Science, Strategy and War The Strategic Theory of ... - Boekje Pienter

meaningless. Interestingly for understanding Boyd, this condition of SIC offers yet anotherdimension of fundamental uncertainty. As Gell-Mann notes,chaos gives rise to effective indeterminacy at the classical level over and above theindeterminacy in principle of quantum mechanics 15 .For many purposes it is useful to regard chaos as a mechanism that can amplify tomacroscopic levels the indeterminacy inherent in quantum mechanics 16 .Phase space, stable basins and forks in the roadWith new methods offered by computers non-linear equations describing this behavior couldbe solved but these solutions are of a very distinct kind 17 . Near equilibrium one findsrepetitive phenomena and universal laws. As we move away from equilibrium, we movefrom the universal to unique, toward richness and variety. The result of these non-linearequations will not be a formula describing the behavior of the system under investigation. Instead the result will be an idea of the “phase space” of a system, which describes the range ofpositions a system can occupy.The problem with chaotic systems is that, unlike a clock’s pendulum (which willslowly return to a stand still), they never pass through the same point, i.e. the system neverrepeats itself so that each cycle of a pendulum (to continue the example) covers a new regionof phase space. It will not be possible to predict which point in phase space the system willpass through at a certain time but it will be possible to map the phase space, for in spite theseemingly erratic motion, the points in phase space are not randomly distributed. Togetherthey form a complex, highly organized pattern (aptly named “attractor”) which computers areable to visualize. These are sometimes also referred to as basins because they betterconnotate a system settling into something. Interestingly, a chaotic system may have fiftyvariables, but its motion may be restricted to a strange attractor of three dimension. Thusspaces of stability are known as attractors, and if the system is understood well enough, theattractor space can be predicted albeit only over the short term.When energy is added to the system, the system moves further away fromequilibrium and moves to the edges of the phase space. In the absence of significantperturbations, a dissipative system will usually follow a ‘normal’ linear trajectory. There willbe the usual boundary testing, but in the absence of any sustained increase in environmentalenergy, the system will return to its original point of reference. At some point suchmovement may be due to internal micro-fluctuations or due to external perturbations fromthe environment. This may give rise to a self-amplifying cycle. At the boundary zone, farfrom-equilibrium,systems may bifurcate. The bifurcation point is a crisis point of a system. Atthese bifurcation points, where the variables of parameters fluctuate constantly, the systemfirst fluctuates between two or more new points, and as the oscillations continue it willabandon the original path and ‘chooses’ between two possible new trajectories on the basisof very small differences in the values of the controlling parameter(s) at the point of change.Figure 9 depicts a bifurcation diagram for chemical reactions far-from-equilibrium 18 .15 Gell-Mann, p.26.16 Ibid, p.27.17 This description is based primarily on Prigogine and Stengers (1984), Chapter V.18 Adapted from Peter Coveney & Roger Highfield, The Arrow of Time, Flamingo, London, (1991),p.166 (a book also on Boyd's personal papers).127