Complexity, Chaos, and National Security Policy: Metaphors or Tools?

Alvin M. Saperstein


Interactions between traditional nation-states, including the extreme interaction of war, can be likened to the interaction between microscopic bodies in physics. Relatively few variables are required to describe the process; the course of events is basically predictable—between occasional major, contingency-based, bifurcations (e.g., the outcomes of specific battles or collisions). Subnational wars—ethnic or tribal conflicts, guerrilla insurgencies—would then have to be likened to the interactions of meso-physics: fluctuations away from the mean become at least as important as the mean. The descriptive words usually resorted to are chaos, complexity, non-predictability, etc.

In the modern era, the actual and potential destructiveness of inter-nation war has tended to stabilize S.U. (Soviet Union)-U.S. type conflicts—with their nuclear weapon implications. This has allowed the realm of ethnic type war possibilities to grow and with it the attention of policy makers, scholars, and soldiers to the concepts of chaos and complexity—the theme of this conference.

That the paradigm of chaos was intimately associated with battle was certainly well known to von Clausewitz and the earlier Greek military historians. Many of the people at this conference, whose writings I have read with pleasure and profit (Beyerchen 1992, Lane and Maxfield 1996, Mann 1992, Mazarr 1994, Rosenau 1996, Rinaldi 1995, Schmitt 1995), have made amply clear the usefulness of the complexity concept in describing international security strategy.1 But do we gain anything from the visits of the soldier and statesman to the academy of the mathematician and physicist, besides some new, exotic descriptive metaphors (e.g., "strange attractor," "self-organizing criticality")?

Do we gain any useful policy making and/or strategic tools as a result of the concordance of the new metaphors, derived from the physical sciences, with the long recognized chaotic-complex aspects of war and national security in a competitive anarchic world?2 Has anything been gained by the transfer of the growing popularity of these paradigms from "hard" to "soft" scientists or the recognition of the growing prevalence of these "fads" by the military and political elites? A new set of metaphors to describe a world does not imply new or different behaviors of that world—we must be very careful not to confuse changes in an intellectual outlook with changes in world events or patterns which we hope to understand and master.

The role of the policy maker, whether in a domestic or an international system, is to master the system: to be able to take actions now which will lead to desirable events, or avoid undesirable events, in the future. Thus he/she must be able to predict the outcome of current activities: if I do A, A’ will result; if I do B, B’ will result, etc. Prediction is the transfer of knowledge of a system from its present to its future. The ability to make such transfers is usually based upon an understanding of the system—unless recourse is made to auguries or direct communications from a transcendental power. Excluding the roles of divination or divinity, we must help the rational policy maker to understand in order to master.

It is clear that the set of metaphors which underline our thoughts and discussions about the political world determine our responses to matters of war and peace.3 Action often follows theory. (But purely pragmatic responses—not the best, but adequate—are often resorted to by some societies with some success. Non-theoretical societies do survive, sometimes.) Moreover, we also recognize that our metaphors may also shape that political world.4 The "field of endeavor," within which we are trying to find appropriate responses, is not itself fixed apriori; its contours may be molded by our metaphors; the topographic maps relied upon by the competing forces may be altered by the plans and actions of these forces. Hence policy and response are easier and more effective, the more appropriate the available metaphors.

It should also be clear that the new metaphors will be helpful in educating that majority of citizens, soldiers, and statesmen which have not experienced chaos and complexity due to the apparent simplicity of the bi-polar world view of the last half-century. It may be easier to have university freshman and military cadets read modern works on complexity and chaos (e.g., Gleick 1987, Waldrop 1992) than have them study Thucydides or von Clausewitz. Metaphors also determine the social acceptability of presenting ideas publicly, thus subjecting them to criticism and possible action. For example, without the intellectual possibility of the dissolution of nations, i.e., complexity, few conceived of (and thus planned for) the end of the Soviet Union (and even fewer for that of its Cold War partner, the U.S.). The new intellectual paradigms should focus attention on the underlying world political realities—chaos and complexities which have always been there, sometimes obscured to many, but always recognized by some.

