Chaos
Chaos
The word "chaos" is used in mathematics to mean something other than what it means in everyday English. In English, we speak of chaos as a state in which everything has gone awry, there is complete disorder, and there are no rules governing this behavior. In mathematics, chaotic systems are well defined and follow strict mathematical rules. Although chaotic systems are unpredictable, they do have certain patterns and structure, and they can be mathematically modeled, often by an equation or a system of equations.
Chaos theory is the study of systems that change over time and are inherently unpredictable. Some systems, such as Earth's orbit about the Sun, are very predictable over long periods of time. Other systems, such as the weather, are notoriously unpredictable. Chaos theory builds a mathematical framework to account for unpredictable systems, like the weather, which behave in a seemingly random fashion.
The last two decades of the twentieth century saw tremendous advances in our understanding of chaotic systems. Chaos theory has been applied* to the study of fibrillation in human hearts, the control of epileptic seizures, the analysis of how the brain processes information, the dynamics of weather, fluctuations in the stock market, the formation of galaxies, the flow of traffic during rush hour, and the stability of structures during earthquakes, to name a few. Wherever there are unstable or unpredictable systems, chaos scientists can be found attempting to reveal the underlying order in the chaos.
*In 1993, Goldstar Company created a washing machine whose pulsator motion exploited chaos theory.
Early History of Chaos Theory
Although chaos theory is very much a latetwentiethcentury development, most chaos scientists consider the great nineteenthcentury mathematician, Henri Poincaré, to be the true father of the discipline. The fundamental phenomenon common to all chaotic systems is sensitivity to initial conditions , and Poincaré was the first to articulate this sensitivity in his study of the socalled threebody problem.
Poincaré had described this problem in 1890 when he attempted to calculate the orbits for three interacting celestial bodies. Poincaré found that Newton's equations for two celestial objects were useless after a short time. The orbits became so tangled that he gave up hope of trying to predict where they would go. No one quite knew what to do about Poincaré's problem and so it remained essentially a mystery for the next eight decades.
Chaos in the Twentieth Century
In the 1950s meteorologists had great hopes that accurate weather prediction was about to become a reality. The advent of computers that could analyze vast amounts of data was seen as a great breakthrough in weather forecasting. Unfortunately, by the end of the decade, the optimism had begun to fade. Even with the rapid analysis of data from weather stations all over the world, forecasting did not seem to improve.
Meteorologist and mathematician Ed Lorenz of the Massachusetts Institute of Technology wondered why and began attempting to model the weather with equations programmed into his computer. The computer used the equations to simulate several months of weather, producing a graph that would rise and fall according to changes in the variables of Lorenz's weather equations.
At one point Lorenz wanted to take a closer look at a certain part of his graph, so he went back to an earlier point on the plot, inserted the values for that point from a printout from the computer, and waited to see a repetition of the graph he had seen before. At first he did, but after a few minutes he was startled to see a new plot diverging from the old plot, creating an entirely different version of the weather. After checking his computer for a malfunction and finding none, he discovered the cause of this unexpected behavior. When he had restarted the simulation, he had entered values from the printout that were rounded off to three decimal places, whereas the computer was using six place decimals for the simulation. This small variation in the initial values had produced a completely different version of the simulated weather.
By accident, Lorenz had rediscovered Poincaré's sensitivity to initial conditions. He realized that this meant that unless one has infinitely precise knowledge of all the initial conditions in a weather system, one cannot accurately predict the weather very far into the future. Lorenz called this finding "The Butterfly Effect"* because his results implied that the flapping of a butterfly's wings in Brazil could stir up weather patterns that might ultimately result in a tornado in Texas.
*The nowpopular phrase "If a butterfly flaps its wings"—which many people use in connection with global ecology—actually came from the study of mathematical chaos.
Fish Population Studies. In 1970, the biologist Robert May was studying the growth of fish populations using a wellknown mathematical model for populations with an upper bound to growth. This upper bound is sometimes called the "carrying capacity " of the environment. The model was the logistic difference equation P (t + 1) = rP (t )(1 − P (t )), where r is a number greater than 1 representing the growth rate, P (t ) is the population as a percentage of carrying capacity at time t, and the factor (1  P (t )) represents the fact that as P (t ) gets closer to 1 (closer to 100 percent of the carrying capacity), the growth rate of the population slows to almost zero.
