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 (1854–1912), 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 principles—the 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 goal—it 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 lines—from 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 phenomena—the 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 theory—not 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 will—all 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 smart—smart 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 exciting—a 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): 339–372.

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): 43–54.

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.

CHAOS THEORY

CHAOS THEORY . In the Principia (1687), Isaac Newton gave an account of mechanics formulated in terms of precise equations of motion. Given the initial conditions of a system, it was possible to predict completely its future behavior and to retrodict its past. Newton himself did not take a purely mechanical view of the world. There was the mysterious force of gravity, concerning whose origin and nature he declined to frame a hypothesis, and he also believed that the maintenance of the stability of the solar system would require occasional angelic intervention. Newton's eighteenth-century successors, however, had different opinions, and they celebrated the triumph of mechanical thinking. Julien de La Mettrie (1709–1751) wrote his book Man the Machine (1748), and Pierre-Simon de Laplace (1749–1827), in his great work on celestial mechanics, believed that he had established the natural stability of the solar system, so that appeal to the hypothesis of divine assistance in its preservation was no longer necessary.

The nineteenth-century development of field theories, inspired by the insights of Michael Faraday (1791–1867) and James Clerk Maxwell (1831–1879), did not essentially change the picture. The partial differential equations of a field theory are as deterministic in consequence as are the ordinary differential equations of Newtonian mechanics. Classical physics, as this whole body of theory is called, appeared to present the image of a clockwork universe, whose Creator could be no more than the Cosmic Clockmaker.

Unpredictabilities

Twentieth-century physics, however, saw the death of a merely mechanical understanding of the world. This came about through the discovery of widespread intrinsic unpredictabilities present in physical process, of a kind resulting from the way things actually are and not simply from deficiencies in experimental or calculational techniques. These unpredictabilities first manifested themselves in quantum theory's account of atomic and subatomic phenomena. In considering, for instance, the decay of a radioactive nucleus, scientists could do no more than assign a certain probability that such decay might occur in a given period of time. Newton's predictive style of reasoning had to be replaced by a purely probabilistic approach. Quantum effects, however, were only directly observable at the level of microscopic process, remote from the macroscopic experience of everyday reality. The latter was still the realm of classical physics.

In the 1960s many scientists began to realize that even classical physics was not as tame and controllable as had been supposed. There are certainly many systems that behave as if they were reliably predictable "clocks," but there are also other systems that behave like "clouds," that is to say their behavior is so sensitive to the fine detail of their circumstances that the slightest disturbance will radically alter their future behavior. One of the pioneers in making this discovery was the theoretical meteorologist Edward Lorenz (1917–). Lorenz had been studying certain equations that corresponded to a highly simplified model of a weather system. He determined that very slight changes in the input of initial conditions into his equations would totally change the character of their output predictions. This exquisite sensitivity to detail has come to be expressed through a serious scientific joke, the butterfly effect : the Earth's weather systems can be in so sensitive a state that a butterfly, stirring the air with its wings in the Amazonian jungle today, could produce effects that escalate until they result in a storm over New York City in about three weeks time! Obviously, according to this model, long-term weather forecasting is never going to work; future storms are intrinsically unpredictable because no one can know about all those butterflies.

Lorenz's surprising discovery had been anticipated by the French mathematician, Jules-Henri Poincaré (1854–1912). In 1889 he published a study of the gravitational three-body problem in which he showed that it did not always possess smoothly predictable solutions of the kind that Laplace had assumed.

Chaos Theory

The analysis of the sensitive and unpredictable behavior of various systems has been called chaos theory. Typically its occurrence is found to arise from equations that have the properties of reflexivity (they turn back upon themselves) and nonlinearity (doubling the input does not double the output, but it changes it in a much more radical way). In principle, the equations are exactly deterministic, in the sense that an absolutely precise input will yield an absolutely precise output, but in practice, because in the real world no initial conditions can be known with arbitrary accuracy, unpredictability results from the uncontrollable effects of residual ignorance. Deterministic chaos, as it is often called, gives rise to apparently random behavior.

