Chaos Theory is applied scientifically it is generally used as a physics aspect; however, the theory can be used as a guideline for all other areas because of Butterfly effect. Butterfly E ffect refers to “sensitive dependence on initial conditions” (Kellert, 1993, p.12). In simpler terms, how a dynamic system starts will affect the outcome. The most common example of Butterfly Effect is “the flap of a butterfly‟s wing could influence the course of a typhoon on the other side of the world” (Pritchard, 1992, p.28). Alterations that seem insignificant can have tremendous affect on the outcome of an event. Lorenz discovered this phenomenon while examining a sequence over to ensure accuracy, and in order to save time he began the sequence in the middle, but instead of using his usual precision of 0.506127, he input 0.506; the outcome was vastly different ( Chaos Theory: A Brief Introduction, n.d). The changes experienced with Butterfly effect are referred to as noise. 5



Figure 3

This is a generic version of the outcome. For the first quarter of the graph it can be seen that the data set ran the same; however, as the graph progresses it is seen that the data is significantly different. Meteorology is the scientific study of the atmosphere and is applied to the real world with weather forecasting. Lorenz‟s findings were based in the mathematical aspects of his field. Specifically the graphical interpretation at the end, but he later wrote an artic le, “Deterministic Nonperiodic Flow,” in the Journal of the Atmospheric Sciences which furthered his original findings. He states, “In this study we shall work with systems of deterministic equations which are idealization of hydro-dynamical systems. We shall be interested principally in nonperiodic solutions, i.e., solutions which never repeat their past history exactly, and where all approximate repetitions are of finite duration” (Lorenz, 1962, p. 130). He began by investigating a system whose state is described as M and the system goes from X 1 to XM:

Figure 4

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where t is the single independent variable, and the functions F i possess continuous first partial derivatives. Such a system may be studied by means of phase space — an Mdimensional Euclidean space T whose coordinates are X 1 ∙ ∙ ∙ XM. Each point in phase space represents a possible instantaneous state of the system. A state which is varying in accordance with (1) is represented by a moving particle in phase space, traveling along a trajectory in phase space. For completeness, the position of a stationary particle, representing a steady state, is included as a trajectory (Lorenz, 1963, p. 131). This is the equation which Lorenz began with for his explanation. It basically analyzes a system from 1 to M where M

ε

+

Z and M is the endpoint. Furthermore, the system‟s values are input in

the function Fi and in a three dimensional space. In the end, the equation will generate a three dimensional graph with the variable on the x-axis being 1 to M and it maps the trajectory of a particle moving through the space. However he transforms the equation using the theory of differential equations to create:

Figure 5

Then to be:

Figure 6

This equation starts at X 10 in the system to X MO and t ; however, the system is now run through f i, which is still continuous (and the parameters are still i from 1 to M). These are the equations to

which Chaos Theory utilizes, deterministic, nonlinear, and dynamical. They are constantly changing throughout three dimensional spaces, they do not fit a pattern, and they follow a rule 7



(the F(x) function style). Overall Lorenz‟s work with meteorology was heavily influenced by Chaos Theory and the mathematical used in the theory. Chaotic graphs can be standard two dimensional graphs, Euclidean graphs (three dimensional), or fractals. The two dimensional graph is similar to Lorenz‟s on page six of this essay. A Euclidean graph is one which in three dimensional space like Fig. 6, and a fractal is a chaotic graph that is an irregular geometric shape similar Fig 7. Figure 7

Figure 8

This fractal (Fig. 7) is commonly known as the Mandelbrot Set. The Mandelbrot Set is obtained for the quadratic recurrence

equation: where C is equal to z o (the initial

z value) and the function z n+1 does not go to infinity ( Mandelbrot Set , 1999). For example: if i were input for C the set would go, 0, i, (-1 + i), -i, (-1 + i),-i therefore the set is finite and is part of the Mandelbrot set. However, if C were to equal 0 the set would go 0, 1, 2, 5…and on to

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infinity therefore it is not part of the Mandelbrot Set because the set continually compounds on itself. The function is named for the man who popularized it, Benoît Mandelbrot, and it is commonly referred to in math textbooks as the M set (Pritchard, 1992, p. 192). The M set is a common fractal studied by Chaos Theory enthusiasts. The set effectively demonstrates a function that does not fit a linear pattern, but does follow a rule and is constantly changing, fitting the definition of chaotic equation! The M set is an example of an iterated function (a function which is made by its own properties), which is generally a dynamic system. Therefore, it exists as an example for explaining what chaotic functions look like, and how they operate on a complex plane. The mathematics behind the M set function as a general model for Chaos Theory.

