extended essay draft 1

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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


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How Have Graphical Interpretations and Formulas in Elias Mueller

Chaos Theory Impacted Science 002996-025

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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 Lorenzs 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, Benot 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 isan

attractor that arises in a simplified system of

equations describing the two-dimensional flow of

fluid of uniform depth , with an imposed temperature difference , under gravity g, with

buoyancy , thermal diffusivity , and kinematic viscosity (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

Figure 1


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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 Effect 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 butterflys 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.


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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. Lorenzs findings were based in the mathematical aspects of his field.

Specifically the graphical interpretation at the end, but he later wrote an article, 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 X1 to XM:

Figure 4


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where tis the single independent variable, and the functions Fi possess continuous first

partial derivatives. Such a system may be studied by means ofphase spacean M-

dimensional Euclidean space T whose coordinates are X1 XM. Eachpointin phase

space represents a possible instantaneous state of the system. A state which is varying in

accordance with (1) is represented by a movingparticle 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 Z+

and M is the endpoint. Furthermore, the systems 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 X10 in the system to XMO and t; however, the system is now run through

fi, 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


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(the F(x) function style). Overall Lorenzs 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 Lorenzs 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 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 zo(the initial

z value) and the function zn+1 does not go to infinity (Mandelbrot Set, 1999). For example: ifi

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, ifC were to equal 0 the set would go 0, 1, 2, 5and on to

Figure 7


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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, Benot 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 Lorenzs 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 Butterfly Effect. Still, the noise in any system might be of a small value, but 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 1Lorenz Attractorcourtesy of http://mathworld.wolfram.com/LorenzAttractor.html


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Figure 2Equations on Papercourtesy of http://twenty-firstcenturyhousewife.blogspot.com

/2009_09_01_archive.html

Figure 3 Graph of Lorenzs Findings courtesy of http://www.imho.com/grae/chaos/chaos.html

Figure 4Lorenzs Equations courtesy of Lorenz, Deterministic Nonperiodic Flow, 1962



Figure 7Euclidean Graph courtesy of http://www.replicatedtypo.com/uncategorized/creative-

cultural-transmission-as-chaotic-sampling/3684/

Figure 8Mandelbrot 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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