Chaotic Systems

Summary

Chaotic systems theory is a scientific field blending mathematics, statistics, and philosophy that was developed to study systems that are highly complex, unstable, and resistant to exact prediction. Chaotic systems include weather patterns, neural networks, some social systems, and a variety of chemical and quantum phenomena. The study of chaotic systems began in the nineteenth century and developed into a distinct field during the 1980s.

Chaotic systems analysis has allowed scientists to develop better prediction tools to evaluate evolutionary systems, weather patterns, neural function and development, and economic systems. Applications from the field include a variety of highly complex evaluation tools, new models from computer and electrical engineering, and a range of consumer products.

Definition and Basic Principles

Chaotic systems analysis is a way of evaluating the behavior of complex systems. A system can be anything from a hurricane or a computer network to a single atom with its associated particles. Chaotic systems are systems that display complex nonlinear relationships between components and whose ultimate behavior is aperiodic and unstable.

Chaotic systems are not random but rather deterministic, meaning that they are governed by some overall equation or principle that determines the behavior of the system. Because chaotic systems are determined, it is theoretically possible to predict their behavior. However, because chaotic systems are unstable and have so many contributing factors, it is nearly impossible to predict the system's behavior. The ability to predict the long-term behavior of complex systems, such as the human body, the stock market, or weather patterns, has many potential applications and benefits for society.

The butterfly effect is a metaphor describing a system in which a minor change in the starting conditions can lead to major changes across the whole system. The beat of a butterfly's wing could, therefore, set in motion a chain of events that could change the universe in ways that have no seeming connection to a flying insect. This kind of sensitivity to initial conditions is one of the basic characteristics of chaotic systems.

Background and History

French mathematician Henri Poincaré and Scottish theoretical physicist James Clerk Maxwell are considered two of the founders of chaos theory and complexity studies. Maxwell and Poincaré both worked on problems that illustrated the sensitivity of complex systems in the late nineteenth century. In 1960, meteorologist Edward Lorenz coined the term "butterfly effect" to describe the unstable nature of weather patterns.

By the late 1960s, theoreticians in other disciplines began to investigate the potential of chaotic dynamics. Ecologist Robert May was among the first to apply chaotic dynamics to population ecology models, with considerable success. Mathematician Benoit Mandelbrot also made a significant contribution in the mid-1970s with his discovery and investigation of fractal geometry.

In 1975, University of Maryland scientist James A. Yorke coined the term "chaos" for this new branch of systems theory. The first conference on chaos theory was held in 1977 in Italy, by which time the potential of chaotic dynamics had gained adherents and followers from many different areas of science.

Over the next two decades, chaos theory continued to gain respect within the scientific community, and the first practical applications of chaotic systems analysis began to appear. In the twenty-first century, chaotic systems have become a respected and popular branch of mathematics and system analysis.

How It Works

There are many kinds of chaotic systems, but all chaotic systems share certain qualities. They are unstable, dynamic, and complex. Scientists studying complex systems generally focus on one of these qualities in detail. There are two ultimate goals for complex systems research—to predict the evolution of chaotic systems and to learn how to manipulate complex systems to achieve some desired result.

Instability. Chaotic systems are extremely sensitive to changes in their environment. Like the example of the butterfly, a seemingly insignificant change can become magnified into systemwide transformations. Because chaotic systems are unstable, they do not settle to equilibrium and can therefore continue developing and leading to unexpected changes. In the chaotic system of evolution, minor changes can lead to novel mutations within a species, which may eventually give rise to a new species. This kind of innovative transformation is what gives chaotic systems a reputation for being creative.

Scientists often study complexity and chaos by creating simulations of chaotic systems and studying the way the systems react when perturbed by minor stimuli. For example, scientists may create a computer model of a hurricane and alter small variables, such as temperature, wind speed and other factors, to study the ultimate effect on the entire storm system.

Strange Attractors. An attractor is a state toward which a system moves. The attractor is a point of equilibrium where all the forces acting on a system have reached a balance and the system is in a state of rest.

Because of their instability, chaotic systems are less likely to reach a stable equilibrium and instead proceed through a series of states, which scientists sometimes call a dynamic equilibrium. In dynamic equilibrium, the system constantly moves toward some attractor, but as it moves toward the first attractor, forces begin building, creating a second attractor. The system then shifts from the first to the second attractor, leading to the formation of a new attractor, and the system shifts again. The forces pulling the system from one attractor to the next never balance each other completely, so the system never reaches absolute rest but continues changing.

Visual models of complex systems and their strange attractors display what mathematicians call fractal geometry. Fractals are patterns that, like chaotic systems, are nonlinear and dynamic. Fractal geometry occurs throughout nature, including in the formation of ice crystals and the branching of the circulatory system in the human body. Scientists study the mathematics behind fractals and strange attractors to find patterns that can be applied to complex systems in nature. The study of fractals has yielded applications for medicine, economics, and psychology.