In fact, the term chaos was somewhat ill chosen. Systems of this kind exhibit a kind of interlacing of order and disorder, which can best be illustrated by the case of dissipative systems that feature a degree of friction acting in the process. In this case, the future behavior is not totally haphazard; rather, its possibilities are contained within an extensive though limited portfolio of options called a strange attractor. The word "attractor" expresses the system's rapid convergence onto this range of possibilities, and "strange" refers to the mathematically intricate shape of this form of possibility. It turns out that the latter is characterized by fractal geometry, a "jagged" range of possibilities in striking contrast to the smoothly varying expectations of conventional classical physics. A fractal presents a pattern that appears essentially the same on whatever scale it is sampled—one might think of them as saw-teeth, themselves saw-toothed; the pattern continues to infinity. The interlacing of order and disorder in chaos theory connects with an important scientific insight. It has come to be recognized that the emergence of novelty requires a state of affairs that can be characterized as being "at the edge of chaos." To be too much on the orderly side of that border would correspond to a situation possessing a degree of rigidity that permitted only rearrangements to occur but did not allow the emergence of genuine novelty. On the other hand, to be too far on the haphazard side of the border would correspond to a situation so unstable that no novelty could persist. Fruitfulness requires a subtle balance between order and openness.

Metaphysics

The unpredictability in chaos theory is an epistemological property, telling us that we cannot know beforehand what the future behavior of a chaotic system will prove to be. There is no inescapable connection between epistemology (what we know) and ontology (what is the case). No logical entailment links the two together. Instead, the relationship is a matter for metaphysical decision (and so, for philosophical argument).

Immanuel Kant (1724–1804) maintained that the appearances of phenomena are no guide to the nature of noumena, or the character of things in themselves. On the other hand, most scientists, either consciously or unconsciously, assume a realist position, believing that scientific knowledge gives access to the nature of the physical world.

In the case of the intrinsic unpredictabilities of quantum theory, almost all physicists have adopted a realistic interpretation of phenomena. Werner Heisenberg's uncertainty principle is not regarded simply as an epistemological principle of ignorance, but it is taken to be an ontological principle of actual indeterminacy. The fact that the work of quantum physicist David Bohm (1917–1992) has provided an alternative option, of equal empirical adequacy, which offers a deterministic interpretation of quantum phenomena, shows that the majority position is indeed a matter of metaphysical choice rather than physical necessity.

In the case of the intrinsic unpredictabilities of chaos theory, however, the majority decision has gone the other way. Most physicists disconnect epistemology and ontology, concluding that the theory shows that deterministic equations are consistent with the appearance of random behavior. "Deterministic chaos" is indeed the way they think about the theory. This absence of any willingness to question the assumption of underlying determinism seems to have been influenced by the feeling of deep respect accorded to the historic equations of classical physics.

Yet a different strategy is metaphysically possible. This would involve interpreting the unpredictabilities as signs of ontological openness. In turn, this strategy would necessitate a reinterpretation of the equations of "deterministic chaos." The sensitivity of chaotic systems to the details of their circumstances provides a way in which such a reinterpretation can be accomplished. Because of such sensitivity, chaotic systems can never properly be treated in isolation. Their vulnerability to the slightest disturbance means that they are intimately linked to their environment. Yet the experimental support that "verifies" the laws of classical physics has all been obtained through investigations of situations in which the assumption of isolatability is an acceptable idealization. Otherwise, analysis of what was going on would have been too complex to be feasible. It is perfectly possible, therefore, to make the metaphysical conjecture that the supposed laws of classical physics are actually no more than approximations of the behavior of what may be conceived actually to be a more subtle and supple physical reality. The approximation involved could be called downward emergence, because it would relate to behavior observable only in the idealized circumstance of an isolated system.

