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In the Right Space

The power of chaos theory tools for business success
  • Richard E. Crandall
January/February 2012

The use of chaos theory in management research has been confined mainly to a metaphorical approach. However, it’s becoming more common for operations and supply chain professionals to adapt some of the mathematical concepts inherent to chaos theory for use in the business world.

Chaos is not a condition of randomness or disorder, but a state whereby phenomena that appear to be unrelated actually follow an unknown or hidden pattern (Smith 2002, Tetenbaum 1998, and van Staveren 1999). The relationships among the variables in a chaotic system exist, but are “rather vague and, at best, difficult to discern” (Kiel and Elliott 1996). Chaotic systems possess two characteristics: sensitive dependence to initial conditions and unpredictability in the long run.

Sensitive dependence to initial conditions. Lorenz (1993), a meteorologist, noted that a slight change in the initial input of data brought about vastly different results in his weather model. This now famous occurrence led to the popular “butterfly effect” illustration. Lorenz also discusses another system that is sensitive to initial conditions—the path of sleds descending a snowy slope. In this example, he illustrates with diagrams how seven sleds can end up in different stopping areas at the bottom of a hill, even though they may have started their descent within ten centimeters of each other. The paths the sleds take change depending on the location of small humps or moguls along the route of descent.

Unpredictability in the long run.This characteristic simply explains how we are more likely to accurately predict certain events in the short term. Take, for example, the weather: It cannot be determined on a lasting basis, but can be forecast into the near future (Lorenz 1993).

These two characteristics represent a good starting point; however, there are other components that should be noted: bifurcations, which are points in the behavior of a system where the outcome can oscillate between two possible values in alternating time periods; positive feedback, which moves the system away from its original state to a new state; and attractors, which are patterns that form when the behavior of a system is plotted in phase space.

Phase space is an area in which all possible states of a system are represented and can be used to examine performance history. When points are joined by a line in chronological order, a pattern develops that can resemble a point, orbit, or something more unusual. The atypical pattern is referred to as a strange attractor (Lorenz 1993). Attractors range from being fairly simple to vastly complex.

Four types of attractors have been identified: point, pendulum, torus, and strange (Barton 1994, Hudson 2000, Stam 2003). In phase space, a point attractor is depicted as a single plot on a graph. This is because the system behavior remains consistent over time. The pendulum attractor, also referred to as a period attractor, resembles an orbit when drawn in phase space. The torus attractor is a more complex pattern that forms an orbit and contains points within it, thus resembling a doughnut when graphed. Finally, the strange attractor, sometimes called a fractal, is a complicated pattern that exists when the system is in chaos. 

Examining firm performance through phase space
An application of phase space analysis—a mathematical tool of chaos theory—can be applied to managerial analysis. Time-series data are required, as they are the primary domain for studying chaotic behavior (Haynes, Blaine, and Meyer 1995; Hudson 2000) and are necessary to acknowledge the iterative process that occurs in the system over time. Iterations increase the magnitude of deviations, causing final outcomes to be quite different from starting points. It is the impact of positive feedback that makes the system shift.

In phase space, system properties are plotted at a point in time. With each iteration, another plot is made, which eventually results in a pattern (an attractor) when the plots are joined in chronological order by a line. The pattern of a time series that looks haphazard actually may have a hidden structure to it if looked at it in a different manner. For example, mechanical systems have been examined in phase space using position and velocity, while ecological systems have been studied in terms of the population size of a species (Briggs and Peat 1989). In medical research, Reidbord and Redington (1992) constructed a phase space with heart rate and patient behavior state as study variables. In public administration research, Kiel (1993) constructed an attractor in phase space using time-series data involving labor costs associated with service requests. Priesmeyer and Baik (1989) examined revenue and profit changes among retailers and identified attractors in phase space. 

Understanding phase space 
Total sales and net income can be examined as they appear in phase space, where there is a need to capture variables as the change in total sales and change in net income. (See Figure 1.) To determine the change in total sales (x-axis coordinate), the difference between present total sales for the fiscal quarter and total sales for the previous quarter is calculated. The same procedure is used to determine the change in net income (y-axis coordinate) using the net income (loss) figures.

Figure 1 depicts the two study variables, change in total sales (x-axis) and change in net income (y-axis). Note that the upper-right quadrant would be the most desirable for the firm, as it indicates consecutive periods of increasing total sales and net income.

Suppose a company was able to increase sales for three fiscal periods by $400. Meanwhile, it increased net income by $200. If this oversimplified situation were graphed in phase space, it would be plotted as one single point in the “growth” quadrant. This would be an example of a point attractor. Conversely, if sales declined by $400 and net income decreased by $200 each period, the result would be a point attractor in the “full decline” quadrant.

Now, think about an instance where sales and net income vary, such as in a seasonal business. The phase space plot for this example would show as a series of points oscillating back and forth between the “growth” and “full decline” quadrants. Obviously, this implies that a point in the “full decline” quadrant is only temporary and that movement back to “growth” is imminent.

Consider another example of a period-two attractor using real data. The company (Company A, a major retail firm) exhibits a consistent, two-phase oscillation from the upper-right to the lower-left quadrants, an indication that sales and net profits are moving in a normal cyclical pattern. Figure 2 illustrates Company A’s performance in phase space.

Figure 3 depicts Company A data graphed in the traditional manner. This has two advantages over the phase space diagram in Figure 2: It shows that net income does not result in a loss during any fiscal period, a feature that is unavailable in phase space; plus, the graph is more visually appealing by showing the position of total sales relative to net income.