Emergent Properties. Chaotic systems also display emergent properties, which are system-wide behaviors that are not predictable from knowledge of the individual components. This occurs when simple behaviors among the components combine to create more complex behaviors. Common examples in nature include the behavior of ant colonies and other social insects.

Mathematic models of complex systems also yield emergent properties, indicating that this property is driven by the underlying principles that lead to the creation of complexity. Using these principles, scientists can create systems, such as computer networks, that also display emergent properties.

Applications and Products

Medical Technology. The human heart is a chaotic system, and the heartbeat is controlled by a set of nonlinear, complex variables. The heartbeat appears periodic only when examined through an imprecise measure such as a stethoscope. The actual signals that compose a heartbeat occur within a dynamic equilibrium.

Applying chaos dynamics to the study of the heart allows physicians increased accuracy in determining when a heart attack is imminent by detecting minute changes in rhythm that signify the potential for the heart to veer away from its relative rhythm. When the rhythm fluctuates too far, physicians use a defibrillator to shock the heart back to its rhythm. A new defibrillator model developed by scientists in 2006 and 2007 uses a chaotic electric signal to effectively force the heart back to its regular rhythmic pattern. Other medical applications of chaos theory include optimizing medical resource allocation, aiding the sustainable development of human health, promoting medical research development, and improving accuracy in treating and diagnosing diseases.

Consumer Products. In the mid-1990s, the Korean company Goldstar manufactured a chaotic dishwasher that used two spinning arms operated with an element of randomness to their pattern. The chaotic jet patterns were intended to provide greater cleaning power while using less energy.

Chaotic engineering has also been used in the manufacture of washing machines, kitchen mixers, and a variety of other simple machines. Although most chaotic appliances have been more novelty than innovation, the principles behind chaotic mechanics have become common in the engineering of computers and other electrical networks.

Business and Industry. The global financial market is a complex, chaotic system. When examined mathematically, fluctuations in the market appear aperiodic and have nonlinear qualities. Although the market seems to behave in random ways, many believe that applying methods created for the study of chaotic systems will allow economists to elucidate hidden developmental patterns. Knowledge of these patterns might help predict and control recessions and other major changes well before they occur.

The application of chaos theory to market analysis produced a subfield of economics known as fractal market analysis, wherein researchers conduct economic analyses by using models with fractal geometry. By looking for fractal patterns and assuming that the market, like other chaotic systems, is highly sensitive to small changes, economists have been able to build more accurate models of market evolution.

Robotics. Scientists aim to develop more sophisticated, intelligent, and reliable autonomous robots. Chaotic system–integrated robots may be the future of robotics. Various types of chaotic dynamics can aid the accurate movement planning in autonomous robots. For various search and surveillance tasks during military and other vital operations, autonomous robots integrated with a chaotic controller provide an upper hand compared to the other methods. Chaotic systems integrated with various modern techniques like neural networks and machine learning help make modern autonomous robots smarter and more reliable. The behavior of these autonomous robots is much closer to natural systems compared to past generations of robots. Scientists are increasingly approaching artificial intelligence (AI) with chaos techniques. AI-extended modeling frameworks allow significant improvements in robotics language technology.

Careers and Course Work

Those seeking careers in chaotic systems analysis might begin by studying mathematics or physics at a university. Those with backgrounds in other fields, such as biology, ecology, economics, sociology, and computer science, might also choose to focus on chaotic systems during their graduate training.

There are a number of graduate programs offering training in chaotic dynamics. For instance, the Center for Interdisciplinary Research on Complex Systems at Northeastern University in Boston, Massachusetts, offers programs training students in many types of complex system analyses. The Center for Nonlinear Phenomena and Complex Systems at the Université Libre de Bruxelles offers programs in thermodynamics and statistical mechanics.

Careers in chaotic systems span a range of fields, from engineering and computer network design to theoretical physics. Trained researchers can choose to contribute to academic analyses and other pertinent laboratory work or focus on creating applications for immediate consumer use.

Social Context and Future Prospects

As chaotic systems analysis spread in the 1980s and 1990s, some scientists began theorizing that chaotic dynamics might be an essential part of the search for a grand unifying theory or theory of everything. The grand unifying theory is a concept that emerged in the early nineteenth century when scientists began theorizing that there might be a single set of rules and patterns underpinning all phenomena in the universe.

The idea of a unified theory is controversial, but the search has attracted numerous theoreticians from mathematics, physics, theoretical biology, and philosophy. As research began to show that the basic principles of chaos theory could apply to a vast array of fields, some began theorizing that chaos theory was part of the emerging unifying theory.

Because many systems meet the basic requirements to be considered complex chaotic systems, the study of chaos theory and complexity has room to expand. Scientists and engineers have only begun to explore the practical applications of chaotic systems, and theoreticians are still attempting to evaluate and study the basic principles behind chaotic system behavior.

Bibliography

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