In this metaphysical scheme, epistemological unpredictability is the sign of an ontological openness. Such openness is not meant to imply that the future is some sort of random lottery, but rather that the causal principles involved are more than those described simply by the conventional physical picture of the exchange of energy between constituents. The unisolatibility of chaotic systems means that though such systems may be made up of components, they also must be considered holistically, in the context of their totality. The variety of the different possible patterns of their future behavior, represented by the different ways in which a chaotic system might traverse its strange attractor, are not discriminated from each other by energy differences, but by differing patterns of dynamical behavior, characterized by different expressions of "information." These considerations led to the metaphysical conjecture that chaos theory should be interpreted as affording scope for a new kind of causal agency, having the character of being top-down (influence of the whole upon the parts) and corresponding to an input of information (specification of patterned dynamical behavior). One might summarize the proposal as suggesting the concept of holistic causality through active information.

Two general considerations may be offered in support of such a metaphysical project. One is that the stance taken accords with the realist strategy followed by almost all physicists in the case of quantum theory. There is no apparent reason to treat quantum theory and chaos theory differently in this respect. Second, human experience of agency, of the willed execution of the intentions of the whole person, encourages the belief that an account of the causal nexus of the world is needed, an account that goes beyond simple constituent notions. The concept of top-down causality, operating through the input of information, offers the glimmer of a prospect of how one might begin the task of reconciling the scientific account of process with the human experience of agency.

The theologian may also find these ideas to be of use in thinking about divine providential interaction with creation. If the causal grain of nature is open in the way suggested, there seems to be no difficulty in believing that the Creator also interacts with the unfolding history of creation through the input of active information.

Quantum Chaology

At first sight it might seem that the metaphysical problems discussed above might readily be solved by combining the insights of quantum physics and chaos theory.

The behavior of chaotic systems quite rapidly comes to depend upon the details of circumstances lying at the level of Heisenberg uncertainty or below. It might seem attractive, therefore, to appeal to the generally accepted indeterministic character of quantum process to induce openness in the behavior of chaotic systems. Yet this approach faces serious difficulties, resulting from a lack of understanding of how one might consistently combine quantum theory and chaos theory. Indeed, these theories appear to be mutually incom-patible.

Quantum theory possesses an intrinsic scale, set by Max Planck's fundamental constant. As a result, in quantum thinking it is possible to give a meaning to terms such as large or small. We have seen that chaos theory, on the contrary, is scale-free because of its fractal character, implying that everything looks roughly the same on whatever scale it is surveyed. Clearly some significant modification of thinking would be required to bring the two theories together. Modern physics has only a rather patchy picture of the causal nature of reality, and is far from being able to offer an integrated account, applicable at all levels.

The complexity of the considerations involved can be illustrated by the behavior of Hyperion, one of the moons of Saturn. It is an irregularly shaped piece of rock about the size of New York City, which is observed to be tumbling chaotically. Quantum effects, with their imposition of scale, would be expected to suppress this chaotic motion very effectively for so large an object. Calculations made on this basis indicate that tumbling should last for only about thirty-seven years. However another effect, called decoherence, due to the environmental influence of the radiation that bathes Hyperion, in its turn suppresses the quantum effects and explains why the moon's chaotic behavior can be expected to continue almost indefinitely. The causal nexus of the world is very complex, with a variety of effects interlacing.

See Also

Physics and Religion.

Bibliography

Gleick, James. Chaos: Making a New Science. London, 1988. An excellent introduction to chaos theory for the general reader.

Peitgen, Heinz-Otto, and Peter Richter. The Beauty of Fractals. Berlin, 1986. Lavishly illustrated account of fractal geometry.

Polkinghorne, John. Belief in God in an Age of Science. New Haven, Conn., 1998. Chapter 3 gives an account of divine action based on an ontological interpretation of chaos theory.

Prigogine, Ilya. The End of Certainty: Time, Chaos, and the New Laws of Nature. New York, 1997. A Nobel laureate's account of the openness of physical process.

