ABSTRACT

Chaos theory has only very recently been applied to organizations. Many of the traditional statistical methods of study cannot or should not be used in nonlinear situations. A chaos theory methodology has been established for exploring organizational dynamics using chaos theory tools and techniques. Heretofore, nonlinear dynamics have been explored over time. Priesmeyer suggested that time does not have to be the modulating entity. Data from the 1993 Employee Opinion Survey of a large government service agency are examined using modulating variables other than time. Employees' overall rating of agency performance is used as a modulating variable. The vast majority of 74 survey items exhibit period 1 trajectories indicating that they may be measuring essentially the same phenomena. A few survey items exhibit relationships that are of higher levels of complexity. Interesting examples of these are provided. An example of complete chaos among survey items is presented.

Introduction

This paper will explore an exciting new technique developed to explore nonlinear dynamics in organizations. H. Richard Priesmeyer (1992) has described a methodology for empirically studying organizations using chaos theory techniques. He has also authored the tool The Chaos System Software©⁽¹⁾ which manipulates the data and creates graphs and reports to easily display results. The software generates phase plane trajectories, marginal history charts, and provides built-in interpretation for many common business ratios. Moreover, the software provides a significance test on the goodness of fit measure (an F-test and a coefficient of determination) for the nonlinear equations it develops.

Priesmeyer suggests that marginal history charts can provide the same type of information that an EKG does for a physician. Traditional business analysis tools are the equivalent of blood pressure and pulse readings. Marginal history charts provide all of the dynamic, rich information of an EKG. These charts can be used to establish normal organizational rhythms, and then provide insights if the organization begins dysfunctional rhythms. These comparisons are made over time. This paper, as you will see below, explores the use of other chaos techniques that can be applied to a snapshot of data.

The Shortcomings of Traditional Statistical Approaches for Nonlinear Systems

The tools of chaos theory differ from other traditional methods. Linear regression, for example, cannot be used when nonlinear relationships exist. Although transformations can be made in linear regression by, for example, taking the logarithm of a given variable, all of these transformations still assume a continuous function with a slope that can be calculated at each point. This assumption is also at the core of calculus, another traditional tool for understanding and predicting phenomena. Since nonlinear phenomena may have discontinuities and other oscillations among values, the tools of calculus and linear regression often cannot be used in the real world.

Another traditional statistical tool is chi square, a test of statistical significance for relationships between nominal and ordinal variables. In order to use chi square, each of the data points must be independent. In nonlinear systems, however, feedback from earlier values affect later values. Nonlinear systems are of the form: X_i+1 = CX_i(1-X_i). For values of C between 1 and 3, X always settles down to a steady value, the attractor. Chaos theorists have discovered that when nonlinear systems are in the chaotic zone, however, they are highly dependent on initial conditions. The chaotic zone begins when C exceeds 3.56. Moreover, "in a nonlinear equation a small change in one variable can have a disproportional, even catastrophic impact on other variables" (Briggs and Peat, 1989, p. 24). An apparently insignificant change in initial weather conditions has been shown to significantly alter the path of major weather systems. The equation above is iterated thousands or millions of times in weather prediction so that insignificant amounts become amplified into significant ones.

Thus far, we have eliminated traditional methods by showing how nonlinear systems do not conform to the assumptions required by these methods. Chaos theory also avoids certain methods on philosophical grounds. For example, because of the importance chaos theorists have found in what appeared to be meaningless data, they are loathe to dispense with any observations or data points as unimportant. Consequently, they associate little importance with such traditional statistical pillars as the mean and other measurements of central tendency which strip away variation. Priesmeyer wonders, "with our number-crunching prowess today, why not consider every observation?" (Priesmeyer, 1992, p. 185). Measures of central tendency are useful in stability-seeking situations or when compensating (negative) feedback is present. Nonlinear systems are often marked by the presence of amplifying (positive) feedback. Chaos theorists also disavow ratios and other statistical methods which cause the loss of magnitude and proportionality.

Chaos Theory Methodology

In place of these traditional methods, chaos theory holds out the promise of "a much richer picture of organizational dynamics than that provided by conventional statistics - even the more advanced statistics of multiple regression, analysis of variance, or factor analysis" (Ibid., p. 166). Few places are as dynamic and changing as organizations. Yet most data collected - inventory counts, turnover ratios, employee satisfaction measures - present organizations as static and two-dimensional. Static measures are becoming increasingly less useful in representing the dynamics of organizations. The developing field of non-linear analysis is providing tools to better capture the dynamics of organizations.

