It is the mark of an educated mind to rest satisfied with the degree of precision that the nature of the subject admits, and not seek exactness when only an approximation is possible.
Aristotle in Nichomachian Ethics
Nonlinear dynamical systems theory is holistic to the extent that it studies properties of physical behavior that are inaccessible to microreductive analytical techniques. But it nevertheless proceeds by massively simplifying the models it studies.
Stephen Kellert in In the Wake of Chaos (1)
Think of nonlinearity as a game. There are the "players" and the "place." In baseball, "nines" play on "diamonds"; in football "elevens" play on a "gridiron." Recall the seven basics of complex adaptive systems (cas) covered earlier. The four properties-aggregation, nonlinearity, flows, and diversity, and the three mechanisms-tags, internal models, and building blocks, as cas constitute the players, or "sevens." But what is the "place" that nonlinearity is played on? Well it resembles a tree, or your lungs, cardiovascular system, or the brain. It is called the "period-doubling cascade," which is depicted in Figure 3.1. (2)
As in football, this "place" contains two end zones. At one end there is Equilibrium, and at the other, Chaos. Neither end zone is a particularly attractive place. In Equilibrium end zone, everything is so stable, there is so much order that growth, innovation, and progress are suffocated. In the Chaos end zone, the situation is just the opposite. The turbulence is so severe that human understanding and intervention becomes impossible.
In between these end zones, is the playing field-a region called Complexity. It is
- a kind of 'phase transition' between order and randomness. Water frozen into a simple lattice of molecules is not very complex. Nor is a gas in which the molecules vibrate at random. But between the two extremes is liquid water, which can move in complex patterns that are almost mesmerizing. (3)
Ice represents the region of Equilibrium in the above quote, and gas, or water heated to steam, the region of Chaos. Water, that life-giving substance, is formed in the Complex region. It is essential territory precisely because that is where complex, adaptive systems (cas) thrive. It is an oasis.
A football field is marked off in 5-yard increments from 0 at one end line to the 50-yard line, and then back to 0 at the other end line. In the generalized "game" of nonlinearity, the playing field is marked off in bifurcation points-1 through 4. Each bifurcation, or "splitting into twos," is a fork in the road, or a branching representing choices, possibilities, or paths. The first bifurcation point, which generates two alternatives, is an end line marking the formal boundary between linearity and nonlinearity-the Edge of Equilibrium. Then comes the second bifurcation point generating four. Next comes the third bifurcation point bringing with it eight branches. However, the difference between the second and third bifurcation point is only about 22 percent of that between the first and second.
These bifurcations follow a rule-a rarity in mathematics, such as the value of pi. The rule is that these bifurcations occur in an accelerated fashion. Each succeeding bifurcation happens at an interval, each closer to 22 percent as long as its predecessor, creating a compression effect. Therefore, the fourth bifurcation point, in about 5 percent of that between the first and second points, develops 16 choices. If we were to carry this forward, the fifth bifurcation and its 32 alternatives occur in a fraction of about 1/100; the sixth with 64 in about 1/500; and so on. But we don't carry the playing field further, because we have entered the Chaos end zone, and this is the turbulence you encounter in which the average mind and will turn to mush. While chaos is deterministic, the turbulence obscures the underlying patterns. To us it is just mindlessly random because of our habitual way of looking.
Nevertheless, there are exceptions, and this end line-the Edge of Chaos-is more elusive, lying in a range. The boundary may be based on the application, which is to say that it is situational and depends on circumstances. In computers, which are essentially crude machines compared to the human mind, their ability to mimic life may require a boundary just short of the third bifurcation point. Human agency may be able to hold sway, beyond that of computers, through the barrier of the third bifurcation just barely into the fourth. But for some, the boundaries may be even greater. This, I assert, accounts for the successes of Napoleon, Rommel, and Patton and their forces, each a superb complex adaptive system that attained at times those qualities which only lie at the very farthest reaches, just short of chaos.
In football the object of the game is to get into the opponent's end zone, while keeping the opponent out of yours. The object in nonlinearity is to stay out of both end zones (and get your opponent into either one). What you want to do is to stay in the field of play, moving back and forth in the Complexity region. As long as you can avoid the end zones, you are doing alright, or, at least, surviving.