Ruelle, David. Chance and Chaos. Princeton, N.J., 1991. Accessible survey by a distinguished mathematical physicist.

Russell, Robert, Nancey Murphy, and Arthur Peacocke, eds. Chaos and Complexity: Scientific Perspectives on Divine Action. Vatican City, 1995. Conference proceedings reporting a variety of points of view and focusing on questions of divine action.

Saunders, Nicholas. Divine Action and Modern Science. Cambridge, UK, 2002. A careful and comprehensive survey of current issues.

John Polkinghorne (2005)

CHAOS THEORY

A physical system has chaotic dynamics, according to the dictionary, if its behavior depends sensitively on its initial conditions, that is, if systems of the same type starting out with similar sets of initial conditions can end up in states that are, in some relevant sense, very different. But when science calls a system chaotic, it normally implies two additional claims: That the dynamics of the system is relatively simple, in the sense that it can be expressed in the form of a mathematical expression having relatively few variables, and that the geometry of the system's possible trajectories has a certain aspect, often characterized by a strange attractor.

Chaos theory proper, it should be noted, has its home in classical physics (and other kinds of dynamics that share the relevant properties of classical physics). The extent to which chaotic mathematics is fruitful in understanding the quantum realm is still a matter of debate.

Sensitive Dependence on Initial Conditions

In the popular imagination a chaotic system is one whose future state may be radically altered by the smallest of perturbations—as when the fluttering of a butterfly's wings creates a disturbance whose size is inflated to the point where it tips the meteorological balance on the other side of the globe, creating a tornado where there would otherwise have been none. Though the "butterfly effect" marvelously engages human fear and wonder at the unpredictability of things, it captures rather less completely what is interesting and distinctive about modern chaos theory.

The idea of an inherent unpredictability in human and other affairs due to the inflation of small disturbances is an old one. Swift wrote in Thoughts on Various Subjects (1711) that "A Wise man endeavors, by considering all Circumstances, to make Conjectures, and form Conclusions: But the smallest Accident intervening, (and in the Course of Affairs it is impossible to see all) doth often produce such Turns and Changes, that at last he is just as much in doubt of Events, as the most ignorant and unexperienced Person" (p. 415).

Modern mathematics is able to characterize the sensitivity of initial condition dependence in various ways that lie far beyond Swift's means. Notions such as the Liapunov exponent help to quantify the speed at which the trajectories of systems starting out with similar initial conditions will diverge. Measure theory quantifies something like the chance that a small initial difference will lead to a relatively large difference in outcome, in systems where not every small change makes such a difference. There is nothing here, though, that would have astounded Swift.

Simplicity

The central insight of chaos theory is that systems governed by simple equations, that is, systems whose behavior can be characterized by a small number of variables, called low dimensional systems, are often sensitive to initial conditions. At first blush this realization has a pessimistic cast. Most obviously it leads to the conclusion that even a simple dynamics may be unpredictable in the medium to long term, as which of two significantly different outcomes occurs may depend on such first details of the initial conditions as to lie beyond the resolving power any reasonable observational effort.

Somewhat less obviously certain kinds of sensitivity to initial conditions impede systematic dynamical understanding. A famous example closely connected to the origins of chaos theory is the three body problem, the task of elucidating all the properties of the dynamics of a three body system in Newtonian gravitational theory. In 1890 Henri Poincaré showed that three body systems can tend to chaos in the modern sense of the word, and concluded that a systematic treatment of three body dynamics would be difficult if not impossible.

Chaos can be an impediment to prediction and systematic understanding in low dimensional systems then. However, if low dimensional chaos is bad news for the study of systems known to have low-dimensional dynamics, it is good news for the study of systems known only to have chaotic dynamics. Traditionally such systems were modeled by complex equations, if at all; chaos theory introduces the serious possibility that these systems may be governed by equations with very few variables. Underlying the complex appearances may be a simple reality. The prospect of finding a hidden simplicity in such complex phenomena as turbulent flows, the weather, the movements of financial markets, and patterns of extinction is what most excites proponents of chaos theory. (Much the same prospect animates the advocates of catastrophe theory, the study of cellular automata, "complexity theory," and so on.)