The first step in representing the dynamism is to use marginal rather than absolute values. According to Clemson (1982) "it is more useful to know that something is changing and to have some idea of how it's changing than to have a detailed but static picture" (p. 189). In other words, if sales are $1 million one year, and $1.1 million the next, these are absolute values and the marginal value is a positive $100,000. Moreover, in nonlinear systems subsequent values are affected by the feedback from earlier values, so the marginal values reflect the direction and flow of the variable of interest. "Because marginal values reflect the dynamic evolution of a process, they provide a wealth of insight not available in" (Priesmeyer, 1992, p. 27) static values such as totals.

These marginal values are then plotted on a phase plane which is a set of traditional Cartesian coordinates. The phase plane is an important tool because it allows for visual identification of the attractor in the relationship between the variables under study. An attractor is an underlying pattern of behavior that exists because of inherent structural characteristics. The path of a pendulum is an attractor for a grandfather clock, for example. Wheatley (1992) has speculated that meaning is an attractor in organizations because when employees share the same sense of meaning, although their behaviors will vary significantly, they will not vary outside of the bounds created by their common understanding of meaning. Figure 1. shows an example phase plane plot.

The data points are plotted and a line connects points in chronological order. In this example, the data points are the annual investment returns of the common stock and small stock index for the years 1950-59. This plot has a very distinctive shape, made by points which are either in quadrant 1 or quadrant 3. Reading this graph suggests that when the common stocks have an increasing return, so do the small stocks. When the common stocks have a decreasing return, so do small stocks. These chaos tools do not offer any insight into cause and effect between the two variables, however. One could not conclude that common stock increases will result in small stock increases, or vice versa. Also, recall that since marginal values are plotted, a decreasing return does not necessarily mean a negative investment return. A decreasing marginal return occurs any time the next data value is lower than the previous.

Since this plot visits only two quadrants, it traces a Period 2 limit cycle. Period 1 trajectories visit only one quadrant and period 4 limit cycles visit the four different quadrants before repeating a visit to a quadrant. All other sequences of quadrant visits are referred to as Period 8, which for all practical purposes in organizations, represents chaos (for a more precise discussion of determining the period of a limit cycle, see Priesmeyer, pp 39-40).

The period of a limit cycle is important because it gives us a measure of the complexity, or "amount" of chaos or order between certain variables. Period 1 trajectories represent the least degree of chaos. Both variables always move together in one direction. Period 2 limit cycles represent an increasing level of volatility. The variables do not always move in the same direction, but they both move together in the typical quadrant 1, quadrant 3 Period 2 limit cycle. The traditional relationship between revenue and earnings is a Period 2 limit cycle. When revenues increase, earnings increase; when revenues decrease, earnings decrease. One classification scheme is to consider that any trajectory which visits only two quadrants during four observations is Period 2.

An increase in period level is known as period doubling, and it takes place after a bifurcation, or branching causes a different level of complexity. At bifurcation points, the system has a "choice" of level of complexity. Many organizations have lately experienced an additional level of complexity in the traditional relationship between revenue and earnings. Recently, some organizations have had years of increased revenues, but decreased earnings. Or, because of downsizing, some organizations have decreased revenues and increased earnings. To date, most experts have considered these experiences to be anomalies because they haven't had the tools to identify the patterns of which they are an integral part.

Organizations with a dramatic seasonal impact often exhibit a Period 4 limit cycle. The extra level of complexity occurs because an inventory buildup may occur in a quarter in advance of when actual sales will occur. The inventory buildup will not help revenue in the earlier quarter, but it will depress earnings. For example consider the quarterly revenue and profit marginal vectors for a surf apparel store in a northern beach town.

Quarter Rev. Profit

I. Jan-Mar - +

II. Apr-Jun + -

III. Jul-Sep + +

IV. Oct-Dec - -

This store traces a Period 4 limit cycle as it visits quadrants 2, 4, 1, and 3 year after year.

Chaos theory has not yet developed tools to adequately explain Period 8 limit cycles. Period 8 limit cycles contain all of the relationships that are more complex than Period 4. Researchers who are deterministic believe that even though these trajectories cannot be understood today, some day they will be understood. They do not believe these apparently haphazard trajectories are random; merely, too complex to comprehend.

