Chaos theory describes the motion of certain dynamical, nonlinear systems under specific conditions. Chaotic motion is not the same as random motion. It is especially likely to emerge in systems that are described by at least three nonlinear equations, though it may also arise in other settings under specific conditions. All of these systems are characterized by a sensitivity to initial conditions within bounded parameters. Chaotic systems must also be transitive (any transformation in period t1 will continue and overlap in period t2), and its periodic orbits are dense (for any point in the system y, there is another point with a distance d = y in the same periodic orbit).

The history of this branch of study is a complex, interdisciplinary affair, with scholars in different fields working on related problems in isolation, often unaware of research that had gone before. One of the most important characteristics of chaotic systems is their sensitivity to initial conditions. In 1961, one of the fathers of chaos theory, Edward Lorenz, accidentally discovered this principle while studying a simple model of weather systems constructed from no more than 12 parameters. Wishing to review a certain set of results, he manually reentered values for these parameters from a printout and started the simulation again in midcourse. However, the new set of predictions that the computer made were vastly different from the first series that had been generated. After ruling out mechanical failure, Lorenz discovered that that by reentering the starting values of the parameters, he had truncated the decimals from five places to three. Lorenz and his colleges had assumed small variance in the inputs of a set of equations would lead to a likewise small variance in the outcomes. Yet in a system of complex or chaotic movement, very small variance in initial conditions can lead to large difference in outcomes. This property is popularly referred to as the “butterfly effect.”

**Theoretical Implications**

While chaotic motion may make long-range forecasting of certain systems impossible, it is important to point out that it does not imply randomness. Rather, chaos, as the term is used in experimental mathematics, describes systems that are still deterministic that yield complex motion. Furthermore, patterns of predictable, or recurring, aperiodic motion may emerge out of this chaos.

It then follows that one way to visualize a complex system is by attractors, or strange attractors, which track the motion of the system through a three-dimensional space. In a truly random structure, the value of the system could be at any point in the three-dimensional space. With deterministic chaotic motion, the system’s values are all found within a bounded subset of space. The shape of this bounded space will vary in predictable ways as the values of the initial parameters are increased in a proportionate series characterized by “Feigenbaum numbers.” One of the most famous of these shapes is the “Lorenz attractor,” which has been characterized as a set of linked concentric spirals that resemble either the eyes of an owl or butterfly wings. This was one of the first strange attractors characterized and is often remarked upon for its beautiful and complex fractal pattern.

**Implications of Chaos Theory for the Social Sciences**

While chaos theory is often characterized as a branch of experimental mathematics, it has important implications for many other fields of study. Early researchers in the development of chaos theory addressed issues in areas as diverse as physics, meteorology, ecology, and even understanding variation in cotton prices over time. Biologists and anthropologists working on ecology or population dynamic problems are no doubt already aware that simple logistic models of population growth can yield chaotic outcomes when starting parameters are moved beyond certain levels.

Evolutionary biologists have wondered if certain strange attractors have phenotypical expression. For instance, it’s interesting to note that not all possible forms can be reached through evolution. While there are certainly great variations in physical form between creatures, not all possible variations are expressed. The giant pangolin from Africa and the giant armadillo from North America look quite similar, but it is not because they are closely related. Rather, the phenomenon of convergent evolution raises the question of whether there are strange attractors for body types.

The social sciences could benefit much from chaos theory precisely because its area of study is often characterized by complex, nonlinear systems. Obviously, this has implications in formalized areas such as economics and game theory. It is also interesting to note that this may also have implications for other sorts of theorizing as well. For instance, teleological theories, such as those advanced by Hegel or Marx, predict that systems move toward a future steady state in which actions may happen but the nature of the system is not disrupted. These future states might be conceptualized as strange attractors for social systems.

One of the traditional misconceptions about teleology stems from the role of the end states. Objections are often raised to the idea that future end states can cause effects in the present (known as “backward causation”). Chaos theory provides a systematic argument for how the same parameters that cause the emergence of a strange attractor also select for a certain set of behaviors in the agents. Therefore, it is not the end state itself that generates change. This also has implications for how agency is conceived in social systems. The movement of a system from a state of random, or indeterminate, behavior to deterministic, or chaotic, behavior will affect the scope of individual agency. Nevertheless, a certain amount of agency remains in these explanations, not only in that a wide range of behavior remains possible once a strange attractor has been reached, but the principle of sensitivity to initial conditions means that much room may exist for agents to determine how the system develops.

#### References:

- Alligood, K. T., Sauer, T. D., & Yorke, J. A. (1996). Chaos: An introduction to dynamical systems. New York: Springer.
- Byrne, D. (1998). Complexity theory and the social sciences. New York: Routledge.
- Gleik, J. (1987). Chaos: Making a new science. New York: Viking.
- Kiel, L. D., & Elliot, E. (Eds.). (1996). Chaos theory in the social sciences: Foundations and applications. Ann Arbor: University of Michigan Press.
- Wendt, A. (2003). Why a world state is inevitable. European Journal of Political Science, 9, 491-542.