It is important to recognize that our metaphors, just as our goals, the "fields of competition and endeavor," and the events themselves, are constantly changing as a result of our formulating ideas, exploring our world, and attempting to control events and reach goals. We must be careful not to imbed our ideas and "world-pictures" in stone since the stone of the world is often brittle and ruptures catastrophically, or flows and deforms like lava. "He that will not apply new remedies must expect new evils, for time is the greatest innovator." (Philosopher-statesman Francis Bacon, 17th century)

Metaphors—Old and New

There are two major classes of metaphors, with roots in the history of physics, that are appropriate to this conference on global politics and national security: The Newtonian view is that of a fixed set of elements. They interact, linearly or non-linearly, in a fixed universe. Depending upon the issue under discussion, these elements (and their interactions) may be: nations interacting with each other (via war, negotiation, trade, cultural or terrorist exchange,...) in a world system; economic, bureaucratic, class,... groups "pressuring" each other within a given nation; military divisions, regiments, battalions,..., engaged with each other in battle or along a front; etc. The strengths of the individual elements and of their interactions may wax or wane, their "location" in the "field of endeavor" may change with time, but their continued existence, as well as that of the system of which they are elements, is taken for granted. (In the wars of kings, it was usually assumed that the opposing king would still be there "afterwards," just somewhat diminished.) This Newtonian paradigm of sovereign nations has been the usual framework for discussions of international security during much of the past few centuries.5

In the currently fashionable Prigoginean6 (Prigogene and Stenger 1984) paradigm ("self-organizing criticality"=SOC), elements and their interactions come into and go out of existence as part of the ongoing process; the field of endeavor may change in size, structure, and constituents with time. Thus states, armies, military and civilian units, may be born, grow, thrive, decay, die and disappear, as part of the process which also creates, distorts, and dissolves, the structures of which they are—if perhaps only temporarily—parts and foundations.7 States may be created out of, or dispersed back into, smaller groups of people as a result of war or other interactions between other states or people groupings.8 "Official" or "unofficial" military units form or dissolve as a result of anticipated or actual conflict between existing, nascent, or hopeful nations.9 Economic, political, or other classes, come and go through turmoil engendered by other groupings in the system of nation or nations.10 In sum, the system determines its apparent elements rather than conversely.11

The changing of the elements, their interactions, and the overall structure may occur at vastly different time scales. Consequently, there may be intervals of time in which the system seems to consist of fixed elements interacting with each other under fixed rules, i.e., a Newtonian description may provide a good approximation for some epochs. Conversely, a Newtonian system of small enough elements may provide the conceptual foundation for a Prigoginean system of larger elements: the shifting elements of the latter may "actually" consist of varying combinations of the fixed elements of the former. For example, guerrilla bands, regiments, tribes, nations, states, are all different time-varying combinations of people; the underlying Newtonian system would be the multi-billion member set of the world’s population. (And, of course, each person is a shifting combination of biological cells. And, each cell is a shifting sets of molecules. And so forth.)

Both of these paradigms can be taken with either a stochastic or a deterministic view. In a stochastic model there are no rules connecting the state of the system at one instant of time deterministically to its state at a following instant. Only probabilities connect the two. Within a stochastic Newtonian model, interactions between elements can be likened to the random collisions of molecules. Policy can be framed by comparing the relative probabilities of the outcomes of different policy-choice-paths and maximizing expectation values. Combining the stochastic and Prigoginean metaphors, security interactions would be modeled by "collisions" between elements which may or may not exist. Without resorting to the full apparatus of quantum field theory, there is no obvious simple means of rationally dealing with such models, and so they will be avoided in this paper.

Deterministic systems have rules, which may be ascertained, which uniquely connect neighboring time states of the system (Fig. 1a). In Newtonian systems, these rules would govern the interactions between the permanent elements. Within the Prigoginean paradigm, the rules would also govern the creation and dissolution of these, now perhaps impermanent, elements. Most people act, and have acted historically, as if there are "rules of human behavior." Hence I will stick to deterministic paradigms.

It is important to stress that determinism does not imply predictability. Prediction implies connections of necessity (not of probability!) between non-perfectly well-defined states of the system separated by finite time intervals. In order to rationally predict future behaviors of a system, we must know its present state. If the future knowledge so obtained is roughly comparable in quality to the present knowledge, the prediction is successful. But present knowledge is never perfect. There are always measurement errors in any determination of the present state. The resultant non-perfectly well-defined present state encompasses a number of possible starting states. The rules determining future states must be applied to each of these starting states. Thus, given any deterministic model, implicit or explicit, upon which predictions are to be based, a range of "paths into the future" are possible (see Fig. 1b,1c). Furthermore, any such model depends upon parameters obtained from necessarily imperfect observations. Hence even a perfectly determined initial state of the system allows a range of future outcomes in any reasonable predictive modeling.