“Chaos breaks across the lines that separate scientific disciplines” (Gleick, 1987, p. 5)

Chaos Theory is not only applicable to mathematics. Butterfly Effect (which was covered on p. 5 of this essay) is an important aspect to know about. In review Butterfly Effect is a “sensitive dependence on initial conditions.” It is important for scientists to understand that however the data set is began can affect the outcome. Take into account Lorenz‟s findings. By simply removing 0.000127 from his standard precision his data set began to flow the same but ended completely different. Through this understanding the science community can see that precision in data measurements should not be neglected. However, being over cautious can result in being preoccupied with making sure that the data is perfect which is difficult to achieve resulting in less understanding. Scientists need to be able to coop with Butterfly Effect and produce quality studies. How is that achieved? Through multiple mathematical simulations. Meaning, take the function in question and use the recorded data to produce a result, and mention

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the possibility of Butterf ly Effect. Still, “the noise in any system might be of a small value, bu t it is quite possible for that noise, in a suitably chaotic system, to cause the behavior of the system to change totally what it would be in the absence of noise” (Pritchard, 1992, p. 98). Due to the fact that Butterfly effect is not calculable it cannot be input as a numeric calculated error with experiments. Just knowing about the possibility though allows for the experimenter to say that “X” is what was found but it is possible that if “Y” and “Z” were to occur of the initial condition were altered that “D” would occur. This can done by shifting the initial conditions and presenting the graph, and overlapping all the graphs to show the range of outcomes depending on the start. In the end mathematics has significantly influenced science because it has allowed for an explanation for data that can have multiple outcomes in studies like meteorology, physics, and even economic sciences. “Chaos has created special techniques of using computers and special kinds of graphic

images, pictures that capture a fantastic and delicate structure underlying complexity.” (Gleick, 1987, p. 4)

Figure 9

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Technology in general has made much advancement since Chaos Theory was first discovered in 1960. Fig 8. demonstrates how graphs can be generated using computers which is commonly used know by many scientists, mathematicians, and students writing mathematical based essays. Through the advancements in technology it has become easier to graph and run mathematical simulations making it easier for scientists to explore Chaos Theory, and Butterfly Effect in comparison to their results.

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Conclusion Chaos Theory is a mathematics based study through which all areas of knowledge can benefit. The arts and social science can benefit by knowing about the Butterfly Effect and how small events can impact the outcome. However, the theory in a whole impacts the sciences. Through studying Butterfly Effect scientists are able to understand that shifts in the environment that they are conducting an experiment can result in differences in data. Although this is pertinent information it should not be highly regarded in all experiments because otherwise data collection can be done too cautiously and results may be in disarray due to over thinking the situation. For experiments relating to meteorology, live animals, population projects, and other similar studies, Butterfly Effect should be a forethought because a slight shift can result in large differences, and the graphical interpretations that can be found allow the researcher to determine a range based on several starting conditions. Chaos Theory is important for the general population to know as well. Mainly to allow understanding of how decisions that appear small now can have a large impact in the end.

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Work Cited Beyerchen, A. D. (1993). Clauswitz, Nonlinearity and the Unpredictability of War [Article]. Retrieved May 30, 2011, from http://www.clausewitz.com/readings/Beyerchen/ CWZandNonlinearity.htm Chaos Theory: A Brief Introduction [Article]. (n.d.). Retrieved June 13, 2011, from IMHO In My

Humble Opinion website: http://www.imho.com/grae/chaos/chaos.html Gleick, J. (1987). Chaos Making a New Science. New York, New York: Viking Penguin Inc. Kellert, S. H. (1993). In the Wake of Chaos. Chicago: The University of Chicago Press. Lorenz, E. N. (1962, November 18). Deterministic Nonperiodic Flow [Special section]. Journal of the Atmospheric Sciences, 20, 130-141. Lorenz Attractor [Mathematical Explanation]. (1999). Retrieved November 1, 2011, from

Wolfram MathWorld website: http://mathworld.wolfram.com/LorenzAttractor.html Mandelbrot Set [Mathematical Explanation]. (1999). Retrieved November 1, 2011, from

Wolfram MathWorld website: http://mathworld.wolfram.com/MandelbrotSet.html Pritchard, J. (1992). The Chaos Cookbook: A Practical Programming Guide. Oxford: Butterworth Heinemann. Rosenblatt, R. (1999, February 15). My Arbitrary Valentine. Time. Retrieved from http://www.time.com/time/magazine/article/0,9171,990217,00.html Williams, G. P. (1997). Chaos Theory Tamed . Washington DC: John Henry Press. Figure 1 Lorenz Attractor courtesy of http://mathworld.wolfram.com/LorenzAttractor.html 13



Figure 2 Equations on Paper courtesy of http://twenty-firstcenturyhousewife.blogspot.com /2009_09_01_archive.html Figure 3 Graph of Lorenz’ s Findings courtesy of http://www.imho.com/grae/chaos/chaos.html Figure 4 Lorenz’ s Equations courtesy of Lorenz, Deterministic Nonperiodic Flow, 1962 Figure 5 Lorenz’ s Equations courtesy of Lorenz, Deterministic Nonperiodic Flow, 1962 Figure 6 Lorenz’ s Equations courtesy of Lorenz, Deterministic Nonperiodic Flow, 1962 Figure 7 Euclidean Graph courtesy of http://www.replicatedtypo.com/uncategorized/creativecultural-transmission-as-chaotic-sampling/3684/ Figure 8 Mandelbrot Set in Complex Space courtesy of http://www.miqel.com/fractals_math_ patterns/visual-math-mandelbrot-magic.html Figure 9 Computer Graphing courtesy of http://lifeofaprogrammergeek.blogspot.com /2009/05/3d-grapher-in-clojure.html

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Extended Essay Draft 1

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