continuum dynamics to

eigenvalue typology

bifurcations, attractors, basins

threshhold effects

topological variety to equivalence classes

fuzzy sets to sharp sets ??

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If we inspect the various examples of typological-topological alternation we have already considered, we can make up a list of some of the characteristic inter-level processes by which topological variation is re-organized relative to a higher scale-level as typological information --- and vice versa (next slide).

Thus, to get discrete types out of continuum dyanmics, we have available to us such common processes as the appearance of eigenvalue solutions to the dynamics, meaning that there are discrete values of some macro-system parameter for which there is a consistent solution, and that for arbitrarily near neighboring values there is no such solution. This is most commonly seen in the case of requiring stable or periodic solutions, as with harmonic standing waves, where there are only finite wavelength ratios to some supersystem constrain parameter (size of the resonating volume), and most fractional or irrational ratios produce unstable or decaying solutions. Of course the most famous cases of this are in quantum mechanics, where the energy eigenvalue spectra of many systems with localized interactions are discrete. Continuous spectra also occur (next slide).

More general than eigenvalue dynamics are all the dynamical systems for which there are multiple attractors, so that often continuous variation in some parameter leads to a bifurcation (more than one dynamical possibility, as in catastrophe theory), and one bifurcation can lead to another, and to a cascade (and to trifurcations, and chaos), all discrete. Moreover, each distinct attractor in effect classifies all points in its phase-space basin (all dynamical states destined to end up on the attractor) into a discrete set of types (one type per attractor in the simplest case). The underlying representations of the dynamics here are all continuous, but the outcomes are discrete.

Not only bifurcations and catastrophes, but all discontinuities and singularities potentially give rise to threshold effects in which some continuously varying dynamical parameter reaches a point where qualitatively different system behavior occurs.

We have also encountered the many cases where some SI's selective sensitivity (i.e. limited sensitivity) ignores the details of topological variation and so in effect creates equivalence classes, which form a discrete typology. Pattern recognition and classification may at root not in fact be a process that depends on what a system sees, but on what it ignores.

Finally we should raise the possibility that there are mechanisms at work in self-organizing systems whereby fuzzy sets, those with degrees of membership, and relations among such sets become 'crisp' or 'sharp' sets with all-or-nothing membership criteria.