The result of these two imperfections of observation is that any set of rationally ascertained system rules, which transfer realistically obtained present knowledge of the system into the future, will result in a range of possible outcomes—a range of uncertainty. If this future range of uncertainty is large compared to the range of present knowledge, the quality of prediction is impaired. If this future range covers all possible outcomes of the system (Fig. 1c), no knowledge of the future is possible—prediction (and hence rational policy making) is impossible.

If the rules governing the system are "linear,"12 the range of future outcomes is always comparable to the range of input uncertainties (Fig. 1b): prediction is possible, and therefore useful to the policy maker. If the system rules are non-linear13 (as are most systems involving competing human beings, wherein the policy of one party must not only include the desired goals of each party but also the response of the other parties’ progress toward those goals14), the system may display extreme sensitivity to small changes in input or system parameters (Fig. 1c). This behavior, called "chaos," (see, e.g., Schuster 1988) makes prediction—and hence control of future behavior of the system—difficult or impossible. However, it may be possible to predict whether or not a system will display chaotic behavior. This possibility, as shown in the following section, allows the policy maker to avoid dangerous behavior. Hence the ability to predict unpredictability is a very useful tool in policy making (Saperstein 1986).

Crisis Instability and Chaos

In pre-WWI Europe, the assassination of two people in the Balkans was enough to ignite a carnage that swept all of the continent and involved all other continents, left millions dead, vast territories desolate, wiped out existing nations and governments, and created new ones. In post WWII Europe, the murder of hundreds, or perhaps thousands, in these same Balkans left most of the rest of the world untouched—except perhaps in their consciences and charitable purses. In the first case, a very small change in the system parameters led to major transformations of the system—the definition of "chaos" if the system were a mathematical/physical system. In the second case, the disturbances effectively damped out as they propagated through the system—the sign of a stable mathematical/physical system. The political scientists have coined the phrase "crisis instability" to describe the first case—extreme sensitivity of the world political system to minor perturbations (see, e.g., Saperstein 1994). In the second case, the world system was "crisis stable." The same world system can manifest crisis instability at some places during some epochs, while displaying crisis stability elsewhere or at other times.

Physical systems, e.g., a moving fluid, can also display chaos (i.e., turbulence) in some circumstances and stability (i.e., smoothly varied, ordered, laminar flow) in others. Mathematical metaphors for these physical systems must be able to manifest both chaos, stability, and the transition between the two, if they are to be a reasonable representation of the physical reality. Furthermore, if it is to be useful, the mathematical model must be able to predict the circumstances under which the system will switch from stability to chaos. For example, the airflow over a given wing design will be laminar for air velocities whose Reynolds’s number15 is below some critical value (Fig. 2). For larger flow velocities, the flow becomes turbulent, dissipating energy in an uneconomical manner and making control of the total aircraft more difficult and perhaps dangerous. The ability to predict the critical Reynolds number, and its variation with changes in aircraft design, is very important for the aircraft designer who wishes to avoid having to find out that his aircraft is unstable via the sacrifice of test pilots or passengers.

Analogously, if it were reasonable to mathematically model the world system of nations, a chaotic mathematical system would be a good metaphor for a crisis—unstable world. Being able to predict the critical "Reynolds number" for such a world model would be very important for the policy maker whose goal was to avoid crisis unstable conditions with their concomitant high probabilities for the outbreak of war (Saperstein 1984).16 (In the modern political/weapons-of mass-destruction world, there are no "test pilots" and we are all potentially sacrificial passengers.)

In a Newtonian world paradigm (or in a Newtonian approximation to a Prigoginean world view), the notion of national security—and the goals of the corresponding policy makers—are fairly straightforward. Policy must be framed so as to either avoid war or to reap the benefits of winning a war (whose win can be "guaranteed" with associated costs less than expected gains). In either case, the prime goal is to maintain control of the future, to retain predictability and hence avoid crisis instability. Given a reasonable mathematical model of the system for which policy is being made, it can be used to explore for system characteristics which allow transition to chaos. The policymaker must then studiously avoid the corresponding behaviors or conditions.

