From: weissert@xxxxxxxxxxxxxxx (Thomas Weissert)
Date: Sun, 19 Jun 1994 13:09:12 -0400 (EDT)
Nick, you asked for feedback, so here is mine.
; 1) Phase-Space is an extensive model of the BWO. It displays machinic
; activity as a synchronic pattern - a line - traced in a space with
; one dimension for each polarity of behavior (hotter/colder,
; faster/slower ...) or - interchangeably - axis of intensification
; (wolfings, becomings woman, cyborgization ...). 'Like' the BWO it is
; a smooth space of systemic process, appended virtually to every
; machine as the whole of its possible range of functions. Complex
; dynamics construes all systems as exploring a phase-space, migrating
; through a BWO.
"Phase" space is representative of a very particular set of conjugate
relationships, namely, the phase relationship between a state and its
momentum, so what is being described here does not characterize a phase
space but more likely a "state space." No loss of generality,
attractors exist in state space, as well as trajectories and many
dimensional objects, such as basins of attraction.
; 2) Systems are ordered (territorialized) to the extent that they
; restrict their behavioral searching to a relatively small zone of
; phase-space, exhibiting attractors. The two classical attractors are
; zero-dimensional points (stasis, perfect equilibrium) and one
; dimensional line-segments (oscillation, periodic equilibrium). Rich
; computer-accessed phase-spaces facilitated the discovery of a third
; kind of attractor (strange ones), of more complex low dimensionality;
; zones of dynamic nonperiodic tangling, in proximity to which
; behavioral trajectories territorialize in a graphic but unsimplifiable
The "line-segments" of classical attractors are necessarily
topologically closed objects, such as circles, where the trajectory
repeats itself in a neverending loop.
Strange attractors are not non-periodic, but are very significantly,
"quasi-periodic" with a global periodicity with local variations.
; 3) The systems captured by strange attractors belong to the phylum of
; cybernetic (nonlinear) machines, for which searching behavior guides
; itself through interaction with its phase-space position, responding
; sensitively to the outcome of its own trajectory (rather than to a
; fixed external influence). In far from equilibrium conditions (where
; there is not a massive point attractor drawing the system
; irresisistably towards machinic stasis) phase-space trajectories can
; sometimes complicate strange attractors, producing a 'dissipative
; structure' or involutionary process that moves ever further from
; both stereotypic regimes (rigid territories, tight and fixed phase-
; space boxes) and random ('ergodic') walks that learn nothing from
; where they have been.
Strange attractors do indeed have massive attracting points, usually
several, but these points are unstable and therefore unreachable, but
they must exist to generate the strange attractor and cause the
Strange attractors are themselves necessarily dissipative structures
already, in that they cannot exist in non-dissipatve systems. The
latter two behaviors described here can exist in non-dissipative, so
called Hamiltonian systems.
Thomas Weissert, PhD ] He's out there, crossing disciplinary lines
Dynamicalogist ] without restraint, synthesizing dynamical
Saint Joseph's University ] theories into some elusive theory of time.
Philadelphia, PA USA ] His methods are ... unstatic.