T. R. Young
Scholar in Residence
Texas Woman's University

L. Douglas Kiel
University of Texas at Dallas

Dec, 1994


No. 006
Distributed as part of the Red Feather Institute Series on Non-Linear Social Dynamics. The Red Feather Institute, 8085 Essex, Weidman, Michigan, 48893.



Any of the administrative sciences wishing to guide firms, agencies and governments through the cycles of uncertainties in an increasingly complex and changing environment might well consider the findings of the new science of Chaos. Chaos theory deals with the changing relationship between order and disorder in the behavior of natural or social systems. Chaos theory is so fundamentally different from previous understandings of social and natural dynamics that an entirely new paradigm for the knowledge process is required. The chief elements of the new paradigm include 1), the nonlinearity but self-similarity of systems dynamics, 2), qualitative transformations to new dynamical states, 3), progressively more complex outcomes as well as 4), the appearance of new forms of order out of even the most chaotic regimes.

The experimental tasks energizing such a management science consists of a) generation and display of data to reveal the hidden patterns of dynamics in phase-space b)identification of the key parameters which drive a system from one dynamical state to another, c) reflection on the implications for a firm or an agency which are close to change points at which entirely new modalities of behavior emerge, d) adoption of flexible strategies of management as causality opens and closes, e) concern for interacting exigencies inside and outside the firms which bend upon employees, suppliers and clients and which affect their dynamics, as well as, f) the quest for new organizational forms in a time of advanced chaos. Corollary to such work is the need to cultivate a mathematics and a geometry with which to study such dynamics.

The majority of the paper is devoted to three nonlinear dynamic states and three transformations from 1-valued causal basins to much more complex fields together with their challenges to management science. A hypothetical application is provided to help illuminate its principle ideas. Lastly, a brief exploration of the larger implications of chaos theory for a "postmodern" management science is offered.

ELEMENTS OF CHAOS THEORY Chaos theory is a set of ideas about the transformation from order to disorder as the generation of new forms of order from turbulent, nonlinear dynamics. A system exhibiting nonlinear behavior may appear quite random over time, yet studies of chaotic regimes in phase-space reveal underlying patterns. Such patterns are called "attractors" since a system appears to be "pulled" toward a region in an outcome field during its cycles or periods (Mandelbrot, 1977). Some attractors are called 'strange' attractors since a system behaves in ways not expected by Newtonian physics, propositional logic, rational numbering systems or euclidean geometry. 1

Chaos research maps the geometry of system dynamics in phase-space. 2 There are five such patterns. They include two linear and very stable equilibria called a) the point attractor and b) the limit attractor (see Glossary). There are also three generic nonlinear regimes which we will discuss in some detail here. They are called 1) the torus with one outcome basin, 2) the butterfly attractor which can bifurcate into a 16n outcome field, and 3) full chaos with an infinity of outcomes to which persons, groups and firms might move.

Management science, as a discipline and profession, must learn how to generate and display the data of systems dynamics in such a way as to reveal the geometry of the deep structures hidden in those data. One of us, L. Douglas Kiel, presents this technique below. Knowledge of those patterns /attractors help one connect individual workers, firms and national economies to key parameters in the larger society the values of which may either stabilize a system or drive it into far from stable equilibrium. While Chaos theory is a science of surprises and transformations, it is also a science of wholeness and connectedness that reunites a given firm with the social and natural environment in which it is found.

Chaos research, as mentioned, tracks the transformations of dynamical systems from one behavioral regime (attractor state) to another. In such transformations, management science has much to learn and much to ponder. As key parameters of systems reach each one of four feigenbaum numbers (F1-F4 discussed below), the system displays an orderly procession from one dynamical state to another. The procession ceases to be orderly and becomes very chaotic at F4. As a system becomes more chaotic, i.e., it transforms from a simple outcome basin to a much more complex causal field.

Fractal Geometry The behavior of a chaotic system takes on a fractal geometry. This contrasts to the ordinary assumption that social structure has a euclidean geometry, i.e., it fills completely the space it occupies. We think of points as having no dimension; lines having one dimension, planes having two and solids having three dimensions. A major finding of Chaos research is that natural objects seldom fill the space available to them. What appears to be a solid in nature is shown to be sponge-like and fill up but a fraction of the space available. There are several valuable techniques with which to measure fractals and thus measure the degree of order amid disorder. 3

A fractal is an estimate of the degree to which a system occupies a region in an outcome basin. In point attractors, the region occupied is very small; the order great. In full blown chaos, disorder almost displaces order; a system may end up almost any place in an outcome field. When attractors occupy but a fraction of the space available to them; they are said to have negentropy or are 'orderly.' In the diagrams below, one can 'see' the degree to which the five generic attractors fill their phase-space.

Outcome Basins In exploring the implications of Chaos theory for everyday management problems, we will look at two near-to-stable chaotic regimes and one far-from-stable regime. We will draw attention to the reader of that region in a larger causal field in which pattern and predictability are lost since it is while in those regions that it is difficult for particular persons or firms to act rationally in the pursuit of given system goals. The first fractal of interest to management science is the torus since it exhibits near-to-stable dynamics. Compared to the two other kinds of change, these variations are tractable indeed to planning and prediction.

