Physics
Chaos Theory
Chaos theory is a branch of mathematics and physics that studies the behavior of dynamical systems that are highly sensitive to initial conditions, leading to unpredictable and complex outcomes. It explores the concept of deterministic chaos, where seemingly random behavior is actually governed by underlying patterns and nonlinear dynamics. Chaos theory has applications in various fields, including physics, biology, economics, and meteorology.
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11 Key excerpts on "Chaos Theory"
- J. Parker(Author)
- 2007(Publication Date)
- Palgrave Macmillan(Publisher)
I focus on Chaos Theory as practiced within physics, wherein it occupies a particular subdiscipline called “dynamical systems theory.” 2 Although “dynamical systems theory” more accurately desig- nates the scientific modeling I hereafter describe, I nevertheless use “Chaos Theory” to acknowledge the phrase’s greater cultural resonance. 3 Chaos Theory has changed the way in which we conceptual- ize so-called chaotic structures in the natural world. Once regarded as “poor in order,” chaos has come to be seen as “rich in information,” according to N. Katherine Hayles, one of the first literary scholars to draw on Chaos Theory. 4 Once seen as aberrant, the nonlinear and the random are now understood as prevalent, and physical behaviors once disregarded and dis- missed are now considered legitimate areas of inquiry. The most far-reaching insights that Chaos Theory offers us are that patterns of order emerge spontaneously out of random behav- ior, that deterministic systems can generate random behavior when small uncertainties are amplified as the system develops through time, and that time itself can operate differently at local levels. Models of chaotic systems demonstrate the entangle- ment of system and systematizer in generating meaning, a feed- back loop thus running between the subjective observer and the object under observation. By looking through a chaos-theory lens, we can gain new insights into narratives whose structures display chaotic qualities. Such a reading enables us to apprehend how their form is their meaning, which emerges from the particular social, cultural, and 2 NARRATIVE FORM AND Chaos Theory historical circumstances, and how their meaning is dynamical, entangling the reader in the interpretive process. Through the perspective afforded us by Chaos Theory, we can discern the disorderly order—the complex yet simple elegance—of these narratives.
- Geoffrey Peruniak(Author)
- 2017(Publication Date)
- University of Toronto Press(Publisher)
Towards Defining Chaos and Complexity Chaos Theory refers to that collection of ideas whereby underlying the apparent random nature of a system there is actually an order or a de-veloping order. Often it is at a local level that randomness is apparent while at a global level there is a pattern that reveals order. Not all physi-cists agree on the precise formulation of Chaos Theory. Some scientists do not believe that Chaos Theory is even a science (Bütz, 1997). Others, such as superstring theorist Brian Greene, are sceptical that Chaos Theory represents something new (1999: 17): Developments such as Chaos Theory tell us that new kinds of laws come into play when the level of complexity of a system increases. Understanding the behavior of an electron or a quark is one thing; using this knowledge to understand the behavior of a tornado is quite another … opinions di-verge on whether the diverse and often unexpected phenomena that can occur in systems more complex than individual particles truly represent new physical principles at work, or whether the principles involved are derivative, relying, albeit in a terribly complicated way, on the physical principles governing the enormously large number of elementary con-stituents. My own feeling is that they do not represent new and independ-ent laws of physics … I see this as a matter of calculational impasse, not an indicator of the need for new physical laws. With seven dimensions being necessary to underpin superstring theory in addition to space, distance, and time, it is evident that even a reduc-tionist position such as Greene’s (1999) yields a picture of a very com-plex universe even without Chaos Theory. According to James Gleick: ‘Chaos breaks across the lines that separate scientific disciplines … it is a science of the global nature of systems ...
eBook - PDFSystems Science
Methodological Approaches
- Yi Lin, Xiaojun Duan, Chengli Zhao, Li Da Xu(Authors)
- 2012(Publication Date)
- CRC Press(Publisher)
117 5 Chaos The study of nonlinear science has touched on reconsiderations of many important concepts and theories, such as determinism and randomness, orderliness and disorderliness, accidentalness and inevitability, qualitative and quantitative changes, whole and parts, etc. These reconsiderations are expected to affect the accustomed ways of how people think and reason and have been involved in addressing some of the most fundamental problems of the logic system that underlies the entire edi-fice of modern science. It is commonly believed that nonlinear science consists mainly of the chaos and fractal theories, the latter of which will be studied in the next chapter. Discrete cases of chaos are often seen in chaotic time series, while chaotic time series contains rich information on the dynamics of the underlying system. How to extract this information and how to practically employ the extracted information represents one important aspect of applications of Chaos Theory. In this chapter, we will study some of the fundamental concepts of the Chaos Theory and related methods. 5.1 PHENOMENA OF CHAOS Bertalanffy (1968) once pointed out that the principles of general systems, although they are origi-nated from different places, are clearly similar to those of dialectical materialism. What is described in this statement is equally applicable to the Chaos Theory, a new research area. The development of the Chaos Theory has led to many dialectical thoughts. The phenomena of chaos were initially discovered by E. N. Lorenz (1963) in his study of a three-dimensional autonomous dynamic system. To investigate the prediction of weather, he simplified the equation system of atmospheric dynamics.
