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					     Jennifer Buck


     Chaos Theory


       Math 111


Tuesday Thursday @ 12:35




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                                     Chaos Theory

Introduction

        Chaos theory has many applications both with in mathematics and outside of

mathematics. Chaos theory is complex and has many applications. The idea that many

simple “nonlinear deterministic systems” can behave in an apparently unpredictable and

chaotic matter was first noticed by French mathematician (Jules) Henri Poincare

(Cornick). Other pioneers in the field of chaotic dramatics were found in the

mathematical writings of Birkhoff, Cartwright, Littlewood, Levinson, Smale, and

Kolmogorov (Cornick). The field of chaos theory has only ganged recognition and

growth within the last decade, as the many different applications are discovered

(Cornick). Some of those applications include topography, psychology, and the stock

market.



Definition

        Chaos theory has many different definitions. One misconception is that Chaos

theory is a study of disorder, but it is not that at all. Instead chaos theory is the “very

essence of order” (Ho). It is because of this misconception that many people do not

understand how chaos theory can be a useful tool in mathematics. Chaos theory can be

defined as the study of complex nonlinear dynamic systems (Ho). It is also the study of

how non related systems or items interact (Beckwith). Chaos theory is a branch of

mathematics that deals with systems that appear to be order but “harbor chaotic

behaviors” (Beckwith). According to Andrew Ho a very generally chaos theory is the




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study of forever changing complex system based on mathematical concepts of recursion

(Ho).




How do we create Chaos?

The most common way to create chaos is to iterate a function. The most common model

for chaos theory is based on an iterative process with simple quadratic equation

f ( x)  x 2  c in the complex domain (Bogomolny). This Process led to computer

models of “unusual richness, beauty, and appeal” and constituted one of the fundamental

pieces in the foundation of the new science of fractals (Bogomolny). Fractals are rough

shapes that look the same whether viewed from far away of close up (Father of Fractals).

A fractal discover by Benoit Mandelbrot called The Mandelbrot set is an example of this

process (Father of Fractals).




                         This is an example of what the Mandelbrot set looks like. If one

was to zoom he or she would see that it look the same form far away or from close up.

        The study of the quadratic equation f ( x)  kx(1  x) is a cornerstone in related

development of the science of chaos (Bogomolny). This equation is a well know logistic

equation (Bogomolny). The process x n 1  kxn (1  x n ) often describes the population

change form year n to the next year n+1 (Bogomolny). The first term (kx) reflects the

reproduction tendency which is supposed to be proportional to the present population

(Bogomolny). The term (1-x) represents the need to coexist and share resources

(Bogomolny). Robert may, an Austrian mathematician engrossed in Biology, studied the


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global behavior of the process in its dependency on the selection of the coefficient k.

(Bogomolny).

        This example in the real domain, is from Alexander Bogomolny. We start

with x o . Draw a vertical line until it intersects the graph of y=f(x). At the point of

intersection y=f( x 0 ). Take this y as a new iterate: x1  f ( x 0 ) . Since we are accustomed

to have x values in the horizontal axis, recall that the equation of the diagonal in the first

quadrant is y=x. Trace a horizontal line until its intersection with the diagonal; and the

downwards toward x-axis. Now starting with x1 you obtain x 2 in a similar way. A

vertical line up to the graph, then a horizontal line towards the diagonal. The illustration

below shows this process.




        On the diagram subsequent iterates become closer and closer to the point of

intersection of the graph and the diagonal. The point of intersection is a solution of the

equation x=f(x). Such a solution is called “stable” because iterates { x n } tend to converge

there. Not all solutions of x=f(x) are stable. If f ( x)  kx(1  x) then obviously x=0 is a

solution of x=f(x) (Bogomolny). However not all solution for x=f(x) are stable. If

f ( x)  kx(1  x) then obviously x=0 is a solution of x=f(x). This is an example of a



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stable one. However if one selects a value closer to 0 then the first approximation of x 0 ,

and x1 will be father away form 0 than x 0 , and the next iterate x 2 will be even father

away and so on (Bogomolny).



Application of Chaos Theory

               There are many applications of Chaos theory. Some of those applications

are topography, psychology, weather, and the stock market. One application is in

analysis of the stock market. The stock market provides trends which can be analyzed to

predict how small changes can have massive outcomes (Linden).

       There exists objects that can not be classified as one- two- or three-dimensional,

but some fall somewhere in between (Father of Fractals). An example of this is

determining the length of the coastline (Father of Fractals). As you zoom in on a

coastline, more detail is revealed, so that it appears to get longer (Father of Fractals). “It

is mathematically possible to have an infinitely long coastline contained within a finite

space” (Father of Fractals). Using Dr. Mandelbrot‟s theories, curves could be assigned

“fractal dimensions” (Father of Fractals). For example the coastline of Britain has a

fractal dimension of about 1.24- dimensional (Father of Fractals).

