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Chaos Theory

Chaos Theory

Chaos theory is a scientific principle describing the unpredictability of systems. Heavily explored and recognized during the mid-to-late 1980s, its premise is that systems sometimes reside in chaos, generating energy but without any predictability or direction. These complex systems may be weather patterns, ecosystems, water flows, anatomical functions, or organizations. While these systems' chaotic behavior may appear random at first, chaotic systems can be defined by a mathematical formula, and they are not without order or finite boundaries. This theory, in relation to organizational behavior, was somewhat discounted during the 1990s, giving way to the very similar complexity theory. Complexity theory differs primarily in that it tries to find simple ways to explain and control multifaceted business systems, while chaos theory is more concerned with predicting changes and understanding the random parts of systems.

Organizations respond to the ideas of chaos theory in a number of different ways, from eagerness to skepticism. As Lyndon Pugh's 2007 Change Management in Information Services says, The key points about chaos theory are that it affirms the need for environmental sensitivity, and that uncertainty represents opportunities.

ORIGINS OF CHAOS THEORY

One of the first scientists to comment on chaos was Henri Poincaré (18541912), a late-nineteenth-century French mathematician who extensively studied topology and dynamic systems. He left writings hinting at the same unpredictability in systems that Edward Lorenz (b. 1917) would study more than half a century later. Poincaré explained, It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible. Unfortunately, the study of dynamic systems was largely ignored long after Poincaré's death.

During the early 1960s, a few scientists from various disciplines were again taking note of odd behavior in complex systems such as the earth's atmosphere and the human brain. One of these scientists was Edward Lorenz, a meteorologist from the Massachusetts Institute of Technology (MIT), who was experimenting with computational models of the atmosphere. In the process of his experimentation he discovered one of chaos theory's fundamental principlesthe Butterfly Effect. The idea is named for its assertion that a butterfly flapping its wings in Tokyo can impact weather patterns in Chicago. More scientifically, the Butterfly Effect proves that forces governing weather formation are unstable. These unstable forces allow minuscule changes in the atmosphere to have major impact elsewhere. More broadly applied, the Butterfly Effect means that what may appear to be insignificant changes to small parts of a system can have exponentially larger effects on that system. It also helps to dispel the notion that random system activity and disturbances must be due to external influences, and not the result of minor fluctuations within the system itself.

Another major contributor to chaos theory is Mitchell Feigenbaum (b. 1944). A physicist at the theoretical division of the Los Alamos National Laboratory starting in 1974, Feigenbaum dedicated much of his time

researching chaos and trying to build mathematical formulas that might be used to explain the phenomenon. Others working on related ideas (though in different disciplines) included a Berkeley, California mathematician who formed a group to study dynamical systems, and a population biologist pushing to study strangely-complex behavior in simple biological models. During the 1970s, these scientists and others in the United States and Europe began to see beyond what appeared to be random disorder in nature (the atmosphere, wildlife populations, etc.), finding connections in erratic behavior. As recounted by James Gleick (b.1954) in Chaos, a French mathematical physicist had just made the disputable claim that turbulence in fluids might have something to do with a bizarre, infinitely tangled abstraction he termed a strange attractor. Stephen Smale (b. 1930), at the University of California, Berkeley, was involved in the study of dynamical systems. He proposed a physical law that systems can behave erratically, but the erratic behavior cannot be stable. At this point, however, mainstream science was not sure what to make of these theories, and some universities and research centers deliberately avoided association with proponents of chaos theory.

By the mid-1980s, chaos was a buzzword for the fast-growing movement reshaping scientific establishments, and conferences and journals on the subject were on the rise. Universities sought chaos specialists for high-level positions. A Center for Nonlinear Studies was established at Los Alamos, as were other institutes devoted to the study of nonlinear dynamics and complex systems. A new language consisting of terms such as fractals, bifurcations, and smooth noodle maps was born. In 1987, James Gleick published his landmark work, Chaos: Making a New Science, chronicling the development of chaos theory, as well as the science and scientists fueling its progress.

There are many possible applications for chaos theory, in and out of the business world, and many are still being studied. In Amita Paul's 2008 article on chaos theory and business practices, Paul gives several examples of data that can be predicted by studying complexity models, such as epileptic seizures, financial markets, manufacturing systems, and weather systems. In order to capitalize on chaos theory, Paul suggests a three-tiered approach:

  1. A goal should be created, a particular state to be aimed at by the business. This could be the solving of a problem or reaching a certain state of productivity.
  2. The organization and its structure should be capable of reaching its goalit should be achievable.
  3. The means to influence the systems should be known by the organization, and the leaders should be willing to put the plans into motion.

Paul admits the many applications possible for chaos theory, but questions its usefulness in the business world. Complexity analysis is always very complicated, requiring a large amount of data and intense mathematics, which can be open to error. A business must have talent, time, and funds to began a chaos analysis on any of its systems.

