Over the past 20 years, Chaos Theory, Nonlinear Dynamical Systems Theory (NDS), and Complexity Theory have been generating a revolution in mathematics and many sciences. More recently, these theories are being applied to the world of businesses, institutions, and other work organizations. A key question is how innovative concepts originating in mathematics and the physical sciences can be relevant to the everyday world that managers face, a world where pragmatic outcomes are the final judge of success?

One way to approach answering this question is to realize that all managers go about their business of management through the "lense" of how they conceptualize or picture what an organization is. For example, the internationally known management theorist Gareth Morgan has eloquently shown in his popular book Images of Organization (Morgan, 1997), that there are several possible ways of conceiving or envisioning what an organizations is. For example an organization can be pictured as a bureaucrat hierarchy, or a social culture similar to a small country, or as a power and domination system. These pictures of what an organization is are not necessarily mutually exclusive. Instead, a manager can operate from several images either simultaneously or consecutively depending on various circumstances.

But whichever concept is shaping how a manager sees his or her work group or organization, the manager's behavior and practices will be in congruence with that concept. For instance, if a manager sees his or her organization as primarily a power and domination system, the emphasis will be on politics and the accumulation of power. The successful manager according to that particular picture will act like a current day Machiavelli.

The point is not that one concept or picture of an organization is true and others are false. As a professor of one of my colleagues told his class one day: "All models are wrong, but some models are better than others." Indeed, chaos theory, NDS, and complexity theories are offering a new, alternative model of what organizations are, a model that, in important respects, is better than past models. This new model is providing not just new theoretical constructs but has important implications for how managers can manage more efectively. This new model is presenting a way of going about management that can be more helpful than past models.

Consider how a very pervasive, traditional image of managing organizations is based on four functions and presuppositions of what an organization is:

1. planning: based on the assumption that the future can be
predicted accurately;
2. organizing: based on the assumption that task structures are
to be imposed on organizational members;
3. controlling: based on the assumption that deviations from
normative practices should always be diminished;
4. leading: based on the assumption that the leaders are
"experts" and, accordingly, their goals or "visions" are
sufficient for organizational direction and motivation.

Findings from NDS, however, challenge each of the above assumptions of the traditional image of organizations.
In this paper we'll take a look at four features of NDS and how they can be applied to managing organizations (for greater depth on these features and implications see Goldstein, 1994):

1. Nonlinearity
2. Attractors and Bifurcations
3. Self-organization and Emergence
4. Far-from-equilibrium Conditions

Nonlinearity

The new sciences mentioned above are all based on a fundamental shift away from "linear" to "nonlinear" mathematics. Very simply, a linear function or equation is one that can be graphed in a cartesian coordinate plane by a line--hence linear. Whereas a nonlinear function or equation cannot be be displayed by a line--hence it is nonlinear. One way to understand the signficance of this difference is that in a linear function or system, the output or outcome of is proportional to the input or initial conditions, whereas in a nonlinear function or system the output or outcome is not proportional to the input or initial conditions. Later we will be coming back to the organizational implications of this lack of proportionality in nonlinear systems. For now, however, we are confronted with a problem: nonlinearity is being defined not in terms of what it is, but, instead, by what it is not, i.e., nonlinear is what is not linear. The British mathematician Ian Stewart (1989) has pointed to the unhelpfulness of this way of defining nonlinearity--it is like calling all animals which are not elephants, "nonpachyderms." Although true, this way of defining all non- elephants simply doesn't tell us much about these non-elephant animals. It doesn't tell us what they are, only what they are not. Similarly, defining nonlinearity as not being linear doesn't help much in providing an understanding of nonlinearity. So we need something more.

We can say nonlinear systems are systems in which the components or parts are interacting in a way that there is a kind of mutual influence or causal relation which keeps getting fedback, going around and around. For example, we can see this kind of nonlinearity in prey-predator relationship in an ecological region: the more predators there are, the more they will eat their prey, thus diminishing the population of prey. But as the population of prey becomes too low, this will eventually cut down on the population of predators for they will starve from lack of food. But, then, as the amount of predators decline, there will be a tendency for the population of prey to increase since there will be less opportunity for them to be eaten. This mutual influence can be modelled by nonlinear functions.

A feature of nonlinearity in mathematics is that, unlike linear equations, nonlinear functions are difficult to solve "analytically" which is why the physical sciences have tried to approximate cause and effect relationships by using linear functions which are much easier to work with. But by so doing they have sacrificed the complex behavior of the real world by replacing with the simple behavior of linear systems. Furthermore, research in chaos theory has shown that under certain conditions, nonlinear functions can have unpredicable outcomes. Since science has generally proceeded by looking for predictable patterns, nonlinear functions have often been relegated to the back of textbooks.