Capra (1982, p. 101) has suggested that all theories are only approximations which are valid within a certain range. Priesmeyer (1992, p. 30) has speculated that traditional statistical methods remain useful for data with a period 1 trajectory. These musings create the possibility that the chaos theory methodology is simply an extension of traditional methods that become more useful in situations in which a phenomenon exhibits a relationship on a higher level of complexity than period 1.

Replacing Time as the System Modulator

Priesmeyer suggests in his Proposition 25 that "limit cycles typically report the evolving dynamic response of a system over time. However, they are not limited to using time as their third dimension; any other variable may be substituted to provide a descriptive image of a system's response to the chosen variable" (Ibid., p. 166). Rather than using time as the third dimension, we tried as the other dimension several survey items that themselves seemed to encompass, or at least be affected by other survey items. Therefore, rather than examining survey values from different chronological periods, all of the data is taken from a single survey.

The Government Service Agency Data

The database used for this analysis was the 1993 Employee Opinion Survey of a large government service agency. The agency Employee Opinion Survey process identifies employee issues and concerns in a wide range of areas important to sustaining a productive workforce. The survey was developed by the agency headquarters employee relations department with the assistance of two contractors who are recognized in this field. An extensive series of focus groups were held to determine the appropriate content areas of the survey. The focus groups were conducted with both bargaining and non-bargaining employees in ten cities in the summer of 1991. The survey was pilot-tested in five divisions and four departments at headquarters in November 1991 and was first administered to all career employees in April of 1992.

The survey was administered to 657,112 career employees in 1993. Questionnaires were mailed to employees at their work address. Completion of the 78-item, forced-choice questionnaire takes 20-30 minutes, is on-the-clock, and is voluntary. All responses are kept completely confidential. Seventy-eight percent of the employees returned the questionnaire for a total of 512,818 responses. For this analysis, we used a subset of the database consisting of the non-bargaining employees of the processing centers of three major cities. These employees were in the categories of clerical/administrative, professional staff, or management. The subset respondents for each city were 285, 106, and 106, respectively. Total response rates for each city were 73%, 85%, and 84%, respectively; the response rates for this sample of non-bargaining employees were unavailable.

Most of the survey items were on a five-point scale ranging from Very Good to Very Poor, or ranging from Strongly Agree to Strongly Disagree. We were provided with a total of 74 items, which excluded items having to do with sexual discrimination, for example. Of the 74 items, 18 were posed so that Strongly Agree, for example, was a "bad" response. A sample of such an item is, "Too many changes of supervisors have caused problems in my work group." In these eighteen cases, the ends of the scale were reversed. Discussion of survey items below with such reversions are noted.

The first step was to conduct a one-dimensional analysis of each contributing factor against the overall rating. We divided all of the surveys into seven categories according to the participant's rating of the agency's overall performance. All who gave an overall rating of 1 are in one category, all who gave an overall rating of 2 are together in a category, etc. up to 7 (a Likert scale from poor to excellent, then reversed). Next, we created an average score for each survey item in each category. The results for the survey item, "I like the kind of work I do" appear in Figure 2. This graph represents the opinion of seven groups of people.

This information is then plotted as follows: Overall performance is on the x axis and a point is created for each of the seven points with the corresponding y (like the kind of work I do) values. Figure 2. contains such a plot.

Figure 2 suggests that employees who rate the overall performance of the agency higher agree that they like their work (1=strongly agree, 5=strongly disagree) These calculations were made for each survey item using the seven overall variables listed in Table 1.

Period 1 Trajectories

Table 1. presents perhaps the most interesting findings in the agency data. Column A shows the percentage of the 73 survey items that would be plotted exactly as Figure 2. above. The seven items listed in Table 1. were each, in turn, used as the X axis variable. Column B shows the percentage of the 73 survey items that would be plotted exactly as Figure 2. above if one data point were adjusted by 0.1 or less. In other words, there is one very small kink in an otherwise continual downward slope. Column C contains the percentage of the 73 survey items that had one kink caused by a data point increasing by more than 0.1. Column D is the summation of columns A, B, and C.

Whenever two variables with a constant increasing pattern are combined in a phase plane, the result is a Period 1 trajectory as shown in Figure 3.

The agency data suggest that approximately 90% of the survey items trace approximately Period 1 trajectories. The conclusion is perhaps surprising because of the breadth of the survey questions. For example, the survey item, "I am given the opportunity to see agency-produced videos on the clock" is in column C above for overall performance item 13-1. The data suggest a direct relationship between whether employees are given the opportunity to see work videos and how they rate the overall performance of the agency. Another example is the survey item, "Rate your employee benefits." This item is in column B above for modulating item 13-1. The data suggest a direct relationship between how employees rate their employee benefits and how they rate the overall performance of the agency. Further study should be conducted to determine whether approximately 90% of the survey items are measuring essentially a single dynamic.