THE SCIENCE OF CHAOS THEORY

As stated by James Gleick, chaos is a science of the global nature of systems, and so it crosses many disciplinary linesfrom ecology to medicine, electronics, and the economy. It is a theory, method, set of beliefs, and way of conducting scientific research. Technically, chaos models are based on state space, improved versions of the Cartesian graphs used in calculus. In calculus, speed and distance can be represented on a Cartesian graph as x and y. Chaos models allow the plotting of many more variables in an imaginary space, producing more complex imaginary shapes. Even this model assumes, however, that all variables can be graphed, and may not be able to account for situations in the real world where the number of variables changes from moment to moment.

The primary tool for understanding chaos theory (and complexity theory as well) is dynamic systems theory, which is used to describe processes that constantly change over time (e.g., the ups and downs of the stock market). When systems become dislodged from a stable state, they go through a period of oscillation, swinging back and forth between order and chaos. According to Margaret J. Wheatley in Leadership and the New Science, Chaos is the final state in a system's movement away from order. When a system does reach that point, the parts of a system are manifest as turbulence, totally lacking in direction or meaning. Wheatley quotes researchers John Briggs and F. David Peat explaining the process of oscillation:

Evidently familiar order and chaotic order are laminated like bands of intermittency. Wandering into certain bands, a system is extruded and bent back on itself as it iterates, dragged toward disintegration, transformation, and chaos. Inside other bands, systems cycle dynamically, maintaining their shapes for long periods of time. But eventually all orderly systems will feel the wild, seductive pull of the strange chaotic attractor.

In simpler terms, every system has the potential to fall into chaos.

The above strange attractor is the very same that a French mathematical physicist identified in the early 1960s. In complex systems, where all should fall apart, the attractor comes in, magnetically pulling system variables into an area and creating a visible shape. Because previous efforts to graph such phenomena could only be

completed in two dimensions, this effect could not be visualized. However, computers now allow the phenomena of strange attractors to become visible, as images of multiple dimensions representing multiple variables can finally be created.

Part of the difficulty in studying chaos theory arises because complex systems are difficult to study in pieces. Scientists' efforts to separate pieces of dynamical systems often fall apart. The system depends on each minute part of that system and the way it interacts with all other components. As Briggs and Peat state, The whole shape of things depends upon the minutest part. The part is the whole in this respect, for through the action of any part, the whole in the form of chaos or transformative change may manifest.

In the same breath, it is important to establish the importance of the autonomy the smallest parts of a system possess. Each component of a complex system has the ability to fluctuate, randomly and unpredictably, within the context of the system itself. The system's guiding principles (the attractors) allow these parts to cohere over time into definite and predictable form. This runs contrary to the impression many have of chaos theory, believing there is no order to be had in such a system. However, chaotic movement does possess finite boundaries, within which is the capacity for infinite possibility. Even lacking direction, parts of a system can combine so that the system generates multiple configurations of itself, displaying order without predictability. These systems never land in the same place twice, but they also never exceed certain boundaries.

PRACTICAL APPLICATION OF CHAOS THEORY

By the early 1980s, evidence accumulated that chaos theory was a real phenomenon. One of the first frequently-cited examples is a dripping water faucet. At times, water drops from a leaky faucet exhibit chaotic behavior (the water does not drip at a constant or orderly rate), eliminating the possibility of accurately predicting the timing of those drops. Scientists took advantage of applications using chaos to their benefit; chaos-aware control techniques could be used to stabilize lasers and heart rhythms, among multiple other uses.

As Kolb and Overdahl point out in their 2007 Futures, Options, and Swaps, chaos theory was also considered to have great potential in predicting the future changes of stocks and bonds. Although no set analysis has been able to use chaos theory infallibly to find patterns in the futures market, much work in chaos mathematics is focused on finding ways to predict stock prices. The idea is that the fluctuations of the trading market resemble the drip of a faucet or the fractals of tree branches; this implies they are similar enough that someone could conceivably find the proper mathematical points and predict changes correctly.

Another arena within which chaos theory is useful is that of organizations. Applying chaos theory to organizational behavior allows theorists to take a step back from the management of day-to-day activities and see how organizations function as unified systems. An organization is a classic example of a nonlinear system (i.e., a system in which minor events have the potential to set off grave consequences or chain reactions, and major changes may have little or no effect on the system whatsoever). In order to exploit the chaotic quality of an organization, one needs to try to see the organizational shape that emerges from a distance. Instead of pinpointing causes in the organization for organizational problems, the company is better served, according to chaos theory, by looking for organizational patterns that lead to certain types of behavior within the organization.

Organizational expectations for acceptable behavior, and the degree of freedom with which individuals are allowed to work, shape the way a company's problems and challenges are handled by its members. By allowing people and groups within an organization some autonomy, businesses encourage the organization to organize itself, enacting multiple iterations of its own functioning until the various pieces of the organization can work together most effectively. An organization that encourages this type of management has been termed a fractal organization, one that trusts in natural organizational phenomena to order itself.

In the guide CIMA Learning System (2007), Botten and Sims discuss what this means for organizations trying to apply complexity theories in a useful manner. How does a business know when chaos theory applications have not done enough or have done too much? After taking a practical look at the results of chaos research, Botten and Sims created a concept of three separate states an organization can find itself in.