I. The Torus and Self Similarity The simplest nonlinear system is a torus; it has one loose but stable outcome basin. The dynamics of a torus are marked by self-similarity. Self-similarity, as a concept, means that while the behavior of any nonlinear natural or social system, including individuals, firms and entire societies, may be similar from day to day, year to year or generation to generation, no one embodiment in any given cycle or iteration of the behavior any given system is precisely like a previous embodiment. Thus variation is the natural state of social forms which take the geometry of a torus. One can predict that a system will be somewhere inside the boundaries of the torus but one cannot predict exactly where the system will be.

Routine dynamics inside a factory, an office, a hospital, a school or a prison have the character of a torus; they are familiar but are never the exactly the same from day to day...similarity displaces sameness in the paradigm now before us. Quality control experts such as W. E. Deming (1986) are sensitive to the self similar dynamics found in chaos research. Deming does not argue for "zero defects" in manufacturing output nor does he argue for perfectly replicable products, but instead accepts statistical variation within defined parameters as preferred management policy. Management policy may well work to restrict entropy to the fractal geometry of a torus yet no set of administrative controls can generate that precise behavior of a point attractor which is so much a part of modern science and its preferences for linear dynamics. 4

But innovators such as Deming have much more to offer a postmodern management science than merely nonlinear production dynamics. In the past, management science was seen as a political tool working more on behalf of owners than oriented to quality, efficiency, and rationality. The chief contribution of Deming to postmodern management science has been his ability to get workers to invest their rare genius in setting and meeting production goals. This democratic self management approach has taken some of the prerogatives of ownership away and has redistributed profits but those firms which want to succeed in a global economy must look to the workers as does Deming and all those who follow his ways.

Figure 1. Three Views of a Torus as Dynamics transform into Structure.

The geometry of a torus is seen in more detail in Figure 1. In reading a torus, the first form in Figure 1 represents the dynamics of one system over one iteration. It might depict the income pattern of a person over a lifetime or the profit behavior of a firm over a year.

The second form in Figure 1 might depict either one system over a dozen cycles or a dozen firms over one such cycle in terms of some attribute of the system of interest to the researcher. The near to stable geometry of any given torus emerges after thousands of iterations of such cycles. At some point between the first iteration and those thousands of iterations, process transforms into structure. The theoretical point is momentous...such regularity becomes part of the biome in which it is found; other systems in the area come to match their own structure and dynamics to this regularity. Causality opens and closes depending upon the region of phase-space in which we look.

One can see that the system (or systems) do not fully occupy the space available to it. Indeed, if it were to occupy all the space available, one would never know just where to look to find it. Dynamics of the system would be fully chaotic; other systems would not be able to map their own behavior to that of the person, firm, species or society...causality would disassemble at that point.

Given the semi-stable dynamics of a torus, management policies generally work with sufficient efficacy to warrant the resources allocated to them. If we want workers or students or shipments to arrive within a close approximation to the time of last arrival, then there are policies which will work to effect that close (but never precise) approximation. The unique aspects of each iteration reinforces contingency based approaches in management science (Koontz, 1961).

In brief then, self similar systems such as a torus exhibit near to stable dynamics. In the observations of thousands and millions of iterations of natural and social systems, the pattern which emerges is a chain of variations with infinite variety, infinite length, and infinite detail rather than a stable, natural point or pattern. This similarity-but-not sameness is of considerable interest to management inquiry. It sets variation at the foundation of the administrative process and decenters precision and conformity as goals of management.

Feigenbaum Numbers When a key parameter of a torus increases in value, one region inside it can expand to form a tongue; when it exceeds a given value, that tongue expands to form another wing of the attractor. Figure 2 offers a close up view of an emerging tongue (arrow) of a torus which might expand with a small increase in an already enlarged key parameter (e.g., price of raw materials, labor costs, capital costs, taxes or such) to form a new wing in an outcome basin.

Figure 2. The Birth of a New Attractor

The Feigenbaum number for the transformation of a torus into a butterfly attractor is 3.0. That is, when a key parameter exceeds 3 times its value in a previous iteration, it forces a outcome basin to expand into two distinct causal fields.5 The point is significant. Most researchers expect systems with similar initial conditions to behave similarly. Chaos theory teaches otherwise. The fact that there are two natural outcomes for a system whose internal parameters have not changed is foreign to most scientific thinking. Given changes in external parameters, water drops, gypsy moths, and perchance business firms undergo phase transitions without changing the nature of water, the physiology of moths or, perchance, the policies of a firm.

More than that, at the margins of the two outcome basins, it is not possible to predict to which outcome basin a moth or a firm or, perchance, a customer/client will go. Causality loosens and solidifies as one samples different regions of a 2n+ outcome field. This fact has profound implications for generalizability, replicability, and falsifiability of research findings. It may be the case that, in the Chaos paradigm, there are no samples, only 'universes;' no statistics, only parameters.