Managerial Implication: There is tremendous leverage present in these Period 1 trajectories. If managers can make improvements, it is likely that these improvements will be reflected throughout the employee opinion survey. Managers need not worry about focusing improvement on one or a few survey items. They can be confident that if they achieve general improvements, the survey item they want to improve likely will.

We tried nine different survey items as the replacement for time as the modulating variable. Two items have very little variation in response. The other seven survey items are listed in Table 1. The survey item, "Please rate the agency on its overall performance" was selected for use for the rest of this article. It captures a global concept and has a seven-point scale from 1=Poor to 7=Excellent. This item is one of the 18 which was reversed. The seven-point scale allows for six marginal data points (the other items tested were all on five-point scales yielding four marginal data points). The number of responses for each scale item are as follows:

n rating description

5 1 poor

10 2

24 3 fair

125 4 good

103 5 very good

88 6

28 7 excellent

Total 383

Period 1 Bifurcating to Period 2

The first increase in complexity occurs as a Period 1 trajectory bifurcates and becomes a Period 2 limit cycle. When "The agency is serious about quality work" is plotted with "How would you rate your job security?" the relationship is Period 1 in the low and mid ranges of overall performance. Then, in the upper range, the relationship becomes more complex as there is no longer a simple relationship of both variables continually increasing. "Serious about quality" continues to increase, but "job security" decreases. Therefore, the last two points are in quadrant 4. This graph is portrayed in Figure 4. In the range where the Period 1 trajectory persists, as an employees' views of the overall performance of the agency increased, so did their beliefs about quality work and job security.

Managerial Implication: Employees who report high ratings of overall performance also report increasing beliefs about the seriousness of quality work, but decreasing confidence in job security. Proposition: Perhaps employees who rate the agency highly and have increasing beliefs about the seriousness of quality work suspect that the agency will have to downsize over time.

Velocity

Figures 3. and 4. reveal additional interesting information. Notice how both trajectories begin near the origin, and then increase in distance away from the origin. These cases both indicate an increasing velocity in these relationships. The velocity is obtained by multiplying the two marginal values of the axis variables. If one or both of the variables is changing slightly, the velocity will be small. Major changes in velocity occur when both variables are experiencing significant change. Velocity can be viewed as a measure of the combined energy of the relationship between the variables. Velocity changes, because they measure the intensity of a limit cycle, can signal a bifurcation point. In the same manner that cardiologists determine pathologies from EKGs, for example, a bifurcation is imminent in quarterly data "if velocity becomes more negative during quarters 2 and 3 relative to positive velocity measures during quarters 1 and 4" (Priesmeyer, 1992, p. 39). Figures 3. and 4. both suggest a major increase in energy and intensity between the respective variables at the high end of the scale.

Period 2 Limit Cycles

Figure 5. portrays the graph of a Period 2 limit cycle. It cycles between quadrants 1 and 3. The variables are "The amount of stress in my job is a problem" (reversed) plotted with "I have enough authority to carry out my job effectively." Lack of adequate authority is often linked with the amount of stress that an individual experiences on the job. The agency data suggest that the relationship is not of the simplest variety. The two variables do move in the same directions as evidenced by the quadrant 1 (+,+) and 3 (-,-) visits but as employees' overall performance rating increases, there is not always a corresponding increase in stress relief and adequate authority. At certain points as performance rating increases, the values of both variables decreases. Consequently, stress on the job is a more complicated phenomenon than simply having enough job authority.

Managerial Implication: Since level of problematic stress always moves together with adequate authority, it seems that employees experience stress because of the authority.

Period 2 Dissolving into Chaos

Another factor often linked with the amount of stress that an individual experiences on the job is the individual's perceived job security. The agency data suggest that this relationship is also not of the simplest variety. Figure 6. shows "The amount of stress in my job is a problem" (reversed) plotted with "How would you rate your job security?" The relationship starts off as Period 2 with visits to quadrants 1 and 2. As employees' overall agency performance rating increases, they basically report less stress and greater job security. However, as the overall rating increases in the "very good" range, the relationship between stress and job security becomes chaotic. Quadrant 3 is visited for the first time followed by a visit to quadrant 4.