Chaos can either: (1) not affect a simple system at all, leading to consistent and accurate results without change; or (2) it can affect a system with a slight amount of error, so that some calculations will always be wrong and some results unexpected; or (3) chaos can reign uncontrolled on a complex system without predictability or order. These three possibilities apply to everything chaos theory deals with, from weather to business planning. Botten and Sims define the first state as stable equilibrium, the second as bounded instability (chaos under control), and the third as explosive instability (uncontrolled chaos).

While explosive instability can be very dangerous for a company, leading to loss of profit and the implosion of its systems, stable equilibrium can also pose a threat to

organizations. Business tends to suffer without change and without at least a little chaos to produce the unexpected, thereby forcing the adaptation and innovation that make organizations successful. In a static environment, organizations lose the ability to react well. Therefore, Botten and Sims suggest that the best state for business systems is the middle state, bounded instability, where an amount of controlled chaos is making the system unpredictable enough to be valuable, but not so unpredictable that it falls apart. In bounded instability, the system is not so repetitive that employees grow bored and lose the ability to function, but instead change occurs based on understood phenomenathe different seasons, for instance, or popular opinion, or the advances of technology. When a business reaches the third stage, explosive instability, it has gone too far; from a chaos theory perspective, it resembles a wildfire or war more than a functioning part of an organization. The middle road should be aimed for when applying chaos theorynot complete predictability and not utter confusion.

However, applying chaos theory to organizational practice tends to go against the grain of most formal management patterns. Order can be confused with the more popular notion of control. Defined by organization charts and job descriptions, traditional management does not generally seek to add disorder to its strategic plan. As Wheatley states, It is hard to open ourselves up to a world of inherent orderliness. Organizations are focused on structure and design. Charts are drawn to illustrate who is accountable to whom or who plays what role and when. Business experts break down organizations into the smallest of parts. They build models of organizational practice and policy with hope that this atomizing yields better information on how to improve the organization's functioning. However, chaos theory implies that this is unnecessary, even harmful.

Self-organizing systems are those enabled to grow and evolve with free will. As long as each part of the system remains consistent with itself and the systems's past; these systems can harness the power of creativity, evolution, and free willall within the boundaries of the organization's overall vision and culture. In this respect, chaos theory shows the need for effective leadership, a guiding vision, strong values, organizational beliefs, and open communication.

WRITING ON CHAOS THEORY

During the 1980s, chaos theory did begin to change decision-making processes in business. A good example is the evolution of high-functioning teams. Members of effective teams frequently recreate the role each member plays, depending on the needs of the team at a given point. Though not always the formally-designated manager, informal leaders emerge in an organization not because they have been given control, but because they have a strong sense of how to address the needs of the group and its members. The most successful leaders understand that it is not the organization or the individual who is most important, but the relationship between the two. And that relationship is in constant change.

One of the most influential business writers of the 1980s and 1990s, Tom Peters (b. 1942), wrote, Thriving on Chaos: Handbook for a Management Revolution in 1987. Peters offers a strategy to help corporations deal with the uncertainty of competitive markets through customer responsiveness, fast-paced innovation, empowering personnel, and most importantly, learning to work within an environment of change. In fact, Peters asserts that we live in a world turned upside down, and survival depends on embracing revolution. While not explicitly concerned with chaos theory, Peters's focus on letting an organization (and its people) drive itself is quite compatible with the central tenets of chaos theory.

As the global economy and technology continue to change the way business is conducted on a daily basis, evidence of chaos is clearly visible. While businesses could once succeed as non-adaptive, controlling institutions with permanently-installed hierarchical structures, modern corporations must be able to restructure as markets expand and technology evolves. According to Peters, To meet the demands of the fast-changing competitive scene, we must simply learn to love change as much as we have hated it in the past.

Organizational theorist Karl Weick (b. 1936) offers a similar theory to that of Peters, believing that business strategies should be just in time supported by more investment in general knowledge, a large skill repertoire, the ability to do a quick study, trust in intuitions, and sophistication in cutting losses. Though he did not articulate his theories in terms of the explicit ideas offered by quantum physics and chaos theory, his statements support the general idea that the creation and health of an organization (or a system) depends on the interaction of various people and parts within that system. However, as Wheatley states in her book:

Organizations lack this kind of faith, faith that they can accomplish their purposes in various ways and that they do best when they focus on direction and vision, letting transient forms emerge and disappear. We seem fixated on structures and organizations, or we who create them, survive only because we build crafty and smartsmart enough to defend ourselves from the natural forces of destruction.

As Botten and Sims theorized with their ideas on bounded instability, some chaos is considered good for an organization. Anthony Walker calls this the edge of

chaos in his 2007 book, Project Management in Construction. While the edge of chaos is far from a stable environment, it is also very excitinga place where the organization's predictable legitimate system tries to keep structures unchanged, and its dangerous shadow system attempts to undermine and challenge the status quo. When both are in healthy competition with each other, at the edge of chaos, they provide the power for change and required innovation.

SEE ALSO Complexity Theory; Trends in Organizational Change

BIBLIOGRAPHY

Botten, Neil, and Adrian Sims. CIMA Learning Strategy 2007: Managment Accounting Business Strategy. Oxford: Butterworth-Heineman, 2006.

Chen, Guanrong, and Xinghuo Yu, eds. Chaos Control: Theory and Applications (Lecture Notes in Control and Information Sciences). New York: Springer-Verlag, 2003.