This kind of change, from 1n outcome state to 2n+ outcome states requires a whole new philosophy of management science. It is the kind of dramatic change which heretofore has been seen as inimical to good management but a 2n causal field offers choices not found in limit or torus attractors. For example, advantages accruing from a 2n gender division of labor are not possible in a unisex society; some societies invent 4n+ gendering patterns in order to combine certainty and flexibility in labor routines. In industrial societies, gender free occupational specialties replace gender divisions altogether as a way to organize the work process since 2n, 4n, or even 8n gendering systems interfere with flexibility in role allocation; in marketing strategies and in accessibility to unused talents and knowledges.

Management policies which had been adequate, even highly successful in a 1n causal basin now no longer cope with the variety found in the new basin. Innovation, flexibility, and spontaneity become assets in such a causal basin replacing stability, uniformity and formality as preferred management policy.

Figure 3. Two Views of a Butterfly Attractor

The Butterfly Attractor Strange attractors with more than one loose but predictable outcome basins are called butterfly attractors. Such a form is considered an attractor since, while a system may vary markedly from one iteration to the next, still it is very probable that it will wind up in one of two fairly stable if very different outcome regions. Figure 3 offers two idealized views of the same butterfly attractor; one view is presented in the well known time series format and the second view in a topological display which reveals its changing causal geometry in phase-space. The first such attractor was identified by Edward Lorenz in his meteorology simulations in the early 1960s (Lorenz, 1963).

And, as we see in the anatomy of a butterfly attractor, when a parameter of the larger environment changes beyond a given value, a system (or set of systems) can fluctuate between two outcome basins. Should a new wing develop from a torus, one might see one set of firms which succeed by virtue of internal policies and one set of firms which inevitably fail even if they adopt the same policies. There are now two outcome basins in that causal field. In between the two basins is a region most interesting to management science since, in that area, linearity is lost. There will be some firms which skirt the margins of both basins and may be drawn into one or the other; there is no way to predict.

It is important to note that it may not be the internal dynamics of the business that leads it toward one or the other wings of an outcome basin, but rather a small change in external parameters. The logic of this point is that, in the future, administration of a given firm must be open to policies which consider the stability of the whole system. When one adds connectivity to uncertainty, the best strategy is to reduce uncertainties for every connected system in the field rather than maximizing the short term advantages of one's own firm.

When a causal field has some fraction of its basin occupied by two or more attractors and some fraction in which great disorder is found, administrative policies which presume one and only one outcome field lose efficacy; indeed, causality, prediction, and control become casualties to new dynamics. For example, if external conditions change to open up new attractors in the labor market, workers would have more than one choice about the conditions of employment. Policies which had worked to attract and/or motivate workers lose efficacy in such a complex field. The difficulty of developing policy as to reasonable labor relations is resolved by a commitment to contingency theory (Koontz, 1961).

Figure 4. Labor Effort Attractor: a Federal Agency

Most dynamics are not so tidy as those in Figure 3. Figure 4 displays the kind of nonlinear order that exists in most actual nonlinear systems. 6 In Figure 4, data points are mapped on a cartesian graph which represents the phase space of work activity. In this phase space map, the current week's percentage of labor required to meet outside requests in a public agency (t), is mapped on the y axis (vertical), while the previous week's percentage of total labor required (t-1), is mapped on the x axis (horizontal). This attractor thus reveals the dynamic structure of the system as a function of the relationship between successive adjoining time periods.

In Figure 4, one can see two areas to which most time demands tend to cluster (bottom left and center right). Then, periodically, the agency labor demands loop from low labor time needs to much higher as requests from outside agencies arrive. The 'erratic' behavior in Figure 4 is entirely natural to nonlinear systems. If one were to try to control the twisting, turning, looping dynamics in Figure 4 to more closely approximate those in Figure 1, one would have to control more and more of the environment in which the system moves. Such efforts are unwise for two reasons; first, the costs of control efforts would affect profits (or wages if it were a worker owned and operated firm).

Secondly, as management piles layer upon layer of controls, control tends to displace other goals. Taking the U.S. economy as a whole, investments in control apparatus eat up ever increasing chunks of federal, state, local budgets as well as that of private firms. Both higher prices and higher taxes are required thus reducing competitive advantages and neglecting essential common social needs. Private security forces now outnumber public police and these are increasing each year. Add to that the various layers of control activity within and outside a firm and one can appreciate the need to adopt new goals in the management of nonlinearity rather than more controls.

If the experience of the former Soviet Union has taught us anything, it has taught us of the futility of trying to plan all activity of a productive system. If one considers the fact that all private and public systems now operate in a global economy, the task of controlling all key parameters in order to obtain stability of behavior of a firm (or a nation) becomes daunting. Even the most powerful nation cannot control all other nations and firms in the global economy now emerging. There are, however, solutions to the problem of order. 7 We discuss the management of Chaos below.