May had noticed that fish populations sometimes maintained the same population from one year to the next and sometimes would fluctuate from year to year. He decided to study what happened with the logistic equation when he varied the values of the parameter r, which essentially represent the growth rates of the population. The logistic difference equation is an iterator . This means that one inputs a certain initial population called P (0) into the equation, does the calculation, and receives as the output P (1), which is the population at the next time interval. Then this output value P (1) becomes the new input into the equation, yielding a new output P (2), and so on.
Using a simple hand calculator, May carried out hundreds of iterations (repetitions) using different values for the initial population, P (0), and different values of the growth parameter r. He discovered that when the value of r was between 1 and 3, the iteration produced a string of numbers that would ultimately settle on a single value no matter what initial value was used. This implied a stable population year after year. When r was raised slightly above 3, however, the output of the iteration cycled back and forth between two values.
This remained true for all initial values until May raised the parameter beyond 3.45 and observed that the values began oscillating among four values, again independent of the initial population. At about 3.54, the oscillation doubled again to eight distinct values, then to sixteen when r was increased passed 3.556, then to thirtytwo and sixtyfour, and so on. When the parameter value reached about 3.56994, this "perioddoubling cascade," as May would call it, ended and the values seemed to jump about randomly.
At this point May noticed that the values produced by an iteration for a given r value were no longer independent of the initial value. For r = 3.6, for example, if the initial value were 0.1, the twentieth iteration was about0.7977, but if the initial value were 0.09, the twentieth iteration was about0.8635. The discrepancy became greater with additional iterations. Once again Poincaré's sensitivity to initial conditions had been rediscovered. In this case, May had discovered chaos in the logistic difference equation.
Feigenbaum's Constant. May's perioddoubling cascade became the subject of intense study by the physicist Mitchell Feigenbaum in the mid1970s. Whereas May had concerned himself with the qualitative aspects of the logistic system, Feigenbaum wanted to understand the quantitative basis of this perioddoubling route to chaos, as it would come to be called. Starting with a handheld calculator and eventually making use of more powerful computers, Feigenbaum discovered a completely unexpected pattern associated with the parameter (r ) values at which each new period doubling occurred. He called these bifurcation values .
Taking the computations far beyond where May had taken them, Feigenbaum came up with exceptionally high accuracy for the bifurcation values. For example, the first six are b _{1} = 3, b _{2} = 3.449490…, b _{3} = 3.544090…, b _{4} = 3.556441…, b _{5} = 3.568759…, and b _{6} = 3.569692….So, for example, when the parameter is raised past b _{5} = 3.568759, the logistic equation's output changes from an oscillation among sixteen values to an oscillation among thirtytwo values. Feigenbaum had the computer draw a graph showing parameter values on the horizontal axis and population values on the vertical axis. The result, known as the Feigenbaum plot, is now one of the icons of chaos theory.
The Feigenbaum plot shows how the bifurcations come increasingly closer as the parameter is increased from left to right. When the parameter value passes 3.56994…, the period doublings are no longer regular. This is the onset of chaos.
As fascinating as this was, Feigenbaum found something even more amazing in these numbers. He found that if he calculated ratios of the form (b _{k + 1} – b _{k })/(b _{k } – b _{k – 1}) for larger and larger values of k, these ratios would approach the number 4.669201609…. Furthermore, he found that if heused other functions, such as sine or cosine , completely unrelated to the logistic equation, the result would be the same. The number 4.669201609… is now known as Feigenbaum's constant and is considered to be one of the fundamental constants of mathematics.
The Link to Fractals. The Feigenbaum plot exhibits selfsimilarity, meaning that any one of the branches looks like the entire plot. Figures that exhibit selfsimilarity are called "fractals ," a term invented by Benoit Mandelbrot in 1975. Just as Euclidean geometry is the geometry of Newtonian mechanics, fractal geometry is the geometry of chaos theory.