Farazmand, Ali. Chaos and Transformation Theories: A Theoretical Analysis with Implications for Organization Theory and Public Management. Public Organization Review 3, no. 4 (2003): 339372.

Gleick, James. Chaos: Making a New Science. New York: Penguin Books, 1987.

Kolb, Robert W. and, James A Overdahl. Futures, Options, and Swaps. Blackwell Publishing, 2007.

Paul, Amita. Chaos Theory. 12Manage: The Executive Fast Track, 2008. Available from: http://www.12manage.com/methods_lorenz_chaos_theory.html.

Peters, Tom. Thriving on Chaos. New York: HarperCollins, 1987.

Pugh, Lyndon. Change Management in Information Services. Ashgate Publishing Ltd., 2007.

Sullivan, Terence J. The Viability of Using Various System Theories to Describe Organisational Change. Journal of Educational Administration 42, no. 1 (2004): 4354.

Walker, Anthony. Project Management in Construction. Blackwell Publishing, 2007.

Wheatley, Margaret J. Leadership and the New Science: Discovering Order in a Chaotic World Revised. San Francisco: Berrett-Koehler Publishers, 2001.

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Chaos Theory

Chaos Theory


Chaos Theory (CT) is a mathematical theory about nonlinear dynamical systems that exhibit exquisite sensitivity to initial conditions, eventual unpredictability, and other intriguing features despite the inevitably deterministic character of mathematical equations. CT has been used to model processes in diverse fields, including physics (quantum chaos, nonequilibrium thermodynamics), chemistry, ecology, economics, physiology, meteorology, zoology, and the neurosciences.

Basic research in mathematics and physics during the twentieth century produced CT. Felix Hausdorff (18691942) made essential contributions in mathematics when he created spaces with fractional dimensions. When Benoit Mandelbrot (1924) applied these spaces to geometry, he discovered new objects that he called fractals. These ideas were combined with the study of recursive and iterative mathematical formulas. The simplest formula of this kind, which was explored in great detail by Mitchell Feigenbaum (1944), is the logistic equation x n+1 = ax n (1 x n), where a is a tuning constant for the system. The system evolves recursively for n = 0, 1, 2, 3, . . . .

In 1963, meteorologist Edward Lorenz (1917) used differential equations with chaotic properties to model a physical system, the first time this had been done. In physics Henri Poincaré (18541912) used features of CT to demonstrate the stability of the solar system, a result that Isaac Newton (16421727) and many other scientists had not been able to achieve because of the potentially chaotic behavior of systems containing three or more bodies. Ilya Prigogine (1917), who did research in thermodynamics, examined nonlinear systems that are far from equilibrium and showed that such a system could generate novel structural features.

All these developments were independent of each other, but they merged in the new concept of CT in the 1970s. The term chaos theory was coined by mathematician and physicist James Yorke around 1972 and was introduced to the scientific literature in 1975 by the mathematician and biologist Robert May. Robert Devaney gave the first mathematical-technical definition of chaos in 1989, although this definition does not cover all features of interest to mathematicians who study chaos. In this technical sense, CT is not to be understood as being opposed to order, and it should not be confused with the metaphorical and colloquial use of the word chaos. Rather it describes how order breaks down and reemerges on many levels of complexity within dynamic systems.


Features of chaos theory

There are four essential aspects of CT. First, because of its recursive and iterative character, a chaotic system is exquisitely sensitive to its initial conditions, which means that the slightest variations in the parameters of a system may result in tremendous differences in the system's development. This feature is known as the Butterfly Effect.

Second, within the various modes of a chaotic dynamical system, there are certain levels of stability, especially when movements or changes come to an end. These levels of stability form the mathematical concept of an attractor. The eventual point of rest of a pendulum's movement is an attractor for the mathematical model of the nonchaotic pendulum system. Similarly, in classical thermodynamics the state of maximum entropy can be regarded as an attractor within nonchaotic mathematical models of fluids. Such nonchaotic attractors can be represented geometrically by a single point or a toroid. An attractor is distinguished from a strange attractor, the latter being used only in CT. The strange attractor is a fractal, of which the best known are the Hénon, Rössler, and Lorenz attractors. Dynamical systems in chaotic modes stabilize on strange attractors.

Third, the essential difference between the development of a nonchaotic system and the development of a chaotic system has to do with determinism and predictability. Although determinism and predictability are mutually entailing in nonchaotic systems, determinism does not entail predictability in chaotic systems. Chaotic systems possess a certain degree of predictability, measured by the so-called Lyapunov exponent, but all chaotic systems are unpredictable in the long run. Because of this astonishing mixture of determinism and nonpredictability, CT is also called the theory of deterministic chaos.

Finally, in contrast to a nonchaotic deterministic system, a chaotic deterministic system is not reversible due to progressive information loss as the system evolves. Thus, it is not possible to trace a system backwards to its initial conditions. If this mathematical form of CT is applied in physics to open systems that are far from equilibrium, additional features are revealed:

  1. Autopoietic systems, which are self-generating, can be described by CT.
  2. In order for a system to evolve in a chaotic manner, it is necessary constantly to supply it with energy, and the input of energy prevents it from entering a state of stationary equilibrium.
  3. Due to this constant input of energy, chaotic systems can evolve new features, such as those used in certain chemical clocks.
  4. Because chaotic systems are not static, they can adapt to new environmental conditions.
  5. The application of CT to evolving systems that are far from equilibrium requires a refinement of the concept of entropy.