III. Exploding Attractors There is a third kind of change of great interest to those in management science. Chaos theory offers an explanation and description of the transformation of strange attractors from quasi-stability toward far-from-stable chaos. Feigenbaum (1978), discovered that a system with eight outcome basins is stable but one with 16 possible outcome basins quickly tumbles into deep chaos. The remarkable thing is that this precipitous rush into an infinite number of outcome basins is universal over all natural systems so far studied. The Feigenbaum constant gives us the possibility of making a prediction of the onset of full chaos when the first few period doubling parameters are known. It defines a key point in postmodern management strategy.

Figure 5. Bifurcation Points from Order to Deep Chaos

Feigenbaum taught us that when the parameters of a causal field change from previous values, an outcome field expands with elegant regularity even though there may be great disorder in the whole field. Figure 5 shows the 'cascade' toward full chaos. As one can see in Figure 5, after one bifurcation (or forking), a system has two outcome basins in which it might be found (Region A). After two bifurcations, there are four possible destinies which any normal system may take (Region B). After three, eight outcome basins which are available (Region C). After four bifurcations, the number of attractors possible for similar members of a set explode to fill any phase-space available to it (Region D).

The Feigenbaum numbers for phase transitions begin with 3.0 discussed above which produces a 2n outcome field. The second bifurcation point (3.4495) produces a 4n field, the third bifurcation (3.56) produces an 8n outcome basin. An outcome field with 16 attractor basins develop at F4 (3.596) to which any given firm, customer, supplier, or worker could move. After F4, firms are in a situation where a very small change can produce, not just one or eight new outcome basins but a veritable flood of entirely new and unexpected, unpredictable end-states toward which a person, a group, a business or a society might go (Rasmussen and Mosekilde, 1988). That happens when a key parameter reaches 3.56999 a previous value at which it was semi-stable.

From Chaos to Order One of the more interesting implications of Chaos theory for management science is found in Region D above. One will note that there are several regions of order in Region D as a whole. The arrows next to E, upper right, point to those windows of order. Those regions denote the emergence of entirely new organizational forms. In biology, such forms may be new species giving one a nonlinear theory of evolution. In sociology, a qualitative increase in food in the early 1700s may have kicked off the industrial revolution by permitting bifurcations in the division of labor due to the new food reserves. James Burke offers an excellent PBS documentary (27 Oct., 1991) on that process (without reference to Chaos theory). Given a division of labor made possible by food reserves and freed from feudal bondage, new technologies were developed.

Ilya Prigogine took a Nobel prize in 1977 for his work on the emergence of order from disorder. Such work resolves the contradiction that, while the second Law of Thermodynamics predicts an increase in disorder, the opposite has occurred in the past four billion years. The tendency is toward more complex forms. Chaos theory and Prigogine's explanation (along with Isabelle Stengers, 1984) offers much food for thought for management science. We return to some of the implications later. Right now we would like to pause along the route to chaos to offer a mock application of such phase transitions for heuristic purposes.

APPLICATIONS Chaos theory would suggest that, as key parameters in the external environment change, then people, firms or societies move to occupy differing regions/attractors in an outcome field. In order to illustrate some of the implications of Chaos theory for management science, we can take a common problem for university administration: parking policy.

Chaos and Parking Basins If some parking places are reserved for faculty and staff and some for students, there are two outcome basins to which drivers could go. If places are reserved for handicapped students, there are three outcome basins to which any given driver could park.' If a fourth area is reserved for visitors, a fifth for seniors and graduate students; another for occupational therapists and/or engineering students and another for maintenance vehicles, the outcome basin available to all drivers is very complex. Chaos theory would suggest that a parking regime with more than 6 or 8 such basins would be close to a bifurcation point at which 'illegal' parking would begin to increase qualitatively.

When key parameters are stable, the system might work--i.e., knowledgeable driver might go directly to the 'proper' parking area. There would, in the normal course of events, be some small and random violations of such policy. Chaos teaches us that uncertainty accumulates with small changes in key parameters. Three examples serve to make the point.

A. Let's assume that most drivers can handle 2 or 3 inches of snow. Three or four inches of snow would delay some students and force a nonlinear increase in violations as students, staff and faculty scrambled to get where they were going in time. Two more inches might greatly increase the number of parking violations; then even 1/2 inch more might throw the whole system into full blown chaos.

B. Most universities allow ten or fifteen minutes between classes. A large change in the time allowed between classes might not change the number of parking violators greatly. If, for example, there were 15 minutes between classes, students and faculty in one class period could get to their car and vacate a place before another set of arriving students began to look for places. If the time between classes were reduced by, say two minutes, there might be a small increase in parking violations. If three minutes, another relatively small increase might be seen. If one more minute were removed, an otherwise adequate parking policy might fail and students might begin parking everywhere; on the sidewalk, double parked in the street, blocking entrances to reserved lots and so forth. Full chaos would have developed.