What does this have to do with work organizations? Well, businesses and institutions, as examples of social systems, are ideal candidates for nonlinear systems since they are made-up of the complex interaction of components: people and materials and technologies. Therefore, businesses and institutions can be modelled by nonlinear equations and, accordingly, will exhibit behaviors congruent with what we are now finding out about nonlinear systems in general.

Attractors and Bifurcations

A crucial aspect of certain nonlinear functions is that they exhibit the emergence of new "attractors," the next feature of NDS we will be looking at. "Attractors" are patterns in an abstract mathematical space which describe the behavior of systems over the long run. Attractors shape the possible range of behaviors possible in a system according to the conditions affecting a system. For example, consider a child's swing. If a father just pushes once on a swing his son is sitting on, the swing will go back and forth for a brief time and then settle dow to a position of rest. The long term behavior of the swing that is only pushed once, therefore, can be described as a "fixed point" attractor, this "fixed point" signifying the state of rest of the swing. However, if the father keeps pushing the swing every time it comes back, and the child maneuvers his body to get the most out of the swing, the long term behavior of the swing is now an oscillating pattern which is called a "limit cycle" attractor.

The oscillating behavior of the swing is, of course, caused by the conditions of the father pushing and the son leaning forward and stretching back at the appropriate periods of the swing cycle. These factors then have allowed the original fixed point attractor of the swing (the rest state) to move to the new limit cycle attractor (the oscillating behavior). The transition from one attractor to another is called a bifurcation.

In terms of our work organizations, the concept of attractors can be very useful for describing and understanding why employees behave the way they do. Everyone in an organization is always acting according to the attractors present in these nonlinear organizational systems. This, however, leads to the implication that employees are never "resisting" change, they are simply acting according to whatever attractors are dominating the system. As a result, the issue is not how to overcome employee resistance. Instead, the challenge facing managers is how to set-up the conditions where new, more effective attractors can emerge. Good leaders have always known how to do this intuitively. NDS can be of help in making explicit what these effective leaders have done by instinct. Good leaders somehow set-up organizational conditions where employees are "attracted" to work productively and with high quality, and with good job satisfaction. And when there are impediments to effective organizations, good leaders know how to aid the organization in bifurcating into new, more helpful attractors.

Self-organization and Emergence

The next feature of nonlinear systems we will examine is self-organization, which refers to a process thought to run by itself instead of one organized by the imposition of an outside force.
Recent studies of self-organizing systems have shown how new, unexpected patterns and new structures can spontaneously emerge out of the interacting components of a nonlinear systems when the right conditions are met. These new, emergent structures are characterized by greater coordination or coherence among the system components than the previous "equilibrium" state. Self-organization represents the emergence of new attractors shaping the behavior in a system.

It needs to be emphasized that self-organization is re-organization of pre-existing patterns, policies, and structures. This reorganization is made possible through various methods that challenge currently existing patterns and structures. Moreover, one of the pioneers in the study of self-organizing systems, the Nobel Prize winner Ilya Prigogine, has pointed to the way that the new structures emerging in self-organizing systems are partly the result of the amplification and incorporation of random events. Nicolis & Priogine, 1989). That is, self-organization brings about new emergent structures through taking advantage of accidents, unexpected events--in other words, "serendipity."

There is another crucial factor involved in self-organization: the need for a containment of the process of self-organization. A system undergoing self-organization needs to be an intact system, a system with some kind of closure containing it. Thus, the coherence found in emergent structures during self-organization take place across the system within the container or boundary defining and demarcating the system as the particular system that it is. Paradoxically, self-organization requires both a firm boundary to keep the system intact and also ways to breach the boundary, at least, in so far as environmental contact is called-for. Therefore, NDS is showing that two key roles of managers is to simulataneously firm-up the boundaries keeping the various systems intact as well as continually connect the various subsystems to other subsystem both within and outside the organization. The latter is why so many companies are finding it beneficial to bring customers inside the organization and send employees out to the market place.

Indeed, a challenge facing managers is how to work on the "containers" or boundaries that harness the tremendous power of self-organization. That is, the manager needs to consistently firm-up boundaries consisting of rules of interaction, lines of authority, and awareness of the work group's mission in the organization. This is particularly important in the face of the tendency in many modern corporations to dismantle traditional, hierarchically imposed boundaries. A manager wanting to facilitate self-organization needs to adopt a kind of boundary "negotiation" in which employees and managers work together to establish new boundaries. The paradox of boundary work is that boundaries need to be firm enough to contain the process of self-organization, yet permeable enough to allow a vital exchange with the environment.

In spite of the wide-spread belief that our businesses and institutions operate according to structures and plans imposed from higher-up in the management hierarchy, actually self-organizing processes are taking place all the time: employees spontaneously help each other out when a member of a work group is absent, or, a supervisor has an accident and an impromptu leader emerges from the group to act as the temporary leader; or people from different departments meet at lunch and come up with a new idea to improve the machinery. In all of these cases of self-organization in the workplace, direction and motivation do not need to be imposed from without. Rather, motivation is self-motivation and direction is self-direction.