Theological implications

The fact that determinism does not entail predictability in chaos theory means that knowledge of the future of a complex physical system that can be modeled with a chaotic dynamical mathematical system is severely limited in practice. This limitation of knowledge of the future may seem undesirable, but it turns out to be useful when CT is used as a conceptual tool for studying evolutionary and autopoietic systems. If philosophical reasoning is used to relate natural science to theology, then this new distinction between determinism and predictability has to be respected. There are three predominant options when relating CT to theology.

Ontology. The distinction between the mathematical theory of CT and its physical application raises the question of how to relate divine action to CT. If one interprets the eventual unpredictability of CT as an epistemological clue to an underlying openness in nature, as does John Polkinghorne, one can speculate whether the world is open to divine influences by the concept of "divine information input without energy transfer." On the other hand, if eventual unpredictability is judged to be merely an epistemic limitation with no ontological implications, then CT is not immediately useful for interpreting the natural-law-conforming action of an intentional divine being, though Robert John Russell and others have invoked it to explain how divine action at the quantum level might be amplified to macroscopic dimensions.

Autopoiesis. If CT is linked to the theory of autopoietic systems, the independence of creatures is emphasized rather than their dependence on God. This interpretation is adopted in some contemporary kenotic theologies and it tends to challenge traditional theological teachings such as providence and omnipotence. Generally, CT leads to the conclusion that it is more plausible to think of God as a cooperative partner in a panentheistic way, rather than as an almighty ruler, if God is to be thought of as a being at all, which is itself theologically controversial.

Unpredictability. The eventual unpredictability that is intrinsic to CT offers the possibility of reinterpreting the concept of divine providence. Rather than conceiving of God's knowledge as a deterministic prescience, one can interpret that knowledge as a knowledge of different options within an open future that is vulnerable to the possibilities of failure and error. In light of CT, one could also argue that in God predictability and determinism are again fused. This third interpretation does justice to human freedom. The use of CT in neuroscience invites attempts to relate CT's distinction between predictability and determinism to neurological interpretations of human free will. However, the deeper problem is whether mental phenomena, such as the will, can be reduced to neural activity, and here CT seems to offer no new insights.


See also Chaos, Religious and Philosophical Aspects


Bibliography

crain, steven d. "divine action and indeterminism: on models of divine agency that exploit the new physics (chaos)." ph.d. diss., university of notre dame, 1993.

devaney, robert l. an introduction to chaotic dynamical systems. redwood city, calif.: addison-wesley, 1989.

dinter, astrid. vom glauben eines physikers: john polkinghornes beitrag zum dialog zwischen theologie und naturwissenschaften. mainz, germany: matthias grünewald verlag, 1999.

gregersen, niels henrik. "providence in an indeterministic world." ctns bulletin 14 (1994): 1631.

kellert, stephen. in the wake of chaos: unpredictable order in dynamical systems. chicago: university of chicago press, 1993.

polkinghorne, john. the faith of a physicist: reflections of a bottom-up thinker. princeton, n.j.: princeton university press, 1994.

russell, robert john; murphy, nancey; and peacocke, arthur r., eds. chaos and complexity. scientific perspectives on divine action. vatican city and berkeley, calif.: vatican observatory publications and center for theology and natural sciences, 1995.

singe, georg. gott im chaos: ein beitrag zur rezeption der chaostheorie in theologie und deren praktischtheologische konsequenz. frankfurt am main, germany: peter lang verlag, 2000.

smedes, taede. "chaos: where science and religion meet? a critical evaluation of the use of chaos theory in theology." in studies in science and theology, vol. 8, eds. niels henrik gregersen, ulf görman, and hubert meisinger. aarhus, denmark: university of aarhus, 2001.

smith, peter. explaining chaos. cambridge, uk: cambridge university press, 1998.

wolfgang achtner

taede a. smedes

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Chaos Theory (meteorological aspects)

Chaos theory (meteorological aspects)

Chaos theory attempts to identify, describe, and quantify order in apparently unpredictable and/or highly complex systems (i.e., atmospheric dynamics, weather systems, etc.) in which, out of seemingly random, disordered (e.g., aperiodic) processes there arise processes that are deterministic and predictable.

Complex phenomena are those generally regarded as having too many variables (or too many possible conditions or states) to yield to conventional quantitative analysis. The motion of molecules in swirling smoke or the turbulent hydraulics of a river current, for example, are systems that exhibit such chaotic complexity.

According to the laws of thermodynamics, all natural processeswhen considering both system and surroundingsexhibit a tendency toward net movement from the ordered state to a more the chaotic (disordered) state. Conversely, according to some chaos theory models, chaotic, unpredictable, and irreversible processes may, evolve into or produce ordered states. Entropy is a measure of thermodynamic equilibrium used to explain irreversibility in physical and chemical processes. The second law of thermodynamics specifies that in an isolated system, increasing entropy corresponds to changes in the system over time and that entropy tends toward (a statistical mechanical concept) maximization. The second law of thermodynamics dictates that in natural processes, without work being done on a system, there is a movement from order to disorder.