C. A parking policy might work on a campus with tightly clustered buildings but after a few years of expansion, increase in time it takes to walk from car to class has the same effect as reducing the time between classes. As the sum of the square of each distance between all classrooms increases, one can expect the onset of chaos in both parking policy and classroom attendance.

Again, notice that nothing has changed in the design of the car, the parking lot or the genetics or physiology of the students. The only thing that has changed is a parameter outside control of students and faculty and a compelling need to park and get to class. In such a population of students, there will be some who are inconsiderate; they don't care whether they block others or not. There will be some rich enough to pay for any number of tickets. These will be the first to ignore policy, but these and all other students who are otherwise well socialized and distressed about 'illegal' parking will ignore policy. Taken as a group, all violators will be self-similar to the student body the day before when each set of parkers went to the 'proper' parking lot.

POSTMODERN MANAGEMENT SCIENCE Chaos theory represents a theoretical paradigm which serves both to critique and supplement Newtonian models of management founded on linearity, coherence, uniformity, prediction and hierarchical control. The assumptions of modern science, i.e., assumptions which have been central to the knowledge process for the past 300 years are greatly modified by new techniques and new insights. In brief, the linear ontology upon which modern science was grounded has been replaced by a nonlinear ontology subsumed by the Chaos concept. Chaos theory undermines the logic of tight control and stability of existing routines that traditional management tends to prefer.

Chaos theory offers management science insight about where and when management control is reasonable or possible and at what scale of organization such control efforts are best directed. In short, the dialectics between freedom and necessity vary across causal fields; given wisdom and judgment, semi-stability can be maintained if key parameters are modulated such that they do not destroy the connections which hold the system together or overwhelm the carrying capacity of the environment.

Chaos theory thus adds understanding and support to the logic of many new management ideas. Current concerns aimed at democratizing decision-making inside the work place find considerable support in the logics of Chaos theory; no one plan or policy is adequate to cover all exigencies for all time. Postmodern management science must yield up space to spontaneous and timely response by workers on the spot. The interest in decentralizing operations and decision-making also fits within the theoretic envelope of Chaos theory. Up to a point, an increase in flexibility adds to the overall rationality of a political economy. 2n, 4n, 8n and 16n outcome basins offer options as well as creative interactions. After 16n outcome basins, order and prediction are so difficult that even the best designed systems fail.

Management Policy and Nonlinear Feedback The source of semi-stability in any attractor is to be found in nonlinear feedback. The curious thing is that, if any management group wishes to fit a firm or agency to the inevitable patterns of change, it must couple linear change with nonlinear response. Linear amplification causes a torus to "...blow apart...' to fill the space available to it (Briggs and Peat, 1989:37). One operative question emerges for management science here. What constitutes a nonlinear response with which to obtain the quasi-periodic behavior essential to stability in preferred attractor(s)?

When the environment is fairly stable, then low-level nonlinear feedback suffices. Within the system, flexible management policies in which rules are moderated by wisdom and judgment, in which exceptions are made, variations tolerated and competing principles weighed against each other; all this constitutes low-level nonlinearity. It is the heart of the kind of quality control which the Deming people encourage.

Objective, uniform and impersonal applications of policies seem necessary in a management science which privileges linearity and coherency. They also appear to be reasonable in a society oriented to democratic values. However if such policies amplify deviancy beyond a Feigenbaum point, nasty surprises may await. Chaos theory implies that differential and situational policies may promote both stability and democratic values if wisdom and judgment supplement reason and rationality.

Managing Chaos Low level Chaos is not to be managed but to be savored. Postmodern management science brings into question traditional efforts oriented to tight management control tactics. Low level chaotic behavior may simply be evidence of a system's adaptive response to its environment. Low level Chaos implies learning as a system continuously reaches new points without retracing previous steps. Such behavior is compatible with the notion of a "learning organization". Conversely, typical repetitive linearized organizational processes inhibit learning by negative feedback which promotes an extreme stability and, in the same moment, inhibits the generation of fluctuations which generate new modes of work or production. One might argue that traditional management practices hinder, if not eliminate, learning in organizations.

However valuable some low level chaos might be for any system, there comes a time when chaos interferes with both production and social life. The question of controllability presents itself with compelling force. At first glance, it seems as if Chaos, by definition is uncontrollable. Yet recent work suggests that, if we can accept a different standard for controllability, then human agency is possible (Young, 1992).

Only Chaos Can Cope with Chaos In conventional systems theory, open systems can maintain stability through the incorporation of order from their environment (Bertalanffy, 1968; Buckley, 1968). When a system is mismatched to its environment, there are three generic solutions by which rematch can be obtained: 1) change the system, 2) change the environment or 3) recourse to a third system by which to bridge the two (Young, 1969, 1977). As we will see, in the Chaos paradigm, the solutions are a bit more complex but still depend upon change. H. Ross Ashby (1968:135) had posited that only variety can cope with variety but what is added to Ashby's Law of Requisite Variety by Chaos theory is the need to know when and how much variety is needed. The kind and timing of such variety depends on the characteristic set of complex cycles exhibited by a system. Hübler, (1992) calls this set the dynamical key to management of chaos.