On the other hand, the phase space graph has two advantages over the traditional graph. First, each plot represents the state of the two variables—total sales and net income. The plot is a representation of the state of the organization at a particular time or phase. Second, it shows more sensitivity to changes in system behavior, as evidenced by the shape of the attractor.

It’s interesting to note that, of the three examples of phase space attractors described previously, none exhibits a state of chaos. Both period-two attractor examples are not sensitive to initial conditions, as their consistent orbits bring back the system into a predictable pattern. This disproves the second condition—that it is unpredictable in the long run. Hence, there is no chaos.

Identifying the chaos
Consider a fictional pizza chain that uses chaos theory to track its performance variables of sales and income. The business identifies a strange attractor in its 10-year operating period. When leaders examine the evolution of the system in two-year increments, the graph displays a different pattern (attractor) from the previous phase space graph. It is impossible to predict the quadrant in which the next fiscal period will fall and the pattern the system will take in the future. In an original study, it was concluded that the time-series data display the characteristics of chaos (Crandall, Crandall, and Parnell 2011).

This conclusion meets the two key chaos criteria: sensitive dependence to initial conditions and unpredictability in the long run. Proving the first condition is impossible, as it would necessitate the assessment of two almost completely identical pizza chains. Only a few differences between the two chains would be permissible, such as different managers and employees. Then, over some period, the chains would diverge and evolve into very different entities. It is the iterations that would make the chains different and thus sensitive to their initial conditions. Indeed, only in a computer simulation can such a scenario be evaluated.

As a proxy to isolate sensitive dependence to initial conditions, one could consider case studies of actual companies. Firms are constantly being formed, but only a few survive for a substantial period. Many pizzerias have come and gone. Virtually every chain started with only one restaurant, some growing large (Pizza Hut), and some not (our fictional pizza company). Most pizzerias are single proprietorships. Yet, they all began the same way—with an idea, some capital, and a single store. The sensitive dependence to initial conditions comes into play when we see how the thousands of pizza restaurants have evolved into different forms. Our example company started as other restaurants did, but it emerged differently.

As for the second condition—unpredictability in the long run—the data support this conclusion with the fictional pizza restaurant. The haphazard or strange shape of the attractor would indicate a pattern of firm movement in phase space that is difficult to determine. It is impossible to predict what its attractor in phase space would look like over the next year. Compare this with the period-two attractor of Company A. With this company, we can say with some degree of confidence that its phase space cycle is likely to continue its pattern.

What does a chaotic system tell us?
There are three reasons why it is beneficial to understand that a time series is chaotic:
  1. The beginnings of a chaotic system are easier to identify in phase space. Because phase space is sensitive to system changes, performance is haphazard in terms of which quadrant it will land next. This pattern is clear in a normal time-series data graph.
  2. A chaotic system may reveal that something unusual is occurring. The pizza store example illustrates a number of points that would fall in the “unusual” quadrant, indicating a decrease in sales and an increase in income. While not necessarily dysfunctional, this situation is out of the norm, as one would expect income and revenues to be positively correlated. Using phase space diagrams would enable managers to reveal the unusual more clearly. 
  3. A chaotic system can serve as an early warning. Traditional time-series data can indicate when there is a performance problem. Because of its sensitivity to shocks, phase space can help identify early on if a problem is emerging in a way that traditional time series cannot.

Look back at Figure 1. The “growth” and “full decline” quadrants are intuitive. Over the long term, income tends to rise and fall with revenue, and most companies seek a position in the “growth” quadrant. They attempt to develop their businesses by increasing sales and, ultimately, profits. Remaining in this quadrant could be viewed as inherently desirable for most firms.

In contrast, “full decline” is nearly always the least desirable position. Income steadily declines, ostensibly due to decreases in revenues. Firms in this quadrant might seek to increase income by cutting costs. But this can be counterproductive if doing so negatively affects product quality, service, or other factors that drive revenues.

The “unusual” quadrant is counterintuitive. Businesses here enjoy an increase in income while revenues decline. This could be a short-term phenomenon that results from aggressive cost cutting. The relationship between revenue and income is a critical one, as managers must struggle to discern effective and ineffective approaches to reducing expenses.

For example, a cut in the advertising budget might be appropriate if promotional efforts do not result in sufficient increases in revenue, but inappropriate if sales decline precipitously as a result. Distinguishing between these possibilities is a complex challenge, but one that executives must tackle.

The “partial decline” quadrant is intriguing. This space involves problems similar to the “unusual” quadrant, but from a different angle. A number of culprits could explain why increases in revenues are not translating into profits—from ineffective cost controls to a boost in the intensity of competition that requires price reductions. The key here is for strategic managers to identify the causes so that remedial action can be taken.

Examining each quadrant independently is not a difficult exercise. The complexity emanates from the movement of an organization across quadrants over time. Identifying performance shifts in a visual manner is perhaps the greatest contribution of phase space, while interpreting their collective meaning is the greatest challenge to decision makers.

Phase space is not meant to replace traditional approaches to time-series analysis; rather, it supplements understanding of business performance. Hence, the use of chaos theory is not a superior approach to analysis but another tool in a box of techniques to aid management analysis.

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Richard E. Crandall, PhD, CFPIM, CIRM, CSCP, is a professor at Appalachian State University in Boone, North Carolina. He may be contacted at

For a free bibliography of other references on this subject, contact the author at

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