Chaos Theory

Chaos theory, a mathematical notion describing underlying order that exists in some seemingly random events, has been employed by scholars of educational leadership as a metaphor or model for understanding the social conditions of schools. Sometimes also called “deterministic chaos,” chaos theory explores how apparently simple phenomena, describable with differential equations, can generate complex, unpredictable behavior. Put another way, some phenomena that are well understood and can be described mathematically still can produce apparently random effects. These “chaotic” processes essentially defy conventional mechanistic understandings of the universe that assume randomness exists only where humans do not possess full knowledge of factors at work. Instead, chaos theory holds that phenomena can be fully described by mathematical expressions and still exhibit random behavior in the sense that it cannot be known in advance.

The French mathematician, Henri Poincaré, developed the theoretical underpinnings for chaos theory in the late 1800s as he attempted to create a model for predicting the motions of objects within the solar system. By the mid-1900s, scientists and mathematicians capitalized on Poincaré's insights by identifying other phenomena that similarly could be described by differential equations, yet yield seemingly random behavior. Edward Lorenz, an early pioneer of the field, in 1961 analyzed weather data and noticed that resulting patterns, previously attributed to randomness or simple error, could be described by such equations—and furthermore, were sensitive to subtle initial differences in conditions. Essentially, Lorenz noted that weather fluctuations contained underlying order, though not predictability. Widespread use of computers in the 1970s and 1980s greatly facilitated the popular development and application of chaos theory by enabling the analysis of large quantitative data sets describing similar phenomena.

Two important implications follow from chaos theory. The first is the notion that some phenomena can be completely well understood yet ultimately remain unpredictable. This challenges the view that the universe behaves mechanistically, with linear causeeffect relationships. The second implication is that phenomena previously thought to be random or unexplainable now might yield to analysis that reveals chaotic behavior. Scholars in disciplines as diverse as economics, fluid dynamics, epidemiology, astronomy, and sociology have analyzed large quantitative data sets only to discover chaotic phenomena.

Books such as James Gleick's 1987 Chaos: Making of a New Science have stimulated broad public interest in the emerging subfield of mathematics. Such works on chaos theory have popularized several notions, chief among these, the “butterfly effect”: Chaotic phenomena have in common that they behave according to well-understood principles but also that they are highly sensitive to initial conditions. Theoretically, weather, which behaves chaotically, can be affected by subtle starting conditions—so that a butterfly beating its wings on one side of the planet could contribute to [Page 113]eventual major storms on the other side. The relationship between one event and another is nonlinear and unpredictable, yet connected in that both are part of a system with underlying order. Fractal geometry illustrates aspects of some chaotic behavior. In a fractal image, which is infinitely complex, any two portions are self-similar when compared; that is, they resemble one another but do not mirror or replicate any pattern exactly. This is true regardless of the sizes of the portions. Apparent order exists, yet the qualities of one part of a fractal image cannot be predicted by knowing about those of another part.

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