A major point of interest for administrative science is that, while system dynamics may, at times, be understood best in terms of the characteristics of individual persons or firms, qualitative change from one organizational form or process to another depends upon the ways in which the whole system interacts with its environment. In brief, there are times when administrative efforts are best oriented to motivate and to control individual behavior and times when such efforts are best directed at system parameters. Chaos theory helps illuminate the regions in phase-space in which those very different management concerns are best focussed. 8

Formal control
Formal control tactics take two forms. First there are tactical efforts such as the Hübler protocol (1992:18) as well as that of Stephen Guastello, below. Hübler says that nonlinear physical systems respond to a special class of aperiodic driving forces. In his words:

In the Hübler protocol, one identifies the dynamical key of a chaotic system, calculates the difference between that key and semi-stable frequencies or oscillations of such a system and then adds such values to the key parameters which drive the system. Such work has been successful in creating solitons efficiently (29). The very sophisticated mathematics and rationale for such technical control tactics are given in the Hübler paper. One should note that whatever controllability of chaotic systems is now in place, it is for physical systems: liquids, lasers, metals and such. Control of nonlinear social systems may surpass the ingenuity of science since the key parameters are so global and the world increasingly so interconnected.

Reinventing Rationality In a very creative paper, Stephen Guastello (1992) showed that a specially derived chaotic tactic was more successful in controlling hiring rates in the construction industry than was a reactive tactic based upon actual labor needs. Guastello studied 90 hiring 'generations' of a contractor in the building industry. When the contractor used rules for hiring based upon fixed reactive rules, his workforce ended up with an unstable and insufficient number of workers. Guastello developed a nonlinear predictor, Nt = Bnt-1e(-aN). 9 Guastello calculated the variations of the workforce with both a linear reactive tactic (actually used) and a nonlinear tactic based upon his nonlinear predictor. That predictor, in turn, is based upon the natural variability of the industry modeled by the values indicated. He found statistical advantages to the nonlinear tactic. In both cases, the chaotic predictor of employment needs was superior (for reactive management, R2=.77 for the chaotic predictor function and .57 for the linear comparison model. For long term planning, R2 =.32 for the chaotic function and .14 for the linear comparison).

The implications of the Guastello tactic are profound. If his predictor can be polished and generalized, it is possible to reinvent rationality in such a way as to obtain better results in medium term management practices. Add to that, the Hübler work on control of disorder in deep chaos. If we go back to Figure 5, we will see shadows which cut across field C. These shadows define the points at which a very careful touch can secure stable dynamics in even deep Chaos. The ability to re-rationalize management science depends upon a very different view of rationality than comes from newtonian, euclidean, aristotlean understandings of rationality. In brief, rational behavior in a chaotic world is loose, holistic and changeable. Each region of an outcome field requires variable tactics for whatever intervention is possible for human beings.

Strategic Control A second approach for the management of chaos is much more strategic. It involves a quest for an optimal array of dynamical states available to workers, firms and economies. This means that one scans the data of performance workers, firms and whole economies to map out the deep structural patterns hidden there. With this knowledge, one can identify key parameters which push a person, a firm or an economy into creative or into destabilizing change. The task of the postmodern manager is to expedite bifurcations which produce desirable attractors and, at the same time, to control key parameters to stabilize such outcome states for a firm while framing organizational policy in such a way as to not destabilize the larger political economy.

If we can begin to see, that rationality is fractal and is located as much in the settings of key parameters of a society or a firm as it is in the minds and motives, genes or genomes of particular persons, then we come to an understanding that efforts to bring order, stability, prediction and pattern into the work lives of human beings depends as much upon the system acting sensibly as it depends upon individual socialization and training. In such a paradigm, reward or punishment have limited utility; firms succeed or fail in spite of good efforts or bad. Or, as some quality experts espouse (Deming, 1986), employees will inevitably fail if the processes that guide their work are flawed.

Management scientists, working out of the chaos paradigm, will want to give some attention to exigent uncertainties faced by employees. An employee may be able to handle one or two uncertainties in their social space but the interactive effects of three or more uncertainties will destabilize the life of even the best, most dependable employee (or client or supplier). A worker might be able to juggle the uncertainties of family and work but, given a third uncertainty, say health, interaction between them might escalate into full chaos for that worker. If such uncertainties are common in the labor force, uncertainty becomes unmanageable for the administration of the entire workforce taken firm by firm. Since workers are also customers, clients and suppliers, such collective uncertainties tend to destabilize the whole system. As national and global economies consolidate, such interconnections increase.