Because entropy in natural processes increases over time, even very straightforward linear-type relationships must eventually take on a degree of irregularity (i.e., of seemingly disordered complexity). In accord with the second law of thermodynamics, apparently chaotic phenomena arise from initially ordered (i.e., lower entropy) systems. This dual tendency toward increasing entropy and chaos from an initially stable state can take place spontaneously. Small perturbations in initial conditions intensify these tendencies. Chaos theorists describe such departures as the butterfly effect.

The study of such mathematical irregularities involving chaos and order remained a relatively unnoticed corner of advanced mathematics until the advent of the digital computer. In 1956, Edward Lorenz, a professor of meteorology at the Massachusetts Institute of Technology was studying the numerical solution to a set of three differential equations in three unknowns, a highly simplified version of the types of equations meteorologists then in use to describe atmospheric phenomena. Lorenz came to the conclusion that his set of differential equations displayed a sensitive dependence on initial conditions, a sensitivity of the same type that French mathematician Jules-Henri Poincaré (18541912) had discovered for the Newtonian equations when those equations were applied to celestial dynamics. Lorenz, however, gave this phenomenon a new and highly appealing name, the butterfly effect, suggesting that, in the extreme, the flapping of a butterfly's wings in Kansas might be responsible for a monsoon in India a month later.

Along with quantum and relativity theories, chaos theorywith its inclusive concepts of chaos and orderis widely regarded as one of the great intellectual leaps of the twentieth century. The modern physical concepts of chaos and order, however, actually trace their roots to classical mechanical concepts introduced in English physicist Sir Isaac Newton's (16421727) 1686 work, Philosophy Naturalis Principia Mathematica (Mathematical principles of natural philosophy). It was Newton, one of the inventors of the calculus, who revolutionized astronomy and physics by showing that the behavior of all bodies, celestial and terrestrial, was governed by the same laws of motion, which could be expressed as differential equations. These differential equations relate the rates of change of physical quantities to the values of those quantities themselves. Such calculated predictability of physical phenomena led to the concept of a mechanistic, clockwork universe that operated according to deterministic laws. The idea that the universe operated in strict accord with physical laws was profoundly influential on science, philosophy and theology.

Most physical models are devoted to the understanding of simple systems (e.g., kinetic molecular theories often rely on concepts related to a ball bouncing in a box). From fundamental laws, using easily quantifiable behavior of such simple systems, theorists often attempt to project the behavior of more complex systems (e.g., the collision and dynamics of hundreds of balls bouncing in a box). It was long thought by physicists that, with regard to these types of models, the complexity of a system simply veiled an underlying fundamental simplicity.

For example, according to classical deterministic concepts, the accurate analysis and prediction of complex systems (e.g., the determination of the momentum of a particular ball among hundreds of other balls bouncing and colliding in a box) could be calculated only if the initial or starting conditions were accurately known. The fact that it is usually impossible to predict the exact condition or behavior of a system (especially considering that such interactions or measurements of a systems must also alter the system itself) is usually explained away as the result of a lack of knowledge regarding starting conditions or a lack of calculating vigor (e.g., inadequate computing power).

See also Atmospheric circulation; Weather forecasting methods; Weather forecasting

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Chaos Theory

Chaos theory

Chaos theory is the study of complex systems that, at first glance, appear to follow no orderly laws of mathematics or science. Chaos theory is one of the most fascinating and promising developments in late-twentieth-century mathematics and science. It provides a way of making sense out of phenomena such as weather patterns that seem to be totally without organization or order.

Cause-and-effect and chaos

Scientists have traditionally had a rather strict cause-and-effect view of the natural world. English physicist Isaac Newton once said that if he could know the position and motion of every particle in the universe at any one moment, he could predict the future of the universe into the infinite future. He believed that all those particles follow strict physical laws. Since he knew (or so he thought) what those laws were, all he had to do was to apply them to the particles at any one point in time.

On the other hand, scientists have always realized that some events in nature appear to be just too complex to analyze by the laws of science. One of the best examples is weather patterns. Even though scientists know a great deal about the elements that make up weather, they have a very difficult time predicting what weather patterns will be. The term chaos has often been used to describe systems that are just too "messy" to understand by scientific analysis.

Origins of chaos theory

The rise of modern chaos theory can be traced to a few particularly striking and interesting discoveries. One of these events occurred in the 1890s when French mathematician Henri Poincaré was working on the problem of the interactions of three planets with one another. The problem should have been fairly straightforward, Poincaré thought, since the gravitational laws involved were well known. The results of his calculations were so unexpected, however, that he gave up his work. He described those results as "so bizarre that I cannot bear to contemplate them."

Dutch engineer B. van der Pol encountered a similar problem in working with electrical circuits. He started out with systems that could easily be described by well-known mathematical equations. But the circuits he actually produced gave off unexpected and irregular noises for which he could not account.

Words to Know

Attractor: An element in a chaotic system that appears to be responsible for helping the system to settle down.