It is perhaps more interesting to consider these survey items with the modulating variable of an employee's own performance, which is not a survey item, rather than rating of agency. This substitution of modulating variable can be made without changing the resulting trajectories if both agency overall performance and employee performance are part of the single dynamic that seems to be captured by 90% of the survey items. Under this assumption, these results suggest that in the low ranges of employee performance, as performance increases, feelings of job security increase and stress decreases. However, in the higher ranges of employee performance, at least three bifurcations occur so that while some employees will report greater job security and less stress as performance increases, just as many others may report less job security and more stress as performance increases.

Managerial Implication: Job security and stress relief peak in employees who perform at above average, but not the highest, levels.

Chaos in the Middle of Period 2 Limit Cycles

A few combinations of survey items exhibit low levels of complexity at the lowest and highest levels of overall performance but have a chaotic zone in between. For example, to see if employees felt that competitors were a threat to their jobs, we plotted "Competition presents a serious threat to the agency" (reversed) with "How would you rate your job security?" This graph is depicted in Figure 7.

The graph can be interpreted as follows:

agency performance competitor threat job security complexity

low range increasing increasing period 2

mid range decreasing increasing period 8

high range increasing decreasing period 2

At low levels of overall performance the relationship between competitor threat and job security is relatively orderly. As performance increases, competitors are increasingly seen as a threat but jobs are perceived to be increasingly secure. In the mid range, job security continues to increase while competitors are decreasingly seen as a threat. Finally, in the high range, competitors are increasingly seen as a threat and jobs feel less secure.

Managerial Implication: Those who view the agency's performance as above average are the least worried about competitors and most secure in their jobs. That range is chaotic because it represents a decision point for both variables and a wide range of opinions by employees.

Chaos evolving into Period 2

Chaos theory is most often used to explore orderly situations that for some reason become chaotic. Catastrophe theory, a subset of chaos theory, for example, focuses exactly on this situation. However, chaos theory can also be used to examine chaotic situations which become less complex. For example, prolonged applause at events such as concerts begins as a chaotic roar of noise, but often develops into a synchronized rhythm as people clap, hoping for an encore performance. Another example of orderliness coming from chaos is that "women living in close groups such as prisons, hospitals, and student residences tend to synchronize their menstrual cycles" (Briggs and Peat, 1991, p. 184).

The agency data contain a few instances of relationships that are initially chaotic, but then settle down to a lower level of complexity. Figure 8 illustrates such an example.

The graph depicts "The amount of stress in my job is a problem" (reversed) plotted with "Pay should be based more on performance than it is at present" (reversed). The relationship between stress level and pay for performance is initially chaotic as evidenced by the consecutive quadrant 2, 4, 1, 1 visits. It remains chaotic through the next visit, to quadrant 3. Then the relationship settles down into a period 2 limit cycle with oscillation between quadrants 1 and 3. At the low and mid ranges of overall performance, there is no discernible relationship between problematic stress and pay for performance. As performance increases, the two variables tend to move together in positive or negative directions.

Managerial Implication: For employees who rate agency performance highly, stress level is directly related to satisfaction with performance-based pay.

Complete Chaos

With agency overall performance as the modulating variable, we were able to find two sets of survey item relationships that are completely chaotic across the range of the modulating variable. They are "Pay should be based more on performance than it is at present" (reversed) plotted with "Decisions currently made at a high level could be better made at lower levels" (reversed), and "Employees are reluctant to reveal problems or errors to management" (reversed) plotted with "Many supervisors have given up trying to discipline employees" (reversed). The latter is depicted in Figure 9.

Several factors come into consideration in the first variable alone (an employee's decision of whether or not to reveal problems to management). Recent press coverage has highlighted the harsh treatment that employees in government who reveal problems have received from managers in the organization. One could speculate that a number of factors may be dominant when an employee decides whether to be the one to reveal problems: integrity, assuming responsibility for being the message bearer, perceived implications in rewards, advancement, or job security, or duty to customers and taxpayers. Given these dynamics, it is not surprising that no discernible relationship exists among the three variables of revealing problems, supervisors who have given up, and agency overall performance.

Managerial Implication: Even though the relationship is chaotic, the trajectory still may contain important information. For example, the final move is a dramatic one from quadrant 2 to 4. Why such a huge increase in supervisors having given up, and at the same time a sizeable decrease in reluctance to reveal problems?