In the larger society, affirmative action policies, nonlinear in terms of seniority rules, merit, or equal opportunity may help prevent the amplification of deviancy of the whole system and thus fractal increase in percentage risk for any given system. Adverse effects of income and status inequality for the whole system have been well documented in social research (Currie, Dunn, and Fogarty, 1980). Special educational opportunities for young Black males, on the face of it, nonlinear in terms of equal gender/racial opportunity and merit, may be just the nonlinear response which stabilizes the whole system and prevent key variables from exceeding the fourth feigenbaum number. Head start programs for poor children, apparently inimical to free market ideas, social darwinism and individualism may make the market work better in the future by improving education such that next generation workers might move to other attractor basins (i.e., high tech occupations rather than low tech). Nonlinear policies, often called social justice programs, are nonlinear in terms of free market ideas, but they tend to keep key parameters within limits which satisfy two important needs of any open system; the need for pattern and predictability on the one side and the need for change and creativity on the other.

If we want to maintain the integrity of a market economy with its many advantages, then there must be such forms of nonlinear feedback with which to defeat the transformation of the torus which describes, say, frequency of entry into the market for essential goods and services by race, class and ethnicity. If not, the market itself may split into 2 or more attractors: one oriented to those with discretionary income and those without. As income inequalities between minority groups, economic classes or nations grow, more and more capital is attracted to the basin of production allocated to luxury goods and less and less to the production basin which essential goods are produced. When those with discretionary income determines the price and supply of essential goods and services, more and more low wage earners are driven out of the market. At some point, low wage earners will begin to seek alternative ways to increase income or to reduce costs in order to stay in the marketplace. Some of these alternative ways may be outside the logics of the marketplace; i.e., they may be nonlinear in terms of a cash economy.

Much strategic effort is outside the direct control of even the most democratic management effort. The holistic nature of Chaotic dynamics requires a democratic politics which involve all significant sectors of a population; in a complex society, that means every adult citizen. One cannot plan on abandoning whole sectors of a population without long term costs.

Chaos and Opportunity Even in a fully chaotic field, there are emergent social forms which offer great opportunity to an alert management team. The Chaos in the former USSR, for example, will produce heretofore unimaginable economic forms, some of which may incorporate the advantages of a market system together with the advantages of democratic socialism. Indeed, several American economists are working on 'market' socialism today. There will be much to learn from the chaotic conditions now developing there.

In his Santa Cruz presentation, Hübler made the point that, given stability, large well administered systems are workable. Given full chaos, small systems are preferred since they can be more sensitive to local conditions. One would want to develop a politics in which seed money was used generously to support new practices, products and services. The ordinary criteria for such loans, grants, shared ventures or investments are set aside in a time of chaos. One takes risks; some pay off and some don't. There is no present strategy for knowing which will serve.

The point continues to be made that rationality is as much a feature of the whole system as it is a capacity contained within the mind and thought of particular workers, students, administrators or managers. Within attractors in a 1n-basin, predictability hence control is possible--a loose and tolerant policy of goal assessment and evaluation is sensible. In the margins between the attractors of 2n, 4n, 8n, or 16n basins, predictability is difficult for the businesses and persons pushed out to these regions. The most advantageous ratio between order and disorder is the central problematic of postmodern management science rather than the quest for order per se.

Management science took on new dimensions when cybernetics, information theory, modern systems theory, decision theory and other linear-oriented aids in planning and implementing developed in the 1940s and 1950s as a result of seminal work by Bertalanffy, Boulding, Buckley and others. Now management scientists are called upon to learn of the work of Lorenz, Feigenbaum, Mandelbrot and many others who work out of this new paradigm. The new perspective will inform policy in a great number of areas now untractable. Problems of crime, bankruptcies, drug use, health care, education and homelessness offer other fields in which to pursue the implications of Chaos theory. We now sit at the edge of scientific history and cannot but choose to join.



There are four numbering systems of which only one is rational in the technical sense. The other, simple scaling systems are: nominal, ordinal and interval. The inappropriate use of rational scaling is a source of much mischief in physical and social science. Return


Phase-space is a make believe map of the change of a variable or set of variables over time. It involves placing the value of a given variable on a grid made up by the intersection (0) of n variables of interest to research. The mathematician/philosopher René Descartes (1596-1650) developed the technique hence it is often called a cartesian map. Return


See Briggs and Peat (1989), Chapter 0, for a very accessible treatment of this 'rubber' math by which fractals are measured; see Holden, (1986) Chapters 13 and 14 for a more rigorous treatment of fractals. The Social Dynamicist, V.3, N.2, June, 1992, offers ways to estimate five different fractal dimensions: capacity, information, correlation, near neighbors, and rates of expansion (Lyapunov dimension). Mandelbrot (1977:363) discusses classical and other methods for evaluating the area occupied by fractals. Return


Roger Penrose (1989:152) asserts that, in order for any system to be amenable to the generation of 'superb' theory, each cycle in its behavior must approximate the previous cycle with an accuracy of one part in 1014th parts. Indeed, all of modern science is devoted to the quest for such precision and discards all other patterns of behavior as 'error,' 'faulty observation,' 'inadequate measurement,' or observer bias. Chaos theory honors such variable behavior as the actual dynamics of the system; the subsequent theoretical statements made about such systems has the same validity as do 'superb' theories. Hence we call the study of self similar systems postmodern science in order to distance it from the privilege accorded precision in the knowledge process by modern science. Return