Indeed, we can say that the capacity for self-organization is innate in our organizations only requiring the appropriate conditions to manifest itself. This means that self-organization does not require any extraordinary conditions. However, certain organizational policies or practices can impede the emergence of self-organizing change, e.g., too rigid bureacratic control, not allowing information to flow to all levels, or keeping departments or divisions too isolated from the rest of the organization. Therefore, another challenge facing leaders throughout all levels of a business is to loosen the constraints that are keeping self-organization from taking place.

Far-from-equilibrium or Bifurcation Conditions

Self-organization as the emergence of new attractors can be seen as a process of bifurcation. Another way to understand bifurcation is in terms of far-from-equilibrium conditions. These are the factors, external and internal to a system, which facilitate the self-organizing emergence of new structures. Far-from-equilbrium conditions remove or loosen whatever is causing a system to maintain the status quo, that is, they move a system away from "equilibrium." By so doing, they lead a system to a state of "instability" where it can be influenced or changed by small and random events. Again, the new structures that emerge are the result of the amplification and incorporation of the random events.

An equilibrium condition in a business or institution can be seen
in the isolation between departments, divisions, or functions, or in the various managerial control mechanism including hierarchical chains of command, inventory control methods, strategies to decrease variances, contingency planning, and so on. Far-from-equilibrium conditions arise when there is a change or interference with the equilibrium constraints: rigid demarcations are traversed, the system is challenged in such a manner that previous operational mechanisms, processes, and configurations are not sufficient. When an organizational system or work unit system is in vital, responsive contact with other systems or the environment, it can be said to be in a far-from-equilibrium conditions. In a work environment where equilibrium constraints are not dominating the system, people are constantly self-organizing into task groups to get something done.

Actually, anything that challenges current work processes and organizational practices or structures has the potential of acting as a far-from-equilibrium condition. Furthemore, far-from-equilibrium conditions arise when information is high in a system. Therefore, whatever methods that can increase the flow of information can function as far-from-equilibrium conditions. This can include, for example, methods for encouraging employees to voice their differerences in ideas, perceptions, and opinions instead of keeping quiet to go along with pressures for group conformity (see "difference questioning" in Goldstein, 1994).

Conclusion

To summarize, let's see how NDS is changing the four traditional management functions we mentioned earlier. First, since complex, nonlinear systems have been shown to be vastly more unpredictable than simple, linear systems, the whole notion of planning as
prediction of the future needs to be rethought. Indeed, my grandmother seemed to have fully known this feature of nonlinear systems: she used to tell me "If you want to make God laugh, tell Him your plans." Notice that she didn't say "prayers"--she believed God heard your prayers because they came from your heart, but plans were another matter since they were based in the belief that one could predict the future.

Second, because complex systems have the capacity to selforganize into new structures, the idea of organizing as imposing task structures needs to be supplemented by the possibility of a spontaneous emergence of new structures. The task for leaders is how to set up the conditions to facilitate self-organizing processes. Of course, there is still the need for leaders to impose task structures, but NDS opens up the possibility for a spontaneous emergence of task structures from the work group's interaction.

Third, the fact that self organization results, in part, from the incorporation of deviations from normative behavior means that the management function of controlling as the dampening of departures from the norm must be revised. Successful leaders have always known how to capitalize on whatever happens, accidents, unexpected market changes, and so on. Indeed, military strategists know that losing battles and retreats can be ultimately used to achieve a final victory.

Fourth, in an NDS perspective, leadership expertise and vision is seen as emerging, not from a few "experts" but, instead, from an entire working group and is not the sole domain of a designated leader. How can a vision be motivational if it is imposed on people from someone else. Of course, they will pay "lip service" to such a vision but it will have no real impact, except maybe to increase employee resistance to it. A "vision" can only be motivational if it emerges out of a group and is not imposed on them. Certainly, managers can always pressure their employees to follow the organizational "vision" but this pressure doesn't mean that these employees will be motivated to add that extra bit of effort and creativity that are demanded by businesses and institutions if they want to survive in today's tumultuous and unpredictable business environment.

References:

Goldstein, Jeffrey. 1994. The Unshackled Organization. (Portland, Oregon: Productivity Press).
Morgan, Gareth. 1997. Images of Organization (Second Edition.).
(Thousand Oaks, California: Sage Publications).
Nicolis, Gregoire & Prigogine, Ilya. 1989. Exploring Complexity. (NY: W. H. Freeman and Company).
Stewart, Ian. 1989. Does God Play Dice? The Mathematics of Chaos (London: Basil Blackwell).