       There are some possible applications of chaos theory in psychology. Marshall

Duke gives seven propositions in which he fells chaos theory can be applied I will focus

on three of these propositions (Duke). The seven propositions are:

1. "Errors" or nonlinearities in the study of behavior represent phenomena in need of

study, not in need of control (Duke).




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2. Like physical systems, when behavioral systems are pressured in terms of time or

space, linearity will tend to be replaced by nonlinearity (Duke).

3. Over time, human behavior is both variable and stable. Lack of predict-ability may be

expected at the level of specific behaviors, but regularity and immutability may be the

rule at more global levels of analysis and observation (Duke).

4. The point in developmental time (i.e., the location on the dynamical course of an

attractor) at which a behavior is measured or studied will affect the form of the behavior

observed (Duke).

5. The sources of specific day-to-day lasting behavior patterns as well as the sources of

problems in functioning are not limited only to major events in people's lives, but also

include relatively small events, which, over time, can result in a significant impact on

behavior (Duke).

6. Dependent upon the scale on which behaviors are examined, it is possible to find

different sorts of patterns, which, in turn, may require different methods of analysis and

conceptualization (Duke).

7. Patterns of behavior observed at large scales of analysis (in terms of time, content,

process, structure, etc.) will also be observed in a smaller scale analysis of the same

behaviors (Duke).



       Proposition 1 suggests that, “were psychologists to study those phenomena that

have traditionally been regarded as „error,‟ significant insights and progress may result”

(Duke). Thus, individual subjects who do not "fit" predicted patterns of behavior might,

nevertheless, be behaving in some way that “we ought to be able to explain” (Duke).




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What the author is explaining is that if we look at errors or things that do not fit we may

be able to find out more information than looking at things that do fit.

       Chaos theorists and Physicists have increasingly found that as regular or linear

systems function under less than ideal conditions over extended periods of time or space

or at rates higher than usual; they tend to break down (Duke). Proposition 2 proposes

that schizophrenia may be understood as an irregular nonlinear dynamical pattern that

occurs when the psychological and neurological systems of an individual go beyond their

levels of normal (linear) functioning (Duke).

       In addition to attractors, another concept deriving from physics that may be

applicable to psychology is termed “sensitive dependence upon initial conditions”

(proposition 5) (Duke). In terms of its relevance to psychology, “the phenomenon of

sensitive dependence upon initial condition seems to yield a number of implications”

(Duke). Among these might be the question of the widespread belief that only significant

major events (such as traumas and other salient experiences) can alter behavior

significantly (Duke). Based on the concept of initial conditions, seemingly insignificant

changes that occur at one point in time also can result in significant differences in

behavioral patterns later in time (Duke). I think proposition 5 gets at the very essence of

the application of chaos theory. Part of chaos theory is realizing that small seemingly

insignificant changes can have a major impact on the overall outcome of the situation.



Conclusion

       Chaos theory once it is understood has many different applications. I was

surprised to discover that it was not just one person that discover fractals but several



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different people throughout history. I thin the reason chaos theory can be applied dot so

many different subject areas is because nothing is completely linear, everything even life

has a few curves in it now and then. Marshal Duke‟s study on how chaos theory can be

applied to psychology is very interesting but I only talked about three just to give a few

specific examples of how he applied chaos theory to his ideas. I learned a lot about how

one idea can evolve as technology evolved in the case of the Mandelbrot set, Dr.

Mandelbrot had to wait for the right technology before he could display his ideas. I

realize that chaos theory is more complex that I originally thought and cover a wide range

of subjects and disciplines.




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                                  Works cited

Beckwith, Anthony. Chaos Theory Page. 18 March 2005. Concord-Carlise High School.

       13 Nov. 2005 <Htttp://mail.colonial.net/~abeckwith/chaos.html>

Bogomolny, Alexander. Emergence of Chaos. 14 Nov 2005

       <http://www.cut-the-knot.org/blue/chaos.shtml>

Cornick, Mathew. Chaos @UMD. 2 Aug. 2005. <http://www.chaos.umd.edu/>.

Duke, Marshall. “Chaos Theory and Psychology: Seven Propositions”. Genetic, Social &

       General Psychology Monograph‟s. Vol. 120 (1994). 267-277.

“Father of Fractals”. Economist. Vol. 369 (2003). 35-36.

       <http://search.epnet.com/login.aspx?direct=true&db=crh&an=1616956>

Ho, Andrew. Chaos Homepage. 13 Nov. 2005 <http://twm.co.nz/chaos_intro.html>

Linden, George. Holism: Classical, Cautious, Chaotic, and cosmic”.

       Individual Psychology, Vol 52 No.3 Sept. 1995.




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