Summary

Heretofore, most researchers have explored organizational dynamics primarily by using static techniques, such as traditional statistical analysis of an employee opinion survey. Chaos theory, to this point, has been used almost exclusively to examine the complexity in relationships over time. It has been suggested that chaos theory offers an approach to revealing dynamic relationships from static data.

We have examined agency employee opinion survey data using that approach and have discovered varying levels of complexity among survey items. We have seen, for example, that stress level and discretionary job authority are related in a way that linear techniques cannot reveal. We have also seen that job security and stress relief peak in employees who perform at above average, but not the highest levels. The relationship starts out as Period 2 and then dissolves into chaos.

When chaos theorists use time as the modulating variable, they are often able to get several, if not thousands of data points. When scale values of a survey item is used as the modulating variable, the number of data points is limited to the number of gradients on the scale, minus one. A minimum of four data points are required to determine the period of a limit cycle. Consequently, a Likert scale, for example, would need at least five potential choices (i.e. strongly agree, agree, neutral, disagree, strongly disagree). The more the choices, the greater the understanding of levels of complexity and bifurcation points in a relationship.

Chaos theory researchers are finding more and more instances in which simple models can explain what appeared to be complex phenomena. Organizational dynamics are ripe for such models, because so far we understand very little about, for example, how change occurs in organizations. Many people simply experience change as permanent white water (Vaill, 1991) and have no more insight than that. This seeming random chaos reminds me of an experience with my son when he was three years old. When I first introduced him to the style of music which is in rounds, he thought it was just a cacophony of noise. He did not yet have a framework for understanding this style of music. As far as he was concerned, "Row, Row, Row Your Boat" had just been turned into chaotic noise. Once one has a framework for understanding musical rounds, what appeared to be chaos can actually be seen (or heard) as something very simple.

Glossary

amplifying feedback: circular causality which actively seeks change by building in the same direction as previous iterations; a vicious or virtuous cycle

attractor: an underlying pattern of behavior that exists because of inherent structural characteristics.

bifurcation: a branch point causing a different level of complexity. At bifurcation points, the system has a "choice" of level of complexity. The system may become more or less complex.

chaos theory: the study of phenomena which exhibit more than one level of complexity across the range of the phenomena.

compensating feedback: circular causality which actively seeks stability by counteracting or cancelling out the effect of previous iterations.

feedback: all phenomena in which there is circular causality.

fractal: pattern characterized by infinite detail, infinite length, no slope or derivative, fractional dimension, self-similarity, and that can be generated by iteration.

limit cycle: the plotting and connecting of sequential observations on a phase plane.

marginal value: a measure of the change in a value. Marginal value m_i = o_i+1 - o_i, where o = observation.

nonlinear systems theory: another name for chaos theory.

period: a measure of the complexity, or "amount" of chaos or order between certain variables.

period doubling: an increase in period level which takes place after a bifurcation, or branching.

Period 1 limit cycle: the least degree of chaos. Both variables always move together in one direction.

Period 2 limit cycle: when only two quadrants are visited out of every four data points.

Period 4 limit cycle: when all four quadrants are cycled before a quadrant is revisited. Appears in the financial figures of organizations with dramatic seasonal impacts.

Period 8 limit cycle: at our present level of understanding, chaos. Any limit cycle which is more complex than Period 4.

phase plane: a set of traditional Cartesian coordinates. The phase plane is an important tool because it allows for visual identification of the attractor in the relationship between the variables under study.

velocity: the product of the two marginal values of the axis variables. Velocity is a measure of the combined energy in the relationship between the variables.

BIBLIOGRAPHY

Briggs, J. and F. D. Peat (1989). Turbulent mirror: an illustrated guide to chaos theory and the science of wholeness. New York: Harper and Row.

Capra, F. (1982). The turning point: science, society, and the rising culture. New York: Simon and Schuster.

Clemson, B. (1984). Cybernetics: a new management tool. Tunbridge Wells, Kent, England: Abacus House.

Priesmeyer, H. R. (1992). Organizations and chaos: defining the methods of nonlinear management. Westport, CT: Quorum Books.

Vaill, P. B. (1991). Managing as a performing art: new ideas for a world of chaotic change. San Francisco: Jossey-Bass.

Wheatley, M. J. (1992). Leadership and the new science: learning about organization from an orderly universe. San Francisco: Berrett-Koehler Publishers.

1. Priesmeyer, H. Richard. The Chaos System Software. Management Concepts, Inc., Fair Oaks Ranch, Texas 1994.