Smith, R. D. (1998) 'Social Structures and Chaos Theory'
Sociological Research Online, vol. 3, no. 1,
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Chaos, like Relativity, Darwinism and Mechanism before it, has the potential to be transformed into a metaphor and to have its terminology misunderstood and misapplied. Indeed numerous new titles currently available indicate this trend has already begun. (eg. Savage, 1988). The best defence against this is a well-defined and accepted formalism
In order to have a formal notion of structure which is consistent with the chaos theory paradigm we now need to return to a requirement so far unelaborated. In its most precise rendering chaos can only arise when the possibility of any given state repeating itself is potentially zero. To take the illustration of a strange attractor such as the Lorenz attractor (as in Figure 4) what is needed is a situation in which the orbital pathway of a flow or flux can continue for an indefinitely long period of time (for eternity) without ever passing through the same point twice. If this condition is not met then the orbit is not in fact chaotic but periodic even though highly convoluted. What this in turn means is that the phase-space in which the flux is propagated should be continuous and not quantised. A quantised space, however large, is effectively finite and thus cannot provide for truly chaotic behaviour.
For true chaos to be present the orbital pathway must be a line of infinite length. If it passes through the same point twice then the deterministic nature of chaos informs us the orbit is of fixed periodicity and length. In order for true chaos to exist in nature, then, it may be necessary for space-time to be continuous and not quantised. If it is quantised then the number of possible positions is, while incomprehensibly large, finite. Conservation of energy means that the phase-space for such an orbit would likewise be limited. Lattice-models in which both chaos and solitons arise, however, have been proposed (Sayadi and Pouget, 1991).
Note that in this plot the lines clearly appear to intersect. They only appear to do so because this a two-dimensional rendering of a three-dimensional process. Chaos in a phase-space depends on the dimensions of the phase-space. If the actual dimensions of the connections (the lattice through which the signal or influence propagates) is of evolving or growing dimensionality then apparently finite restrictions may be ephemeral. See, however, for opposing views on themes related to this, Dyson (1988) and Gell-Mann (1994).