An example of interest to the strategist of bipolar nuclear arms races (in the context of the S.U. - U.S. Cold War) is the modeling of the Strategic Defense Initiative, the proposal during the Reagan Presidency to deploy a massive system of ground-based and space-based defenses against strategic-ranged ballistic nuclear missiles. The model (Saperstein and Kress 1988) presumed that each of the two antagonists would deploy similar offensive and defensive systems against the other (Fig. 3). The deployment numbers would be determined in response to the opponent’s deployed weapons numbers; the result is a non-linear interactive system whose stability can be investigated by conventional means: introduce a small disturbance into the system and compute how it grows. As expected, there are starting configuration numbers (of offensive and defensive missiles) for which the perturbations remain small, others for which they grow greatly and rapidly (Fig. 4). The latter configurations are the crisis-unstable systems which are to be avoided by the relevant strategic planners.17

The same paradigm has been used to explore questions of more academic interest. Using a non-linear Richardson18 model of the arms race between competing nations, a comparison (Saperstein 1991) was made of the stability region of three-nation systems (Fig.5a) with that of two-nation systems (Fig.5b). The former was found to be smaller than the latter, indicating that it is more difficult to stabilize a tri-polar world than a bi-polar world, a conclusion which has also been drawn by many "conventional" non-mathematical political scientists. Another concordance between the results of mathematical modeling of international systems and conventional analysis has been that a system of democratic states is less likely to have wars than a system including oligarchic states. The model conclusions (Saperstein 1992a) result from the differing values of the Richardson-type parameters19 stemming from democratic versus oligarchic societies. The differences arise since the (Newtonian) nation entities of the Richardson model, and hence their interactions, result from averages over a larger Newtonian model whose elements are the nation’s decision makers—citizens, politicians, officials—a large class in the democratic state, a small group in the oligarchic state. In the latter case, the interaction parameters resulting from the average are more likely to be large enough to produce an unstable system. Finally, a comparative stability analysis was made of systems of competing nations, each looking out for its individual security, versus systems of alliances, shifting so as to maintain a "balance of power" (Saperstein 1992b). Again, the result—that it is easier to stabilize a balance-of-power system—was expected from conventional political analysis.

In all of the above cases, the chaos metaphor was used to steer policy makers away from potentially dangerous crisis instability situations—away from chaos. Alternatively, when war and its associated chaos is unavoidable, there is the traditional approach to the chaos of battle, an approach used by successful military planners whether or not they recognized or used the chaos metaphor. Since small perturbations can lead to largely different outcomes ("For want of a nail, a shoe was lost,... a kingdom was lost.") one appropriate response (characteristic of the U.S. military since Grant) has been to always deploy overwhelming forces, if they can be made available. (Have more than enough horses, so that the loss of a few would make no difference.) That is, the statistical fluctuations which mimic chaos usually scale as the square-root of N, the number of significant elements. For large enough N, the relative fluctuations are unimportant. An alternative to increasing the sizes of the force units available (the Newtonian elements of the system) is to increase the number of different types, their flexibility and rapid adaptability to changes. Have horses, mules, people, jeeps, well trained and available to carry out the required tasks. Better yet, have available alternative sets of tasks and immediate goals, which will lead to the final desired goal—if you can’t take that hill, take the other one. It is clear here that the new chaos metaphor offers no new tools to the military planner though, as has been previously suggested, it may significantly aid the military educator.

National Security in an "SOC" World

The goals of the national Security policy maker are not so obvious in a Prigoginian ("Self Organizing Criticality") world. Should policy be aimed at encouraging or discouraging the creation of new nations, the breakup of the old? Should new alliances, new armies, new bones of contention be anticipated? All of these are the possible system elements and interactions (between the elements) which may arise and evolve via the life of the system. It is now clear that all of these SOC possibilities must be anticipated as well as the vagaries of dealing with the usual interactions between the Newtonian elements of long-lived nation’s and alliances. For example, should the "West" have encouraged the break-up of Yugoslavia? (There is a long history of eastern European people living at peace with each other in strongly ruled, multinational, non-democratic States.) Are we better off competing with oppressive but strong oligarchies or dealing with fragmented—even worse, fractal—democracies?20

One of the prime reasons for our failure to successfully deal with Iraq—a "sovereign" element in the Newtonian system—is that we fear to deal with its possible break-up. Similarly, there were important confusions in our society in anticipating and dealing with the break-up of the Soviet Union. Our policies towards China have also suffered from these confusions. In the Newtonian scheme-of-things, nations are sovereign states and deal only with each others’ sovereigns. "Infringing upon sovereignty" is severely frowned upon. It is clear that we still speak to such a world, though we no longer live it.