Cause-and-effect: The view that humans can understand why certain events (effects) take place.

Chaos: Some behavior that appears to be so complex as to be incapable of analysis by humans.

Chaos theory: Mathematical and scientific efforts to provide cause- and-effect explanations for chaotic behavior.

Generator: Elements in a system that appear to be responsible for chaotic behavior in the system.

Law: A statement in science that summarizes how some aspect of nature is likely to behave. Laws have survived many experimental tests and are believed to be highly dependable.

Then, in 1961, American meteorologist Edward Lorenz found yet another example of chaotic behavior. Lorenz developed a system for predicting the weather based on 12 equations. The equations represented the factors we know to affect weather patterns, including atmospheric pressure, temperature, and humidity. What Lorenz found was that by making very small changes in the initial numbers used in these equations, he could produce wildly different results.

Generators and attractors

Scientists and mathematicians now view chaotic behavior in a different way. Instead of believing that such behavior is too complex ever to understand, they have come to conclude that certain patterns exist within chaos that can be discovered and analyzed. For example, certain characteristics of a system appear to be able to generate chaotic behavior. Such characteristics are known as generators because they cause the chaotic behavior. Very small differences in a generator can lead to very large differences in a system at a later point in time.

Researchers have also found that chaotic behavior sometimes has a tendency to settle down to some form of predictable behavior. When this happens, elements within the system appear to bring various aspects of the chaos together into a more understandable pattern. Those elements are given the name attractors because they appear to attract the parts of a chaotic system to themselves.

Applications

In theory, studies of chaos have a great many possible applications. After all, much of what goes on in the world around us seems more like chaos than a neat orderly expression of physical laws. The weather may be the best everyday example of that point. Although we know a great deal about all the elements of which weather patterns are made, we still have relatively modest success in predicting how those elements will come together to produce a specific weather pattern. Studies of chaos theory may improve these efforts.

Animal behavior also appears to be chaotic. Population experts would like very much to know how groups of organisms are likely to change over time. And, again, we know many of the elements that determine those changes, including food supplies, effects of disease, and crowding. Still, predictions of population changeswhether of white deer in the wilds of Vermont or the population of your hometowntend to be quite inaccurate. Again, chaos theory may provide a way of making more sense out of such apparently random behavior.

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Chaos Theory

Chaos Theory

BIBLIOGRAPHY

Chaos theory is a theory of systems dynamics; that is, the analysis of the laws of motion of various systems over time. Complex dynamics investigates why certain systems, while evolving in a predictable fashion for some time, may display at other times a behavior, which looks erratic (random), hence unpredictable. In this context, the main thrust of chaos theory has been to demonstrate that behind this seemingly random behavior or disorder, there is, however, a deterministic underlying structure, which can be described and analyzed by means of differential equations that do not involve uncertainty.

Formally chaos is a nonlinear deterministic process that looks random, a case in which a dynamic mechanism yields a time path so erratic that it passes most standard statistical tests of randomness. Chaotic time paths often have the following features: (1) a trajectory that sometimes displays sharp qualitative changes, such as those associated with large random disturbances; (2) a time path that is extremely sensitive to microscopic changes in the values of its parameters; and (3) a time path that never returns to any point it had previously traversed, but which may, however, display an oscillatory pattern in a certain bounded region. The terms chaos, strange attractors, and complex dynamics have been used interchangeably in the literature to characterize these complex processes.

In linear dynamics, small causes give rise to small effects, and large causes produce large effects. Hence there is a certain sense of proportionality in linear thinking. Nonlinearity, on the other hand, connotes lack of proportionality: very small causes (small changes in the initial conditions, for instance) can give rise to very large effects. One implication of chaos theory, in this context, is to show that nonlinearities are not the exception but the rule of nature and life. Weather forecasting, for instance, is difficult because very small fluctuations in the environment give rise to very large-scale changes. Chaotic systems often possess fractal structures and time-dependent feedback mechanisms. A fractal structure consists of two major features: self-similarity (or scale-invariance) and lack of smoothness. Self-similarity refers to a system that always looks the same regardless of how many times the system is magnified. On the other hand, lack of smoothness relates to the disconnected appearance of fractals.

While the origins of chaos theory date back to the seminal work conducted in 1890 by the French mathematician Jules Henri Poincaré on the so-called three-body problem, it is Edward Lorenzs 1963 research on atmospheric dynamics and Benoit Mandelbrots pathbreaking 1983 investigation of fractal geometry that have rekindled interest in chaos theory since the mid-twentieth century. Applications of complex dynamics have found fertile grounds in several fields such as fluid dynamics, plasma physics, chemistry, electrical engineering, signal processing, cardiology, finance, and time series econometrics and economics, some of which are surveyed in Julian Sprotts Chaos and Time Series Analysis (2003).

The roots of chaos analysis in economics can be traced back to the literature on business cycles, that is, the analysis of irregular fluctuations in the output level of an economy. While exogenous business cycle theories have become the orthodoxy during the last four decades, the emergence of chaos theory hinted at the possibility that erratic output fluctuations are due to the complex interaction of economic factors, and hence may be endogenously generated.

