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 (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

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wolfgang achtner

taede a. smedes