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Jennifer Buck Chaos Theory Math 111 Tuesday Thursday @ 12:35 1 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 2 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 3 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 4 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). 5 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). 6 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 7 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. 8 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. 9

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