Chaos studies in theoretical economics have attempted to model economic systems in such a way that chaotic dynamics emerge during the adjustment period to equilibrium or in the evolution of the system itself over time. On the other hand, several empirical investigations have been done in order to identify chaotic behavior in financial and economic time-series data. While earlier empirical studies claimed to have found evidence of chaotic behavior in a number of economic time series, such as U.S. business cycle data, various monetary aggregates, and precious metal prices, subsequent research, such as that conducted by Aydin Cecen and Cahit Erkal in 1996, demonstrated that there is little evidence in favor of deterministic chaos in exchange rate returns.

SEE ALSO Catastrophe Theory; Shocks

BIBLIOGRAPHY

Cecen, Aydin A., and Cahit Erkal. 1996. Distinguishing Between Stochastic and Deterministic Behavior in Foreign Exchange Rate Returns: Further Evidence. Economics Letters 51: 323-329.

Lorenz, Edward. 1963. Deterministic Nonperiodic Flow. Journal of the Atmospheric Sciences 20: 130-141.

Mandelbrot, Benoit B. 1983. Fractal Geometry of Nature. New York: W. H. Freeman.

Poincaré, Jules Henri. 1890. Sur le Problème de Trois Corps et les Equations de la Dynamique. Acta Mathematica 13: 1-270.

Sprott, Julian C. 2003. Chaos and Time Series Analysis. New York: Oxford University Press.

Aydin A. Cecen

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Chaos Theory

CHAOS THEORY

Chaos theory was originally a branch of mathematical physics developed in 1963 by Edward Lopez. It deals with events and processes that cannot be modeled or predicted using conventional mathematical laws and theorems, such as those of probability theory or biostatistics. Chaos theory is concerned with finding rational explanations for such phenomena as unexpected changes in weather. The theory assumes that small, localized perturbations in one part of a complex system can have widespread consequences throughout the system. The vivid example often used to describe this concept, known as the "butterfly effect," is that the beating of a butterfly's wings can lead to a hurricane if the tiny turbulence it causes happens to generate a critical combination of air pressure changes. The key word here is "if," and much of chaos theory is concerned with attempts to model circumstances based on this conditional conjunction. Unpredictable events in medicine, such as the course of certain cancers and the fluctuations in frequency of some diseases, may be attributable to chaos theory.

Toward the end of the twentieth century, humankind began to face a crisis that required comprehension of chaos theory. A complex set of natural and human-induced changes in global ecosystemsglobal climate change, stratospheric ozone attenuation, species extinctions and reduced biodiversity, social and demographic turbulence, economic globalization, technological revolutions in communications, violent regional conflicts, and political instabilitybegan to have far-reaching implications. These factors all interact in complex ways that impact on human health and well-being and could cause dramatic changes in the prevailing patterns of health and disease. Public health policies and long-range plans require certain assumptions about the likelihood that events will occur in accordance with known trends about which data and information exist. Such plans tend to be based on extrapolations of trends, such as population numbers and age distributions, and the impact of specific diseases are projected.

Scenarios in long-rang health planning need to take chaos theory into account if they are to cover all possible contingencies. For example, in long-range health plans for the application of technology to the diagnosis and treatment of cancers or coronary heart disease, one or more scenarios must take into account the possibility of innovative technical breakthroughs. This approach to long-range health plans has benefited from lessons learned by the oil and petrochemical industries in the 1970s, when only one major oil company was prepared for a sudden reduction in the available crude oil supply. This company's plans had included supply change as one possible scenario. In the health sector, long-range plans are flawed if they do not allow for chaotic events.

John M. Last

(see also: Catastrophe Theory; Planning for Public Health )

Bibliography

Kiel, L. D., and Elliot, E. eds. (1996). Chaos Theory in the Social Sciences: Foundations and Applications. Ann Arbor: University of Michigan Press.

Robertson, R., and Combs, A., eds. (1995). Chaos Theory in Psychology and the Life Sciences. Hillsdale, NJ: Lawrence Erlbaum Associates.

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chaos theory

chaos theory, in mathematics, physics, and other fields, a set of ideas that attempts to reveal structure in aperiodic, unpredictable dynamic systems such as cloud formation or the fluctuation of biological populations. Although chaotic systems obey certain rules that can be described by mathematical equations, chaos theory shows the difficulty of predicting their long-range behavior. In the last half of the 20th cent., theorists in various scientific disciplines began to believe that the type of linear analysis used in classical applied mathematics presumes an orderly periodicity that rarely occurs in nature; in the quest to discover regularities, disorder had been ignored. Thus, chaos theorists have set about constructing deterministic, nonlinear dynamic models that elucidate irregular, unpredictable behavior (see nonlinear dynamics). Some of the early investigators of chaos were the American physicist Mitchell Feigenbaum; the Polish-born mathematician and inventor of fractals (see fractal geometry) Benoit Mandelbrot; the American mathematician James Yorke, who popularized the term "chaos" ; and the American meteorologist Edward Lorenz.

See J. Gleick, Chaos: Making a New Science (1987); I. Stewart, Does God Play Dice?: The Mathematics of Chaos (1989); A. A. Tsonis, Chaos: From Theory to Applications (1992); D. N. Chorafas, Chaos Theory in the Financial Markets (1994).

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