It is not evident to me that a single metaphor/tool—like chaos—is available or useful to us in dealing with a world system characterized by "complexity."

Instead of specific new tools, these metaphors can contribute to the development of the new attitudes required for the more complex modern world. They can help sharpen minds dulled by a Newtonian world view so as to be alert to all new possibilities. (It should be obvious that such alertness and openness was always present in some outstanding historic leaders whose minds were, perhaps, not so overburdened with Newtonian simplicities.) Above all, we have to be alert to (and be able to respond to) the possibility of bifurcation21 (Fig.6a) of the existing system into very different possible worlds, containing new and different elements interacting in novel ways. Such bifurcations may occur at national levels—where nations rise and fall, where they are of interest to the strategist, and at local levels—of tactical interest, where military, governmental, or corporate units are created or destroyed. Though these bifurcations are contingent, the probabilities of their occurrence, and their outcomes, are not structureless; familiarity and insight into the fundamental aspects of the system can lead to clues as to when the probabilities of such change are large, and when they are small.


Thus we shall need very flexible diplomats and soldiers at all levels.22 (The metaphors of complexity may be helpful in recruiting as well as in educating them.) They will have to be very knowledgeable about past behavior of the system and its elements—as determining the chances for radical transformation of the system. They will have to be open and adaptable to the new and novel which may confront them - with or without rational anticipation.23 Clearly, the new policy makers will have to be thoroughly cognizant of the relevant elements of anthropology, sociology, and psychology, as well as history. Knowledge of the functioning of existing governments, their departments or military units, will not be sufficient, as these elements may be bubbling-up or dissolving into the inchoate foam of people and groups below.

Not only are flexibility and imagination required for attaining one’s ends in a complex system. The ends themselves will often be shifting and/or unclear. In some cases it may be desirable to fragment competing parties ("divide and conquer"—e.g., the British role in India); in other cases to consolidate them (create alliances or nations—e.g., the creation of Yugoslavia24). Of utmost importance is the recognition that the policymaker can help direct these shifts, by influencing the elements at a lower level than those of the system of interest; e.g., in a system of nations, it may be advisable to attempt to influence their individual citizens.25 So much for the sanctity of national sovereignty!

In mathematical terms, the usual way of seeking the "best" solution to a problem is to look for some maximum value of a function-surface over the space of values pertinent to the problem (e.g., Axelrod and Bennett 1991). The highest maximum (or the lowest minimum) is the best solution—the desired policy—and if the surface is known, that best solution can eventually be found. However, in a "Self-Organizing Criticality" world, the act of moving over the surface in search of its maximum can radically change the surface. It will thus act more as an elastic membrane than as a fixed-function surface. Thus we may not be able to look for the "good strategy" in opposition to the "bad strategy" but may have to settle for the "contextually appropriate strategy."

Conclusion: Chaos and Complexity— Tool and/or Metaphor?

It is clear that successful military and political policy makers have always entertained the potentiality of chaos and have sought the tools of redundancy and flexibility of resources to deal with that possibility. The only new tool to deal with chaos presented here is the engineering tool of attempting to predict crisis instability and then avoid it or be prepared to live with it. Quantitative dynamical models of the system of interest may be useful in making such predictions. If they are inadequate or unavailable, verbal models have a long history, and potentiality, of use.

If the leaders of the pre-WWI European states had recognized that the railroad schedule-dominated mobilization of their troops was a source of great crisis instability (Tuchman 1962, van Creveld 1989), perhaps they would have avoided starting—and being trapped by—the process. But this recognition would have required that the chaos metaphor be more commonly found in the "intellectual air" of turn-of-the-century Europe than was the case in that rapidly industrializing Newtonian-reductionist society.

Given a Newtonian paradigm, the policymaker strives to be efficient in reacting to a given "field of endeavor"; chaos is to be avoided or dealt with by overwhelming force and/or redundant means of force delivery. The present world seems to require a Prigoginean outlook: don’t accept the battlefield or the world system as a fixed given. The complexity, or adaptive self-organizing, metaphor should be very useful for the necessary education, recruitment, planning, and thinking required to deal with and survive our future. However, no obvious specific tool—like predicting crisis instability—comes to mind. The metaphor require that one should always be contemplating the future. And, among these considerations for the future, always include attempts to change the field of endeavor itself.

Hence, it may not be useful for the policymaker to always look for the uniquely "best solution." It may be necessary to settle for a local temporary maximum—a good solution, rather than the best. In the elastic fabric of our present and future world, the "perfect" is often the enemy of the "good."