See Briggs and Peat (1987:58) for a very accessible explanation of the Feigenbaum numbers. Return


Data are from the Communications Division, State of Oklahoma, calendar year 1990, from the work of L. Douglas Kiel, "Nonlinear Dynamical Analysis: Assessing Systems Concepts in a Government Agency," Public Administration Review, (Nov./Dec., 1992). Return


See 'Chaos Theory and Human Agency in Management Science' in draft. Return


Chaos theory has many new concepts with which to grasp its ideas. There is a Glossary to which one may turn for explanations of such terms as phase-space, fractal or nonlinearity. Return


N = 100


by T. R. Young, Patricia Hamilton and L. Douglas Kiel

Attractor: A region in an outcome basin to which the dynamics of a system tends to take it. The size and shape of the attractor depends, sensitively, upon key parameters and the dynamics to which it is driven by such parameters. An attractor may occupy space between dimensions; if so, it is said to be a fractal.

Basin: The region in an larger basin of outcomes to which a set of initial conditions (causes) drives a system or set of similar systems. A system is said to be 'attracted' to that region, hence the pattern of nonlinear dynamics seen in such a basin is called an attractor. Image a saucer inside which spins a marble. The path of the marble is the attractor; the whole region is the basin. Since the path is a nonlinear function of key parameters, that area can be considered a causal basin. There can be n number of such basins/attractors in a larger outcome field.

Bifurcation A doubling of a period of a key parameter. With each doubling, there is distinct change from one behavioral regime to a new one for all systems affected by that parameter. After the third bifurcation in key parameter(s), the system tends to move in ways which fill the space available to it in an outcome basin. This latter state is a far from stable chaotic state.

Chaos Theory: A science which deals with the complex harmonies and dis-harmonies exhibited by natural and social systems. It is the study of the changing ratio of order and disorder in an outcome field. That set of foundational ideas which describe the behavior of complex unpredictable systems. Chaos theory focuses upon states with multiple periods or without predictable periodicity. Chaos research studies the transitions between linear and non-linear states of such dynamical systems. [From Chaos, <L. <G. abyss; from which our word, chasm also comes. The presumed original state of disorder of the unformed world. Funk and Wagnalls Dictionary.]

Dynamical Key: Each attractor has its own characteristic set of cycles. If one can identify that set and match it, then it is possible to affect the behavior of the system (Hübler, 1992:18).

Fractals: From the Latin, fractus, broken (frangere, to break). A measurement of the degree to which a body takes up space available it; an estimate of its efficiency in using the space it occupies. In more simple terms a fractal is a measure of the irregularity of an object. Mandelbrot uses fractal as a generic term applicable to all mappings of system dynamics in phase space.

Key Parameter: Any factor which affects the behavior of a system. Food supply is a key parameter which affect the population growth and decline of animals as are climate and predators.

Linearity A linear system is one in which cause and effect are related to each other in a proportional way. For instance, if one pound makes a rubber band stretch twice its length; a two pound weight will make it stretch four times its length. For many systems, causality is curvilinear; a given cause will have an effect that is smaller with each additional doubling of it. For many systems, the addition of the nth doubling will cause it to transform into non-linearity. Nuclear fission is linear up to the point of a critical mass. Nonlinearity marks the onset of chaos.
Nonlinearity: A pattern of behavior in which a change is out of proportion to the value of a driving force. There are three generic forms of nonlinear change which a researcher may chart. The simplest form is:

Outcome Basin: The range of alternative outcomes available to a nonlinear system at any given stage in its behavior. Simple systems such as a pendulum have a point to which they are attracted; more complex systems such as a thermostat are limited to a slightly larger region in phase-space. See limit attractor.

Ontology That which exists; the study of that which exists. Modern science sees ontology as that which exists in and of itself. The ontological base of modern science was deemed to act as a deterministic system. In modern social science, the ontology of the natural and physical world was thought to exist apart from the will, perceptions or activities of either the people studied or the observing scientist. The geometry of such objects were thought to be euclidean; either they existed as points, planes or solids or they didn't exist. The postmodern scientific view is that dynamics may be and often are nonlinear; that geometries may be and usually are fractal; that the scientist shapes the world as it is studied. Often knowledge reenters the world from which it came as part of a self fulfilling prophecy which changes the reality quotient of such worlds.

Phase-space: A Cartesian map of the dynamics of a system (or a set of systems) created in n-dimensional space by turning time-series data into a picture (above called an attractor) showing its topology.

Postmodern came into use in the 1950s in reference to architectural styles as a protest to the cubes, circles, pyramids and cones that marked modern architecture. The term was extended to literary critique of all pretensions at universal standards for novels, poetry, and composition. It use spread to art criticism; to critique of music and theater. Now it is used to denote an era in which all absolutes; all universals; all claims to objective Truth and all pretensions of perfection are challenged. Such a perspective fits excellently well into the ontological paradigm defined by nonlinear dynamics since there are no centers, absolutes or final states to which systems 'naturally' evolve.


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