In order to prevent entering either the rigid suffocating world of the Equilibrium end zone or the bewildering Chaos end zone, most humans (revolutionaries and anarchists excepted), and the institutions and societies they build, intuitively have practiced a back and forth shuttle within the confines of Complexity-from the edge of equilibrium to the edge of chaos. This amounts to a sort of precarious balancing act, like a gymnast on the balance beam, in order to stay where cas allows learning, adaptation, and emergence necessary to an interesting life and the prospect of progress. We are like the dairyman Tevya's fiddler on the roof, trying to keep our balance while we play a pretty tune..."coping with the bounds."
W. Brian Arthur, a Stanford economist, gives examples of the complexity shuttle in human affairs.
- But, interestingly, even when a system gets lumbered down with complications, there is hope. Sooner or later a new simplifying conception is discovered that cuts at the root idea behind the old system and replaces it. Copernicus's dazzlingly simple astronomical system, based on a heliocentric universe, replaced the hopelessly complicated Ptolemaic system. Whittle's jet engine, ironically, replaced the incurably complicated piston aeroengine of the 1930s before it also became complex. And so growing complexity is often followed by renewed simplicity in a slow back-and-forth dance, with complication usually gaining a net edge over time. (4)
Watching the unconventional but always interesting feminist Camille Paglia addressing the cadets at West Point on C-Span, I caught the following in paraphrase. Paglia talked about what she called the tension between Plato and Dionysus-averting rigidity and authoritarianism, on the one hand, and chaos on the other-an eternal dynamic observed throughout political and social history. Here we have another "slow back and forth dance." In fact, the basic processes of political life can be reduced to these dynamic interactions between liberal and conservative tides in the complexity shuttle. Corporations, congregations, and even families also make these adjustments.
The complexity shuttle historically has been, while instinctive, a clumsy affair. Not well understood or articulated, because the principles of nonlinearity were only dimly perceived, pratfalls have been common. This has led to unintended conflicts, inequities, suffering, and wasted resources. In the future, an acute awareness of nonlinearity, coupled with the refinement of nonlinear techniques, promises to significantly improve our ability to "cope with the bounds." Should we succeed, this may well be the hallmark achievement of the 21st century.
A "play" is represented by the path of a bifurcation (which is actually a complex adaptive system or cas encountering its environment, which includes other cas). The cas senses its situation and collects information about surrounding conditions. It then responds to this information by using a set of internal models to guide its actions. The cas also encodes data about new situations for use at a later date. The key for a path or "play" to become history, and then stay history, is that the seven basics of cas, covered in Chapter 1, work well together providing agility and quick adaptation, especially a "good-enough" set of internal models to stabilize itself.
Now one, both, or none of these branches of a bifurcation may survive. Those that survive are saved by a successful encounter of a cas with its environment, which stabilizes it for an instant or for hours, weeks, months, years, or eons. History itself can be viewed as the human record of these saved branches; an event happened. A condition existed, whether recorded or not. Or consider your ancestry. You can read its bifurcation history in the "family tree," including our own birth. For those paths which are not saved, the event/condition never happens, the sign of failure of a cas to adapt with its environment. Even those branches which are saved, such as a birth, will sooner or later lose their stability because over time conditions change, resulting in death.
A "play" or a bifurcation path representing a cas "expedition," occurring close to bifurcation point 4 is worth more than one near bifurcation point 1. Complex adaptive systems (cas) thrive best at the edge of chaos, or the closer to the boundary with the chaos end zone the better. This play, however, is complicated by the accelerating pattern of the bifurcations, each forming faster and faster. The result is that if we are not careful, we will not have the time to either recognize what is happening or the time for correction. Therefore, pushing the limits of the envelope in order to benefit from the heightened capability of cas at the edge of chaos is a matter to be balanced with the risk of getting too close. History is full of winners, but, perhaps, more losers trying to pull this off. Alvin Saperstein, a Wayne State University physicist, provides a good historical example of losers:
- When all is said and done, the most useful aspect of the chaos and complexity metaphor...is to remind us and help us to avoid falling into chaos.. ..If the leaders of pre-WWI European states had recognized that the railroad-schedule-dominated mobilization of their troops was a source of great crisis instability, 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. (5)
Michael Shermer, an adjunct professor of the history of science at Occidental College, has developed a "contingent-necessity" model of history consisting of six corollaries. (6) These represent a refined playbook in which the generalized concept of the complexity shuttle in Figure 4.1 (see Chapter 4) is customized to deal specifically with the characteristics of history. Even without definitions and detail, these six corollaries have the feel of a nonlinear playbook (and the complexity shuttle).