eBook - ePub- Ching-Yao Hsieh, Meng-Hua Ye(Authors)
- 2016(Publication Date)
- Routledge(Publisher)
Chapter 6Chaos Theory, Philosophy, and Economics
Contemporary physics has redirected most of its research attention to high-energy particles. This focus puts the frontier of research in physics beyond the realm of comprehension for most people. High-energy particles cannot be seen with the human eye and have no direct connection to daily life. Meanwhile, many simple things that we can observe cannot be statistically explained. For instance, such phenomena as the turbulence of fluids, the fluctuating intensity of lasers, some chemical reactions, and cardiac rhythms and arrhythmias, are only poorly understood; or simple figures we see with our naked eyes in day-to-day life, such as the shapes of clouds, or mountains, or coast lines, are never seriously measured and studied. To avoid facing these awkward and difficult tasks, scientists think of them as irregularities full of randomness and assert that few meaningful results can be learned from them.The advance of fractal geometry and Chaos Theory has paved the way for physics to return to the human scale without embarrassment. Once again, physics is becoming the center of scientific development.Chaos Theory deals with objects fundamentally different from those dealt with by relativity and quantum mechanics. Nevertheless, the three theories share the same spirit in questioning the traditional view of reality. It is, thus, appropriate for us to begin this chapter wit h the fall of Laplacian determinism.From Laplacian Determinism to Fundamental Uncertainty
The 18th-century French mathematician Pierre Simon de Laplace once boasted that given the current position and velocity of any particle in the universe, he could predict that particle’s future for the rest of time. For more than a hundred years, most scientists shared Laplace’s determinism, at least in principle. If there are any reasons that make this deterministic prediction impossible, they are rather technical ones and will be overcome sooner or later such that better and better predictions can be expected.
eBook - PDFShaking the Invisible Hand
Complexity, Endogenous Money and Exogenous Interest Rates
- B. Moore(Author)
- 2006(Publication Date)
- Palgrave Macmillan(Publisher)
As chaos sets in we encounter the inadequacy of our methodology, rather than the inadequacy of our laws. The complexity of a system may be defined by the com- plexity of the model necessary to simulate the behavior of the system. Chaos the- ory presents a limitation on the predictability of physical systems. Sensitive dependence requires physically impossible accuracy to perform any predictive task. Chaos Theory 55 By presenting us with examples of tasks so difficult that the fact that we are finite beings makes us unable to accomplish them, Chaos Theory challenges the distinction between theoretical and practical impossibility. 25 Indefinitely small changes in starting conditions can result in dramatically different system outputs. Sensitive dependence undermines the methodological assumption behind the conventional strategy of approximation analysis and calls into question a number of central beliefs about the way of the world and how we should go about under- standing it. If our world contains chaos even the smallest vagueness can blossom into open ambiguity. The challenge of Chaos Theory is to welcome this openness, and not see it as cause for regret. Some scholars regard Chaos Theory as containing within it the promise of the long desired reunification of the sciences. People are genetically hardwired to discover regularities. They instinctively look for the periodicity wrapped in random noise. But by using the nonlinear equations of fluid motion the world’s fastest CRAY supercomputers are incapable of tracking a turbulent flow of a cubic centimeter of liquid for more than a few microseconds. Physicists had good reasons to dislike a model that finds so little clarity in nature.
