Academic journal article McNair Papers

10. Nonlinearity and a Modern Taxonomy of General Friction

Academic journal article McNair Papers

10. Nonlinearity and a Modern Taxonomy of General Friction

Article excerpt

All but one of the historical and conceptual elements necessary for this essay's fourth and final task--recasting general friction in modern terms--have now been introduced. The sole outstanding item is the concept of nonlinearity as it has come to be understood in fields like mathematics and physics since the early 1960s. By revealing how small differences in inputs can make large differences in outcomes, nonlinear dynamics will not only complete the task begun in chapter 6 of building indirect arguments for friction's undiminished persistence in future war, but furnish the last conceptual elements needed to update and extend Clausewitz's original concept.

Nonlinear science has been deferred to the end mostly to avoid burdening the exposition any earlier than necessary with a subject that various readers may find unfamiliar, difficult to grasp, or simply alien to the subject at hand. As mentioned in chapter 4, Clausewitz himself was not the least bit shy about appropriating concepts like friction and center of gravity from the physics of his day to illuminate the phenomena of war. Furthermore, in the winter 1992/93 issue of International Security Alan Beyerchen argued convincingly that Clausewitz himself not only perceived war "as a profoundly nonlinear phenomenon ... consistent with our current understanding of nonlinear dynamics," but that his use of a linear approach to the analysis of war "has made it difficult to assimilate and appreciate the intent and contribution of On War." (1) This author's experience has also confirmed that attempts to apply the ideas of nonlinear science to the study of war continue to be met with resistance, if not incomprehension, for precisely the reason Beyerchen cited: the widespread predominance of linear modes of thought. Hence, it seemed wise to defer nonlinearity until all the other evidence and arguments suggestive of its relevance had been deployed.

What is nonlinear science all about? The core ideas are not hard to describe. Nonlinear dynamics arise from repeated iteration or feedback. A system, whether physical or mathematical, starts in some initial state. That initial state provides the input to a feedback mechanism which determines the new state of the system. The new state then provides the input through which the feedback mechanism determines the system's next state, and so on. Each successive state is causally dependent on its predecessor, but what happens to the system over the course of many iterations can be more complex and less predictable than one might suppose. If the nonlinear system exhibits sensitive dependence on initial or later states, then at least three long-term outcomes are possible: (1) the system eventually settles down in some single state and remains there despite further iterations (long-term stability); (2) the system settles on a series of states which it thereafter cycles through endlessly (periodic behavior); or, (3) the system wanders aimlessly or unpredictably (so-called "chaotic" behavior). In the third case, detailed predictability of the actual state of the system can be lost over the course of a large enough number of iterations. (2) Chaotic behavior, however, should not be confused with randomness. Successive tosses of a coin remain the exemplar of a random process; if the coin is not biased, then the probability of either "heads" or "tails" on one's next toss is 50 percent. The paradigmatic example of a chaotic process, by contrast, is a "flipperless" pinball machine of infinite length. Edward Lorentz has characterized its behavior as being sensitively dependent on a single "interior" initial condition, namely the speed imparted to the pinball by the plunger that players use to put each ball into play. (3) On this view, chaos may be described as "behavior that is deterministic, or is nearly so if it occurs in a tangible system that possesses a slight amount of randomness, but does not look deterministic." (4)

The "mathematics of chaos" that has been used since the early 1960s to explain the sort of nonlinear dynamics exemplified by Lorentz's infinite pinball machine can be easily demonstrated using a personal computer or a programmable calculator to explore a simple nonlinear equation such as the "logistic mapping," x^, = ^nfl - -^n) (where the variable x is a real number in the interval [0, I], and the "tunable" constant k can be set between 1 and 4). …

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