When all is said and done, on a strategic level, the most useful aspect of the chaos and complexity metaphors is to remind us and help us to avoid falling into chaos.26

End Notes

1. "Complexity may be defined as the set of deterministic theories that do not necessarily lead to long-term prediction....The numerical variables are still uniquely related to each other locally in space and time. But...we cannot obtain the future values implied by the theory just as a result of compact, well-defined manipulation of the present values....Complexity theories thus depend on the complete ‘path’ taken by the system between its beginning and end points....Every intermediate instant of time may see the theoretical system diverted from the path it might have taken in the absence of perturbations, which are always present....The system is extremely context-dependent." (Saperstein, 1995)

2. Contrary to popular wisdom, it may not be so bad to be prepared to fight the last war! Last wars have always been chaotic and complex; it is only in the post World War II "cold war" that some serious stategists have believed in a non-complex world paradigm.

3. A "non-appeasement" world view, stemming from the failure of appeasement towards Hitler, has governed our post-WWII policies towards Stalin, Iraq, Bosnia...

4. For example, the Wilsonian ethnic metaphor—that every ethnic nation should have it own state—broke up the European multi-ethnic empires, leading (?) eventually to disasters like the Bosnian conflict.

5. "Do what you wish to your own people and your neighbors will not get involved." "Zaire Under the Gun," New York Times, Nov. 3, 1996, p.E3.

6. I am indebted to John F. Schmitt (1995) for this characterization.

7. In a complex adaptive system, these "emergent properties" or "structures" are the result of contingency, not determinism: you cannot predict when, or if, they will emerge, how long they will endure.

8. Zaire is a national state now—but for how long?

9. From whence did the Taliban militas come; will they last?

10. The Russian "Mafia" may be such an "emergent" "business class."

11. In the perpetual intellectual dispute between "wholeness" and "reductionism" (the whole is different from the sum of its parts vs the whole is equal to the sum of its parts), SOC is in the wholeness camp.

12. Changes in output are proportional to changes in input; equivalently, the output resulting from the sum of two inputs is equal to the sum of the two outputs separately resulting from each of the two inputs.

13. Non-linearity implies that the anticipated response to a planned action modifies the plan.

14. As an example of non-linear behavior, consider a nation, pacific in intent, which only arms itself in anticipation of possible attacks from its explicitly aggressive neighbor. It realizes that the neighboring nation will detect its arms buildup and respond with its own; in fact the neighbor might be inclined to advance its presumably planned attack so as to come in ahead of the determinedly defensive arms buildup. So, in anticipation of this response, the defensively oriented nation launches a supposedly preemptive attack against the presumed aggressor!

15. A dimensionless system parameter which is determined by the characteristic size and flow velocity as well as by the viscosity and density of the fluid and which determines the properties of the fluid flow. When the Reynold number excedes the critical value (determined by the basic characteristics of the system) the system becomes unstable to transition to a chaotic state.

16. Certainly, close thoughtful attention to the developed world’s hungry reliance upon petroleum, imported from regions controlled by closed oligarchies, should have raised the prospect of impending crisis instability.

17. This warning of the possibility of a loss of predictability and control over an escalating arms race came at a time when some optimistic Cold-War strategists were arguing for the practice of precise control over an upward spiraling MAD dance.

18. The usual linear Richardson model of a two-party arms race assumes that the rate of acquisition of arms, by each party, is proportional to ("linear in") the existing stock of arms of its opponent and to its own arms stocks. The non-linear model takes into account the possibility that the opponent’s stocks can become "saturated" and hence of diminished danger.

19. The coefficients of proportionality between the existing arms stocks and the acquisition rates for new arms—hence a measure of the distrust and fear of the opponent and the confidence in one’s own arms.

20. Czechoslovakia fragmented into the Czech Republic and Slovakia. Unfortunately for the people of Bosnia, the different ethnic groups living there have fractal boundaries between them. In the former case, there are clearly two separate areas, separated by a reasonably "smooth" boundary; this is not true in the latter case.

21. Bifurcation (Fig. 6a) represents a choosing (in the usual way) one of several possible futures (which contingently become available), leading to the creation of sets of distinct plans—one for each future. Chaos (Fig. 6b) implies that these different futures are interbraided. Hence plans must constantly be mixed and revised.