eBook - PDFBorrowed Knowledge
Chaos Theory and the Challenge of Learning across Disciplines
- Stephen H. Kellert(Author)
- 2009(Publication Date)
- University of Chicago Press(Publisher)
But if you put a small drop of food coloring in the taffy, the machine will soon stretch and fold the candy so that the color is spread throughout the taffy. 2 Two particles of dye that started out near each other might well end up on opposite sides of the big blob, and it is difficult if not impossible to predict where they will be. A chaotic system obeys simple rules like a clock but displays complicated and un-predictable behavior like a pressure cooker. Chaos Theory offers a new kind of model physical system: simple system, complicated behavior. In brief, Chaos Theory is the study of unpredictable behavior in sim-ple, bounded, deterministic systems. Such behavior is extremely com-plicated because it never repeats, and it is unpredictable because of its C H A P T E R O N E 6 celebrated sensitive dependence on initial conditions: even extremely small amounts of vagueness in specifying where the system starts render one utterly unable to predict where the system will end up. This sen-sitivity is often characterized in terms of the so-called butterfly effect, according to which our inability to know about the flapping of a but-terfly’s wings in some faraway place leads to our inability to correctly predict momentous weather phenomena in our own backyard (Gleick 1987 , 11 – 31 ). The unpredictability of chaotic systems should not be thought of as a temporary inconvenience; instead, the unpredictability represents an in-principle limitation on the ability to make exact pre-dictions about the specific details of the system’s behavior (see Kellert 1993 , chap. 2 ). Chaotic behavior occurs only in nonlinear systems, that is, systems in which the relevant variables are not related one to another according to strict proportionality.
eBook - PDF- Soc For Chaos Theory In Psychology & The Life Sci, W Sulis, A Combs(Authors)
- 1996(Publication Date)
- World Scientific(Publisher)
In an organization's communication research, Chaos Theory highlights the problems involving the measurement of the communication's effectiveness and influence. Observations, measurements and calculations are always imprecise, which also leads to inaccuracies in conclusions and predictions. According to the butterfly effect, mistakes and nonspecifics can grow cumulatively and arbitrarily rapidly. However, the behavior of chaotic systems is at all times completely predetermined. Unpredictability is merely the result of the fact that no finite information is enough to describe the system so precisely that the prediction will not in a short time become worthless. We can say that the more precisely the system's condition is known, the more precisely and long-term a prediction can be made, albeit only to an extent. What can be done? Although the behavior of communication systems cannot be predicted absolutely, it can still be influenced in that direction. With regard to chaos in the development of a system, we can limit it by controlling the parameters. Similarly, by changing the system's structure and properties we can stretch the plan of chaos and create so-called slow chaos. In place of single predictions 204 and precise measurement, for the behavior of the organization we can construct alternative scenarios which are checked and corrected as conditions dictate. Traditional empirical organizational communication research, which tries to find linear causal relationships and numerical indicators, might find Chaos Theory very frustrating. According to the theory, the precise prediction of the behavior of chaotic systems is lost over time. The only possibility is continual checking, calculating and correcting of prepared predictions. According to London and Thorngate 3 0 , the most interesting dynamic ex-pressions of social systems are in traditional social science models, often lost through incorrect terminology.
eBook - PDF- Michael Siek(Author)
- 2011(Publication Date)
- CRC Press(Publisher)
Subsequently, he figured out the reason of round-off effects resulting from the stored values that he used as initial conditions for the second calculations differed slightly from the original values. This led him to conclude that a tiny perturbation of the initial conditions can lead to enormous differences over time (Lorenz, 1963). The perspective of weather prediction provides an interesting metaphor to express the effect that small causes can have big impacts, well-known as Butterfly Effect (Glieck, 1987): Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas? With a modern PC today, one can easily retrace Lorenz’ footsteps and understand the remarkably rich behavior of his simple system as well as the sensitive dependence on initial conditions. 5.2 Basics of Chaos 5.2.1 Dynamical system Theory of dynamical systems tries to understand and describe the changes over time of the physical or artificial systems. Some examples of such systems are: the solar system, weather, motion of billiard balls, stock market, and so forth. Many areas of hydrometeorology, geophysics, economics and physiology involve a comprehensive analysis of the dynamical systems based on the particular laws governing their change. These laws are derived from a suitable theory such as Newtonian mechanics, fluid dynamics, mathematical economics (Tsonis, 1992; Strogatz, 2001). 5.2 Basics of Chaos 69 All these models can be unified conceptually in the mathematical notion of a dynamical system, which consists of two parts: phase space and dynamics. The phase space is the collection of all possible states of a dynamical system. Each state represents a complete condition of the system at a certain moment in time. The dynamics is a rule that transforms one point in the phase space representing the current state of the system into another point representing the state of the system one time unit.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Applications Chaos Theory is applied in many scientific disciplines: mathematics, programming, micro-biology, biology, computer science, economics, engineering, finance, philosophy, physics, politics, population dynamics, psychology, and robotics. Chaotic behavior has been observed in the laboratory in a variety of systems including electrical circuits, lasers, oscillating chemical reactions, fluid dynamics, and mechanical and magneto-mechanical devices, as well as computer models of chaotic processes. Observations of chaotic behavior in nature include changes in weather, the dynamics of satellites in the solar system, the time evolution of the magnetic field of celestial bodies, population growth in ecology, the dynamics of the action potentials in neurons, and molecular vibrations. There is some controversy over the existence of chaotic dynamics in plate tectonics and in economics. One of the most successful applications of Chaos Theory has been in ecology, where dynamical systems such as the Ricker model have been used to show how population growth under density dependence can lead to chaotic dynamics. Chaos Theory is also currently being applied to medical studies of epilepsy, specifically to the prediction of seemingly random seizures by observing initial conditions. A related field of physics called quantum Chaos Theory investigates the relationship between chaos and quantum mechanics. The correspondence principle states that classical mechanics is a special case of quantum mechanics, the classical limit. If quan-tum mechanics does not demonstrate an exponential sensitivity to initial conditions, it is unclear how exponential sensitivity to initial conditions can arise in practice in classical chaos. Recently, another field, called relativistic chaos has emerged to describe systems that follow the laws of general relativity.