To what extent can the nature of this hidden simplicity, if it exists, be divined? Given sensitive dependence on initial conditions, it is difficult to find the simple equation that best predicts the observed phenomena, since small errors in measuring initial conditions can make even the true model look like a bad predictor. More feasible is to infer some of the more interesting properties of the putative underlying law, such as the degree of sensitivity to initial conditions and certain geometrical aspects of the dynamics induced by the law (discussed below).

Under favorable conditions this information can be used to model accurately the behavior of chaotic systems to some extent—or at least that is the hope both of academic chaoticians and of those hoping to use the mathematics of chaos theory to beat the financial markets.

By far the boldest posit made in undertaking such work is the assumption that there is a simple dynamic law lying behind the subject system's complex behavior. For elaborate systems such as ecosystems and economies, the assumption of dynamic simplicity is often no more than a leap of faith; however, Strevens describes some circumstances in which ecosystems and some other complex systems have a low dimensional macrodynamics.

The Geometry of Chaos Trace the trajectory of a paradigmatically chaotic system through the space of possible states and the result is a complicated tangle of looping paths. It is the geometry of this tangle more than anything else—more even than sensitive dependence per se—that is distinctive of chaos (though there is disagreement as to which feature of the geometry is most important).

One especially striking feature of such trajectory tangles is their often-fractal structure: They cut out a shape in the space in which they are embedded so intricate that mathematicians ascribe it a fractional dimension. Such a shape is a strange attractor (strictly speaking an attractor only if it is a set of trajectories that systems starting from some points outside the attractor eventually join).

Many of the more interesting properties of chaotic systems can be understood as arising from the intricate geometry of the trajectory tangle. One well-known example is the appearance of "period-doubling cascades" in systems that are moving from a periodic to a chaotic regime of behavior: As some parameter affecting the system's dynamics is tweaked, the system first oscillates between two states, then between four states, then eight states, and so on, with shorter and shorter times between each successive doubling, until it goes chaotic. What is interesting about this behavior is that it turns up in many physically quite different kinds of systems, and that there are certain aspects of the period doubling, notably the rate at which the doubling increases, that are the same (in the limit) in these otherwise rather different systems. This universality in chaotic systems holds out the promise of understanding the behaviors of a considerable range of systems in terms of a single mathematical—in this case, it turns out, a geometrical—fact. So far however the wider significance of this understanding is unclear.

A more practical part of chaotic geometry is the use of limited data about the behavior of chaotic systems to reconstruct to a certain extent the geometry of the system's trajectory. Suppose that the behavior a chaotic system is characterized by three variables, so that the system's "trajectory tangle" is a subset of three-dimensional space. Suppose also that only a single property of the system's dynamics can be observed, a function of the values of the three variables. In favorable conditions, this single set of observations can be used to recover the geometrical structure of the three-dimensional dynamics. Various predictions, quantitative and qualitative, can then be made from the recovered geometry.

This is a powerful technique, as it assumes no knowledge of the number or even the nature of the underlying variables. However its success does depend on, among other things, the simplicity assumption explained above: The technique supposes that there are no more than a small number of variables.

Chaos and Probability

The disorderly behavior of chaotic systems can be called "random" in a loose and popular sense. Might the behavior of at least some such systems be random in a stronger sense? The suggestion that chaos might provide a foundation for probabilistic theories such as statistical mechanics has been one of the more fruitful contributions of chaos theory to philosophy.

The best scientific theories of certain deterministic or near deterministic systems are probabilistic. Perhaps the most prominent examples are the systems characterized by statistical mechanics and population genetics; the simplest examples are various gambling setups such as a roulette wheel or a thrown die. The probabilistic characterization of these systems is apt because the various events that make up their behavior (die throws or deaths, for example) are patterned in characteristically statistical ways, that is, in ways that are captured directly by one or other of the canonical probability distributions in statistical theory.