At the macro level, where fundamental scientific questions and processes are the focus of investigation-the composition of the subatomic world, the creation of the universe, ecosystem change, and especially biological evolution-certain conclusions are largely accepted. Among these is that organisms encounter and endure chaos in order to evolve. Falling into chaos means submitting to "punctuated equilibrium," a term that describes the way that evolution works. Evolution does not follow a smooth curve. Instead, it is marked by intermittent stutter-step movements, akin to earthquakes and avalanches, triggered by a phenomenon known as "self-organizing criticality" (SOC). Briefly stated, self-organized criticality is based on the principle that
- Large interactive systems perpetually organize themselves to a critical state in which a minor event starts a chain reaction can lead to a catastrophe...a deceptively simple system serves as a paradigm for self-organized criticality: a pile of sand. (7)
Following punctuation, the system loops to the Equilibrium regime. Further, there is no complexity shuffle. The arrow points in only one direction-towards chaos.
But for the commander, statesman, and manager in the day-to-day world of real outcomes for which they are responsible, the chaos end zone is a place to avoid like the plague. He or she is expected to perform, not evolve, at least not in the biological sense.
Apparently, scalar effects are involved here, and they need to be recognized in some framework, similar to the distinction between macroeconomics and microeconomics. Just as economics, in general, has some globally common attributes, it is also recognized that there are differences in behavior and impact at different scales. For example, one does not apply macroeconomics to the family budget, or conversely, microeconomics to an analysis of Gross National Product. So too it is with nonlinearity.
At the micro level, the failure of the complexity shuttle involves falling off the complexity region into, on the one hand, chaos or on the other, equilibrium. This can happen from (1) a lack of agility, (2) risk-taking, or (3) the accumulation of events, resulting in environmental conditions, largely beyond our control, which cause us to lose our balance. The latter condition can be caused by the effects of coevolution. Remember Robert Jervis's words in Chapter 3:
- We usually think of individuals and species competing with one another within the environment, thus driving evolution through natural selection. In fact, however, there is coevolution: plants and animals not only adapt to the environment, they change it. As a result, it becomes more hospitable to some life forms and less hospitable to others.
Further, it appears to be difficult for a system in Equilibrium to get back into the Complex domain without the aid of exogenous, or outside, factors. Consider the Great Depression: the complexity of the 1920s economy sucked into the deep chaos of the 1928 Wall Street Crash lasting days, followed by years and years of the stagnation of the Great Depression. Despite the largely ineffectual attempts to "jump start" the economy by New Deal policies, the economic system was only propelled back into the Complex regime by conditions generated by the impending outbreak of World War II.
The region of Equilibrium at the opposite end of the Complex region is also a threat. The dynamics are different. Instead of losing the reaction time caused by the compression of bifurcations at the other end, one gains breathing space at the edge of equilibrium-what one might overconfidently consider a risk-adverse area to operate in. History is replete with examples of decay, with no sign of a chaotic preamble. Consider, for example, the Soviet Union's command and control economy. Certainly its dynamics were close to the "edge of equilibrium," perking along at best in a mildly nonlinear range to afford it the minimum of innovation, while still providing rigid Five Year Plan control using linear techniques. In the late 1980s, events led it not to a frenzied outburst of bifurcations. It just wound down, not up. It went "metronomic"-the tick-tock of the point attractor. It died, it didn't erupt.
In many works in the field of nonlinearity, chaos is not so much to be avoided as the plague, as is reductionism. There is a school of thought, known as Holism, which does not recognize the scalar effects. Instead, it imposes macro-nonlinear principles on the micro-nonlinear level as a norm. Consider the following paragraph:
- In human and social systems, both linear order and chaos intertwine in varying degrees and alternate throughout the life history of the system. A period of relative order is followed by a period of chaos, which in turn brings forth a new order. The period of deep chaos is a natural and necessary part of the development of every living and social system. It comes at the bifurcation point of discontinuous change. The conditions that are the fertile ground for the creation of the new order are born out of the turbulence of chaos. (Italics added.) (8)
It is obvious that we have a problem here. Holism treats the complex regime as largely a mere transition from Equilibrium to Chaos, and presumably on to better things, whereas in micro-nonlinear terms-that is for humans-it is an oasis! Holism rejects any concession to the use of reductionism in nonlinear affairs. Holism insists that a nonlinear system be dealt with "in the whole." In Holism everything is connected to everything else, and there is no hierarchy. Holism, therefore, is a slippery slope which without "handles," has a tendency to default to "worst case" scenarios, and therefore, advocate extreme measures to avert them. Certain radical formulations regarding environmental concerns come to mind.