eBook - PDF- David A. Turner(Author)
- 2005(Publication Date)
- Continuum(Publisher)
In contrast, a Chaos Theory approach emphasises that there are iterative feedback loops, with teachers influencing students, students influencing students and students influencing teachers, but none of them absolutely determining the responses of the others. Influence without determinism suggests that there is partial autonomy between the different levels within an organisation. The process of modelling using Chaos Theory must go beyond the simple assertion that organisations are complex, or that they demonstrate some of the features of Chaos Theory. Chaos Theory includes an understanding of how those features are to be interpreted, and work is necessary to elaborate the models more carefully and to demonstrate that the insights of Chaos Theory are valuable. In Tsoukas (1998: 305) words Analogies are not discovered; they are constructed. To develop an account of chaotic behaviour, I draw upon ideas I first put forward some time ago (Turner 1992) and which are developed more fully in Hall (1991) and Gleick (1998). 163 THEORY OF EDUCATION Tsoukas (1998: 305) identifies the key concepts which Chaos Theory has to offer: Chaos and complexity theory draw our attention to certain features of organisations about which organisation theorists were, on the whole, only subliminally aware. Notions like nonlinearity, sensitivity to initial conditions, iteration, feedback loops, novelty, unpredictability, process and emergence make up a new vocabulary in terms of which we may attempt to redescribe organisations. The issues of emergence and iteration, sensitivity to initial conditions, feedback loops, and unpredictability, are developed in the following section, in a way which will facilitate the application of those concepts to the study of educational institutions. 164 CHAOS AND COMPLEXITY Emergence and Iteration Figure 10.1: Part of the Mandelbrot Set The root of the ideas of emergence and iteration lie in fractal geometry, graphically illustrated in FigurelO.l.
- Robin Robertson, Allan Combs(Authors)
- 2014(Publication Date)
- Psychology Press(Publisher)
THE MESSAGES OF CHAOS How does chaos connect to order building and an ecological sense of how things work? Chaos provides the foundations of an ecological physics with a few rather simple lessons, as follows: 1. Order is hidden in chaos. There is order in complexity. Strange attractors show that intricately ordered flows can be hidden in what looks like completely erratic behavior. They also suggest that "most" dynamics have a heretofore unrecognized penchant for producing intricately ordered patterns. Mutually af- fecting variables tend to coeffect themselves into stable, ordered patterns. 2. The order in chaos is holistic order and results from mutual effects. The order in chaos is a result of interdependent variables coeffecting each other- push-me-pull-you-fashion-into a coherent pattern. The result is a hidden holis- tic pattern. It does not come from any one variable, it doesn't go in a straight line, and it does not imply a fixed sequence. It is order of the whole. People didn't even see the order in chaos until they looked at how all the system's variables change together over the whole range of the system's dynamics. Chaos is a holistic order that nevertheless arises from mechanical activity. 3. The order in chaos provides a mechanical explanation for "mysterious" hidden global ordering (an "invisible hand"). Adam Smith spoke of an invisible hand at work behind the operation of economies. Hegel described the world evolving through dialectics and order hidden beneath surface vicissitudes. Mutual-effect systems that create order of the whole provide a mechanical basis for this type of observation. The activity of the elements of a mutual-effect system creates global order; this mutually created global order creates a pressure on each individual element toward conformity to the global pattern. A treadmill gives a simple example of this global mutual-effect momentum phenomenon.
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