The mathematics of chaos offers an explanation of the probabilistic aspect of these patterns, and so offers an explanation of the success of probabilistic theories applied to certain sorts of deterministic systems.

The explanation, or rather the family of explanations, is quite complex, but it can be loosely characterized in the following way. A paradigmatically probabilistic pattern has two aspects: A short term disorder, or randomness, familiar to every gambler, and a long term order that is quantified by the statistics characterizing a probability distribution, such as the one-half frequency of "heads" in a long series of coin tosses.

Chaotic systems are capable of producing probabilistic patterns because they are capable of producing both this short term disorder and the requisite kinds of long-term order. The short-term disorder is due to the sensitive dependence on initial conditions; the long-term order to other aspects of the "geometry of chaos," principally chaotic dynamics' resemblance to a "stretch-and-fold" process.

Nowhere near all chaotic systems, it should be noted, generate probabilistic patterns. Indeed this area of investigation is not, in a certain sense, mainstream chaos theory: There are no strange attractors or period-doubling cascades, though there is a characteristically chaotic geometry to the relevant trajectory tangles. As well as explaining the success of probabilistic theorizing in science, chaos has been put forward—for much the same reasons—as a foundation for the metaphysics of probability, on the principle that what explains the probabilistic pattern is deserving to a considerable extent of the name probability.

Philosophical Significance

What is the philosophical significance of chaos? With respect to general philosophy of science, opinion is divided. Some philosophers, for example Stephen Kellert, have argued that chaos theory requires the abandoning of prediction as the touchstone of successful science, a new conception of the nature of scientific explanation, and the end of reductionism. Others, for example Peter Smith, have argued that these conclusions are too extreme, and that insofar as they are justified, chaos theory is not necessary for their justification, though it may well have brought to philosophy's attention problems previously wrongly ignored.

With respect to certain foundational questions about science, the significance of chaos is less controversial. The notion of determinism and (in the context of processes that are deterministic deep down) the notions of randomness and probability cannot be discussed without reference to work on dynamical systems since Poincaré that falls within the ambit—broadly conceived—of chaos theory.

See also Geometry; Philosophy of Physics; Poincaré, Jules Henri; Probability and Chance; Swift, Jonathan.

Bibliography

Bass, Thomas. The Predictors. New York: Henry Holt, 1999.

Belot, Gordon, and John Earman. "Chaos Out of Order: Quantum Mechanics, The Correspondence Principle and Chaos." Studies in History and Philosophy of Modern Physics 28 (1997): 147–182.

Earman, John. A Primer on Determinism. Dordrecht: Reidel, 1986.

Kellert, Stephen H. In the Wake of Chaos. Chicago: Chicago University Press, 1993.

Ornstein, Donald, and Benjamin Weiss. "Statistical Properties of Chaotic Systems." Bulletin of the American Mathematical Society 24 (1991): 11–116.

Sklar, Lawrence. Physics and Chance. Cambridge, U.K.: Cambridge University Press, 1993.

Smith, Peter. Explaining Chaos. Cambridge, U.K.: Cambridge University Press, 1998.

Stewart, Ian. Does God Play Dice? The Mathematics of Chaos. Oxford, Blackwell, 1989.

Strevens, Michael. Bigger than Chaos: Understanding Complexity through Probability. Cambridge, MA: Harvard University Press, 2003.

Suppes, Patrick. "Propensity Representations of Probability." Erkenntnis 26 (1987): 335–358.

Michael Strevens (2005)

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 (1869–1942) 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 (1854–1912) used features of CT to demonstrate the stability of the solar system, a result that Isaac Newton (1642–1727) 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): 16–31.

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

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 Lorenz’s 1963 research on atmospheric dynamics and Benoit Mandelbrot’s 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 Sprott’s 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