# Chaos Theory

Chaos theory is a branch of mathematics that falls under complex systems theory and is concerned with the behavior of a special type of dynamic system. The distinguishing feature of this type of system is its extreme sensitivity to initial conditions in determining the system’s behavior. The well-known term *butterfly effect* describes a chaotic pattern studied by meteorologist Edward Lorenz in his work on predicting weather patterns in the early 1960s. The roots of chaos theory, however, go back to Isaac Newton and the concept of determinism and deterministic systems. In this perspective, a system’s initial conditions completely describe the system’s future behavior. In his studies of weather prediction in the early 1960s, Lorenz found that his very simple weather system models generated wildly different outcomes even though the initial conditions for two identical models differed only slightly. Lorenz summarized chaos theory as “when the present determines the future, but the approximate present does not approximately determine the future” (Danforth, 2013). These conditions have since been found in systems from a broad range of disciplines and have important implications for the prediction and control of complex systems. Systems that are susceptible to chaotic behavior have a number of distinguishing features, including nonlinearity, feedback, strange attractors, and extreme sensitivity to initial conditions. This entry explains these features and discusses their implications for reputation management.

### Central Concepts in Chaos Theory

Nonlinearity describes the nature of the relationship between variables. The common assumption employed in most mathematical system models is that of linearity, or constant proportional changes in effects. This is, however, a simplification of the actual relations among components in the system. In many cases, these relationships are exponential, such that a small input can yield a very large output reaction.

Feedback in a dynamic system is simply a closed-loop cycle of cause and effect in which “causes” in one part of the system lead to “results” elsewhere in the system. These results, in turn, act through the system to change the original conditions. Feedback is of two types: positive and negative feedback. Positive feedback is an amplifying force that can move either in an increasing direction (sometimes called virtuous circles) or in a decreasing direction (or vicious circles). Negative feedback, also called counteracting feedback, occurs when the system’s reaction to change is to dampen or resist the direction of the change. Negative feedback loops tend to be self-stabilizing and resist the tendency to run out of control. The interaction of multiple positive and negative feedback loops that are present in all complex systems can give rise to many different observable behaviors that can span from steady state performance to chaos.

The concept of an “attractor” in a system’s pattern of behavior over time is used to characterize complex system behavior. To understand this concept, chaos theorists use the “phase space” to plot the behavior of the system over time. The phase space of a dynamic system is a hypothetical space that is used to define the state of the system. A phase space has as many dimensions as are required to characterize the system’s state or condition. For example, the location of an item in the natural world is described by a four-dimensional phase space: three dimensions of space and one dimension of time. The dimensionality of an organizational system’s phase space depends crucially on the selection of the variables used to describe the performance of the system.

In chaos theory, the attractor is a set of points in the phase space of the dynamic feedback system that defines its steady state motion. An attractor can be one of four possible types: (1) a single point, (2) a closed curve, (3) a torus, or (4) chaos. Single-point attractors are associated only with linear systems, and closed-curve attractors have repetitive fluctuations that are predictable. The torus and chaotic attractors exhibit nonrepetitive fluctuations. However, the torus attractor fluctuates in a predictable manner, while the chaotic attractor does not.

In deterministic models, a system will behave similarly with even small variations in the initial conditions. However, chaotic systems’ extreme sensitivity to initial conditions is important in attempting to understand the system’s behavior. If the system is chaotic, which may not be known in advance, the same system when subjected to similar but slightly different inputs will respond with dynamic behavior that is significantly different. This has implications for the ability to predict the behavior of this type of system. At best, in practice, this sensitivity allows for only a short-term prediction of system behavior; long-term planning is futile.

### Implications of Chaos Theory for Reputation Management

Given the characteristics of the general business environment, chaos theory is a promising framework that offers new insights into the dynamic behavior of formal systems, such as industries and firms, and for less structured concepts. Reputation falls under the latter category. The theory suggests that different management styles may be required for effective leadership under various organizational conditions as represented by the system’s mapping on the phase space. However, social systems analysis requires the consideration of many more variables than with systems in the physical sciences. Presumably, organizational systems with considerable detail complexity and nonlinear feedback loops would be susceptible to chaotic behavior, making this perspective a valuable contribution to the manager’s toolbox. But, while simple systems with few components can be relatively easily analyzed for chaotic behavior, it remains an open question as to whether a formal analysis of something as complex as a modern business organization is feasible. Establishing a reliable link between practice and the mathematics of chaos theory is a difficult task.

The chaos theory perspective can contribute to reputation management in several ways. First, the application will require that an explicit, operational representation of how reputation is influenced by organizational activities be developed. The purpose of this step is to clarify the drivers of the reputation construct. Second, with a clearer understanding of how reputation is formed, the construct can be mapped on a relevant phase space. Comparing the mapping with the types of attractors will enable the reputation construct’s dynamical behavior to be accurately characterized.

Finally, management styles can be associated with system behaviors as indicated in the phase space mapping. For example, for system behavior in regions of the phase space that are remote from the attractor, leadership that encourages search, experimentation, and learning may be best for moving the system construct in a more desirable direction. Once in a stable location in the phase space, a leadership style that focuses on stability and cohesiveness to maintain the desired state may be more appropriate.

Despite the challenges of linking an abstract theory with messy practice, the insights from chaos theory can make managers more aware of the causes of observed behaviors of their systems. Sensitivity to initial conditions, identification of the primary feedback loops that generate organizational performance, the use of phase space mapping of selected variables (e.g., reputation scores over time), and the issues of not being able to make reliable predictions under some conditions are important contributions to improving organizational understanding and performance, including the firm’s reputation.

Danforth, C. M. (2013). *Chaos in an atmosphere hanging on a wall*. Mathematics of Planet Earth. Retrieved January 11, 2016, from http://mpe2013.org/2013/03/17/chaos-in-an-atmosphere-hanging-on-a-wall/

Gleick, J. (1988). *Chaos: Making a new science*. New York: Penguin.

Loye, D., & Eisler, R. (1987). Chaos and transformation: Implications of nonequilibrium theory for social science and society. *Behavioral Science*, 32, 53–65.

Stewart, I. (1990). *Does God play dice? The mathematics of chaos*. Oxford: Blackwell.