Geometrically, the dynamics of chaotic systems are described by figures called "attractors," of which the Feigenbaum plot is one example. Although these dynamics are unpredictable due to sensitivity to initial conditions, they are geometrically bound to a certain structure, the attractor. It is in this sense that chaos scientists are said to seek order in chaos.
see also Fractals.
Stephen Robinson
Bibliography
Devaney, Robert L. A First Course in Chaotic Dynamical Systems. New York: AddisonWesley, 1992.
Gleick, James. Chaos: Making a New Science. New York: Penguin Books, 1987.
Gulick, Denny. Encounters with Chaos. New York: McGrawHill, 1992.
Kellert, Stephen. In the Wake of Chaos. Chicago: University of Chicago Press, 1993.
Peitgen, HeinzOtto, Hartmut Jurgens, and Dietmar Saupe. Chaos and Fractals: New Frontiers of Science. New York: SpringerVerlag, 1992.
Peitgen, HeinzOtto, and Peter Richter. The Beauty of Fractals. Heidelberg: SpringerVerlag, 1986.
Stewart, Ian. Does God Play Dice? The Mathematics of Chaos. Cambridge, MA: Blackwell Publishers, 1989.
Internet Resources
Fraser, Blair. The Nonlinear Lab. <http://www.apmaths.uwo.ca/~bfraser/nonlinearlab.html>.
The Mathematics of Chaos. Think Quest. <http://library.thinkquest.org/3120/>.
Trump, Matthew. What Is Chaos? The Ilya Prigogine Center for Studies in Statistical Mechanics and Complex Systems. <http://order.ph.utexas.edu/chaos/index.html>.
FEIGENBAUM'S UNIVERSAL CONSTANT
Although initially limited to studies of chaos theory, Feigenbaum's constant is now regarded as a universal constant of nature. Since the 1980s, scientists have found perioddoubling bifurcations in experiments in hydrodynamics, electronics, laser physics, and acoustics that closely approximate this constant.
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nonlinear dynamics
nonlinear dynamics, study of systems governed by equations in which a small change in one variable can induce a large systematic change; the discipline is more popularly known as chaos (see chaos theory). Unlike a linear system, in which a small change in one variable produces a small and easily quantifiable systematic change, a nonlinear system exhibits a sensitive dependence on initial conditions: small or virtually unmeasurable differences in initial conditions can lead to wildly differing outcomes. This sensitive dependence is sometimes referred to as the
"butterfly effect,"
the assertion that the beating of a butterfly's wings in Brazil can eventually cause a tornado in Texas. Historically, in fact, one of the first nonlinear systems to be studied was the weather, which in the 1960s Edward Lorenz sought to model by a relatively simple set of equations. He discovered that the outcome of his model showed an acute dependence on initial conditions. Later work revealed that underlying such chaotic behavior are complex but often aesthetically pleasing geometric forms called strange attractors. Strange attractors exist in an imaginary space called phase space, in which the ordinary dimensions of real space are supplemented by additional dimensions for the momentum of the system under investigation. A strange attractor is a fractal, an object that exhibits selfsimilarity on all scales. A coastline, for instance, looks much the same up close or far away. Nonlinear dynamics has shown that even systems governed by simple equations can exhibit complex behavior. The evolution of nonlinear dynamics was made possible by the application of highspeed computers, particularly in the area of computer graphics, to innovative mathematical theories developed during the first half of the 20th cent. Three branches of study are recognized: classical systems in which friction and other dissipative forces are paramount, such as turbulent flow in a liquid or gas; classical systems in which dissipative forces can be neglected, such as charged particles in a particle accelerator; and quantum systems, such as molecules in a strong electromagnetic field. The tools of nonlinear dynamics have been used in attempts to better understand irregularity in such diverse areas as dripping faucets, population growth, the beating heart, and the economy.
See S. N. Rasband, Chaotic Dynamics of Nonlinear Systems (1990); A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics (1992); S. J. Guastello, Chaos, Catastrophe, and Human Affairs: Applications of Nonlinear Dynamics to Work Organizations and Social Evolution (1995); A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods (1995).