22. In a chaotic situation, every element must be prepared to become a Clausewitzean "center of gravity" if the designated center is knocked out. The German tanks did so well early in WWII, against their technologically equal or superior opponents, because each one was equipped with radio and each understood the goals and rationale of the original plans and so was able to take over and modify plans as necessary.

23. A good football team may have separate offensive and defensive squads, but each must be able to fulfill the role of the other when circumstances (fumble, interception) so require—which is often. In the military, it may be possible to make do with a previously designated and trained "peace-keeping quarterback" and a "peace-making quarterback," etc., each prepared to take over and lead a well trained "general-purpose squad" for the appropriate purposes. We know and expect that ordinary military units can carry out diverse tasks.

24. Note that the same "world system" sometimes finds it useful to consolidate, and sometimes useful to fragment its previous consolidation, e.g., Yugoslavia.

25. Such influence has long been attempted, e.g., Voice of America, BBC Overseas Service, "hidden" subsidies to the political parties, labor unions, business enterprises, newspapers, radio, TV, etc., of other countries, and of course, propaganda to troops on and behind the front lines.

26. The author is greatly indebted to his colleague (and wife) Harriet for her careful reading of the first draft leading to critical, insightful, and productive suggestions.


Axelrod, Robert and Scott Bennett. 1991. A Landscape Theory of Aggregation. University of Michigan preprint.

Bacon, Francis. 17th century. Quoted in Newsweek, Oct. 14, 1996, p.43.

Beyerchen, Alan. 1992. "Clausewitz, Nonlinearity, and the Unpredictability of War." International Security 17:3, 59-90.

Gleick, James. 1987. Chaos: The Making of a New Science. New York: Viking.

Lane, David and Robert Maxfield. 1996. "Strategy Under Complexity: Fostering Generative Relationships." Long Range Planning, 29:2, 215-231.

Mann, Steven R. 1992. "Chaos Theory and Strategic Thought." Parameters Autumn, 54-68.

Mazarr, Michael J. 1994. The Revolution in Military Affairs: A Framework for Defense Planning. Carlisle Barracks, PA: U.S. Army War College.

Prigogen, I. and I. Stengers. 1984. Order Out of Chaos. New York: Bantam.

Rinaldi, Steven M. 1995. Beyond the Industrial Web: Economic Synergies and Targeting Methodologies. Maxwell Airforce Base, School of Advance Airpower Studies, Air University.

Rosenau, James N. 1996. Complex Humanitarian Emergencies: Toward an Integrated Understanding. Cambridge: Center for Population and Development Studies, Harvard University. April 1.

Saperstein, Alvin M. 1984. "Chaos - A Model for the Outbreak of War." Nature 309, 303-5.

Saperstein, Alvin M. 1986. "Predictability, Chaos, and the Transition to War." Bull. Peace Proposals, 17:1, 87-93.

Saperstein, Alvin M. and Gottfried Mayer-Kress. 1988. "A Non-linear Model of the Impact of S.D.I. on the Arms Race." Journal of Conflict Resolution 32:4, 636-70.

Saperstein, Alvin M. 1991. "The Long Peace - Result of a Bi-Polar Competitive World?" Journal of Conflict Resolution 35:1, 68-79.

Saperstein, Alvin M. 1992a. "Are Democracies More or Less Prone to War? A Dynamical Model Approach." Mathematical and Computer Modeling 16:8/9, 213-221.

Saperstein, Alvin M. 1992b. "Alliance building vs Independent Action: A Non-linear Modeling Approach to Comparative International Stability." Journal of Conflict Resolution 36:3, 518-45.

Saperstein, Alvin M. 1994. "Chaos as a Tool for Exploring Questions of International Security." Conflict Management and Peace Science, 13:2, 149-77.

Saperstein, Alvin M. 1995. "War and Chaos." American Scientist, 83, 548-557.

Schmitt, John F. 1995. Chaos, Complexity and War: What the New Nonlinear Dynamical Sciences May Tell Us About Armed Conflict. Quantico, VA.: Concepts & Doctrine Division, Marine Corps Combat Development Command.

Schuster, H.G. 1988. Deterministic Chaos—An Introduction. Weinman, FRG: VCH Verlagsgesellschaft mbH.

Tuchman, Barbara. 1962. The Guns of August. New York: Dell.

van Creveld, Martin. 1989. Technology and War. New York: the Free Press.

Waldrop, M.M. 1992. Complexity. New York: Simon and Schuster.

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