Such a condition is untenable and useless for the responsible commander, manager, and diplomat involved in national security policy, military strategy, and operations. Some form of reductionism-we all understand things as models, or miniatures, each an abridgment of reality-must come to the fore to deal with nonlinearity, and coping with the bounds of the complexity shuttle.
The only qualification required of this reductionism is that it must help us to be better at doing the complexity shuttle. They will not be the kinds of methods or techniques we are used to, or like to use. They are all more tacit than the hyper-overt models we are used to in linearity. These are Aids to Learning.
Tools of analysis are, well, tools. Like wrenches, hammers, oscilloscopes, radars, and differential equations, they are preconceived "artifacts," preassembled and ready-to-go. All their "learning" is built into them by their human designers who are familiar with the nuts, bolts, nails, frequencies, and problems these tools deal with. They are basically linear; they are proportional, additive, replicable, and certainly they are bothered with because they will result in an expected, measurable effect on the cause. And they work well enough in environments which are mildly nonlinear.
But how does one devise a wrench, or radar, for any degree of nonlinearity beyond the mild? Nothing so overt as a tool can be preassembled. One must rely on something more tacit, sometimes even close to spontaneity, like the work of our immune system. Recall John Holland's observation that, "One cannot have a prepared list of rules for all possible situations, for the same reason that the immune system cannot keep a list of all possible invaders." (9)
If a tool is not available, than an "aid" is the next best thing. An aid is something which helps to do the job, which is learning. An aid is not a tool, which is already "learned." But, that is what our current understanding of nonlinearity allows us. In the future we can expect to get better. Someday, there may even be something akin to nonlinear tools.
- Even with the weather, there are building blocks-fronts, highs and lows, jet streams, and, so on-and our overall understanding of changes in weather has been much advanced by theory based on those building blocks. It is still difficult to predict detailed weather changes, particularly over an extended period. Nevertheless, theory provides guidelines that lead us through the complexity of atmospheric phenomena. We understand the larger patterns and (many of) their causes, though the detailed trajectory through the space of weather possibilities is perpetually novel....A relevant theory for cas should do at least as well. (10)
In the mean time, how do we make the most of what we have? We can get good at six things: metaphors, Perrow's quadrants, applying Van Creveld's rule, systems dynamics, genetic algorithms, and pattern recognition. Each of these aids will be covered in the next section. Each is a low-level model, all of them being more tacit than we are used to. But they are cas friendly, and allow insight into their workings. It could be that Aids to Learning work by somehow exploiting the mechanisms of cas to get insights into their properties. Without putting too fine a point on it, there appears to be a rough correlation between the Aids to Learning and the mechanisms of cas, covered in Chapter 2 (See Figure 3.2).
But even if you do get good at nonlinear reductionist techniques, they will still not be as comfortable as a wrench, or hammer, or overt models. The tacit nature of aids to learning are by design "crude looks at the whole." Insights into nonlinear environments are provided by "coarse graining" and " blurred vision." In order to get "past the trees to the forest," we try to look up through the levels in the hierarchy to understand the overall patterns.
Finally, the complexity of the nonlinear and its resistance to predictability gives the phrase "solving the problem" a hollow ring. In actuality, we "cope with the environment." Nonlinear reductionist techniques do not optimize, they satisfice. Their object, like cas, is not the perfect answer, but the good enough, fast enough to ensure survival. They seek the fittest, not the fanciest, avenue. They are inelegant and messy compared to the fastidious, but often ineffective constructs of the linearist. What the aids lose in formalistic symmetry is more than gained by the vibrancy of life. Our brains have been trained; aesthetics, form, and taste are part of our intellectual inheritance. We will have to undergo (using perhaps the most abused term in history) a "paradigm shift."
| Coping with the Bounds Index | Foreword | Acknowledgments | Introduction | Part One Introduction | Chapter 1 | Chapter 2 | Chapter 3 | Chapter 4 | Part Two Introduction | Chapter 5 | Chapter 6 | Chapter 7 | Chapter 8 | Chapter 9 | Chapter 10 | Conclusion | Appendix 1 | Appendix 2 | Appendix 3 | Appendix 4 | Appendix 5 | Appendix 6 | Notes |