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chaos theory
chaos theory Theory that attempts to describe and explain the highly complex behaviour of apparently chaotic or unpredictable systems which show an underlying order. The behaviour of some physical systems is impossible to describe using the standard laws of physics – the mathematics needed to describe these systems being too difficult for even the largest supercomputers. Such systems are sometimes known as ‘nonlinear’ or ‘chaotic’ systems, and they include complex machines, electrical circuits, and natural phenomena such as the weather. Nonchaotic systems can become chaotic, such as when smoothly flowing water hits a rock and becomes turbulent. Chaos theory provides mathematical methods needed to describe chaotic systems, and even allows some general prediction of a system's behaviour. However, chaos theory also shows that even the tiniest variation in a system's starting conditions can lead to enormous differences in the later state of the system. Because it is impossible to know the precise starting conditions of a system, accurate prediction is also impossible.
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chaos
chaos originally, denoting a gaping void or chasm, later extended to formless primordial matter; in current usage, complete disorder and confusion. In Greek mythology, Chaos is sometimes personified as the first created being, from which came the primeval deities Gaia, Tartarus, Erebus, and Nyx.
Recorded from the late 15th century, the word comes via French and Latin from Greek khaos ‘vast chasm, void’.
chaos theory the branch of mathematics that deals with complex systems whose behaviour is highly sensitive to slight changes in conditions, so that small alterations can give rise to strikingly great consequences, as in the butterfly effect.
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chaos
chaos A theory derived from the observation that when the mathematical description of a system includes several nonlinear equations (i.e. equations that cannot be represented by straight lines on a graph), the future behaviour of that system may be unpredictable, because of wide variations that result from its sensitivity to very small differences in initial values supplied to any mathematical model. Chaos was first studied with reference to weather forecasting, but the theory has since been found to have many ecological implications (e.g. in studies of predator–prey relationships and population dynamics).
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chaos
cha·os / ˈkāˌäs/ • n. complete disorder and confusion: snow caused chaos in the region. ∎ Physics behavior so unpredictable as to appear random, owing to great sensitivity to small changes in conditions. ∎ the formless matter supposed to have existed before the creation of the universe. ∎ (Chaos) Greek Mythol. the first created being, from which came the primeval deities Gaia, Tartarus, Erebus, and Nyx.
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Chaos
Chaos
any confused or disorderly collection or state of things; a conglomeration of parts or elements without order or connexion. See also clutter, confusion.
Examples: chaos of accidental knowledge; of foul disorders, 1579; of green and grey mists, 1878; of laws and regulations, 1781.
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chaos
chaos A state of disorder which is governed by simple and precise laws, but where the outcome is unpredictable and may change greatly with slight variations in starting conditions. Most real systems, such as weather patterns and satellite orbits, display chaotic behaviour. See also FRACTAL.
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chaos
chaos †chasm, abyss (as in Luke 16: 26) XV; primordial formless void XVI; utter confusion XVII. — F. or L. — Gr. kháos vast chasm, void, f. IE. base *ghəw hollow.
Hence chaotic XVIII.
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Chaos
Chaos. The primordial condition from which (or onto which) order is imposed, according to many religions, so that the cosmos can appear.
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chaos
chaos
•across, boss, Bros, cos, cross, crosse, doss, dross, emboss, en brosse, floss, fosse, gloss, Goss, joss, Kos, lacrosse, loss, moss, MSDOS, Ross, toss
•Laos
•Áyios Nikólaos, chaos
•Eos • Helios
•Chios, Khíos
•Lesbos • straw boss • Phobos • rooibos
•extrados • kudos • reredos • intrados
•Calvados • Argos • Lagos • logos
•Marcos • telos
•Delos, Melos
•Byblos • candyfloss
•tholos, Vólos
•bugloss • omphalos • Pátmos
•Amos, Deimos, Sámos
•Demos • peatmoss • cosmos • Los Alamos • Lemnos • Hypnos • Minos
•Mykonos • tripos • topos • Atropos
•Ballesteros, pharos, Saros
•Imbros • crisscross • rallycross • Eros
•albatross • monopteros • Dos Passos
•Náxos • Hyksos • Knossos • Santos
•benthos
•bathos, pathos
•ethos • Kórinthos
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