November 2, 1992            GLOSSARY
January 2, 1994
                               of

             TERMS FOR THE STUDY OF COMPLEX SYSTEMS

                               by 

                           T. R. Young 
                    The Red Feather Institute

                              and 

                        Patricia Hamilton
                        School of Nursing
                    Texas Woman's University

Attractor:  A region in an outcome basin to which the dynamics of a system
tends to take it.  The size and shape of the attractor depends, sensitively,
upon key parameters and the dynamics to which it is driven by such
parameters.  An attractor may occupy space between dimensions; if so, it is
said to be a fractal.

     Attractor, Limit:  A very stable pattern of behavior in which a system
     moves between two points of which an automobile cruise control or
     thermostat are examples.

     Attractor, Point:  The pattern of behavior of a system whose
     dynamics tend to converge to one point in phase-space.  The favorite
     example is that of a pendulum which tends to revisit a given point at
     precise intervals or to come to rest at a specific point each time it
     is perturbed.

     Attractor, Strange:  A strange attractor is simply the pattern, in
     visual form, produced by graphing the behavior of a nonlinear system. 
     Since that behavior tends to be both patterned and unpredictable, it
     is called strange.  If the dynamics are likely to fall somewhere in a
     region of phase-space, it is said to be attracted to that shape.

     Attractor, Torus:  An attractor shaped a bit like a doughnut.  It uses
     more than two and up to three dimensions in phase-space.  A system's
     dynamics [determined by key parameters] can take it anywhere on or
     inside the cylinder of the torus but one cannot say just where.

Basin:  A region in the larger field of outcomes to which a set of initial
conditions (causes) drives a system or set of similar systems.  A system is
said to be 'attracted' to that region, hence the pattern of nonlinear
dynamics seen in such a basin is called an attractor.  Image a saucer inside
which spins a marble.  The path of the marble is the attractor; the whole
region is the basin.  Since the path is a nonlinear function of key
parameters, that area can be considered a causal basin.  There can be n
number of such basins/attractors in a larger outcome field depending on
how many bifurcations have occurred.

Bifurcation  A doubling of a period of system.  With each doubling, there is
distinct change from one behavioral regime to a new one for all systems
affected by one or more key parameters.  After the third bifurcation in key
parameter(s), the system tends to move in ways which fill the space
available to it in an outcome basin.  This latter state is a far from stable
chaotic state.

Chaos Theory:  A science which deals with the complex harmonies and
disharmonies exhibited by natural and social systems.  It is the study of
the changing ratio of order and disorder in an outcome field.  See also that
set of foundational ideas which describe the behavior of complex
unpredictable systems. Chaos theory focuses upon states with multiple
periods or without predictable periodicity.  Chaos research studies the
transitions between linear and non-linear states of such dynamical
systems. [From Chaos, <L. <G. abyss; from which our word, chasm also comes. 
The presumed original state of disorder of the unformed world.  Funk and
Wagnalls Dictionary.]

Complexity; The science of.  Complex systems are always changing, always
'learning,' always observed to have some mix in which it exhibits both great
order internally and varying disorder in its behavior.  See Structural
Features and Dynamics of Complex Systems; an accompanying teaching
protocol.

Dynamical Key:   Each attractor has its own characteristic set of
complex cycles.  If one can identify that set and match it with
countervailing input, then it is possible to affect the behavior of the
system (H�bler, 1992:18).  In theory, chaos is manageable.

Feedback:      When a system acts in such a way to affect other systems
in a causal field, and when those systems, in turn, affect the behavior of
the first system, a feedback process is in place.  A common example used in
the literature for a limit attractor is the switch on a thermostat; when it
approaches a lower preset limit, it activates a source of heat which then
affects the thermostat in such a way that the switch is deactivated when
the temperature reaches a higher preset limit.

     In biology, the number of trout affect the number of pike which, in
turn, affect the number of trout since Pike are predator to trout.  In
sociology, a symbol used by one person will often elicit a similar response
in another person whose response in turn with elicit further response from
the first person.  A greeting is a case in point; if one says, Hello to
another and if the greeting is returned, both parties are open to further
interaction.  If the greeting is not returned, then feedback stops.

     There are three forms of feedback which are of interest to Chaos
theory: positive linear feedback which tends to push a system into full
chaos; negative linear feedback which tends to draw a system down to a
point attractor; and nonlinear feedback which tends to maintain an unstable
system in a given pattern.  This last form of feedback has most interesting
meaning for both the management of chaos and for social policy of any
society which wants to maintain enough disorder to permit adjustment to a
disorderly environment and enough order to permit prediction and control.

Fractals:  From the Latin, fractus = broken; (frangere, to break).  A fractal
is a measurement of the degree to which a body takes up space available it;
it is an estimate of the efficiency with which a given system uses the space
it occupies.  In more simple terms a fractal is a measure of the irregularity
of an object.  Mandelbrot (1977) uses fractal as a generic term applicable to
all mappings of system dynamics in phase-space.  Natural fractals occupy
real time-space.

Linearity    A linear system is one in which cause and effect are related
to each other in a proportional way.  For instance, if one pound makes a
rubber band stretch twice its length; a two pound weight will make it
stretch four times its length.  For many systems, causality is curvilinear;
a given cause will have an effect that is smaller with each additional
doubling of it.  For many systems, the addition of the nth doubling will
cause it to transform into non-linearity.  Nuclear fission is linear up to
the point of a critical mass.  Nonlinearity marks the onset of chaos.

Nonlinearity:  A pattern of behavior in which a change is out of proportion
to the value of a driving force.  There are three generic forms of
nonlinearity which a researcher may expect in natural and social systems. 
Each represents a qualitatively different ratio between order and
disorder.  The simplest form is:

     Self-similarity:  A self similar system tends to occupy about the
     same region in phase-space but never exactly the same.  A torus is a
     good example of a system displaying similarity but not sameness.  This
     is first order change.  But see below.

     Bifurcation  A doubling of a key parameter of a system.  With each
     doubling, there is discernible change from one behavioral regime to a
     new one.  This second order change is different from self-similarity
     since a system now has two (or more) causal basins to which it might
     move.  Each basin is very different yet still one can discern some
     local similarity.

     Deep Chaos  After the fourth bifurcation in key parameters, a system
     tends to behave in ways which fill the space available to it in an
     outcome basin.  This latter state is a 'far from stable' chaotic state
     and constitutes third order change.

Outcome Field:  The range of alternative outcomes available to a nonlinear
system at any given stage in its behavior.  Attractors such as those of a
pendulum have a point to which they are attracted; more complex systems
such as a thermostat are limited to a slightly larger region in phase-
space.  After a second bifurcation, outcome fields have multiple outcome
basins.

Ontology   The study of that which exists; often O. refers to that which
actually exists in and of itself apart from human action or perception.  The
ontological base of modern science was deemed to display linear dynamics
thus requiring rational scaling and logically coherent theory to define its
dynamics.  In modern social science, too, the ontology of the natural and
physical world was thought to exist apart from the will, perceptions or
activities of either the people studied or the observing scientist.  The
geometry of such objects were thought to be euclidean; either they existed
as points, planes or solids or they didn't exist.  Postmodern science
accepts that dynamics may be and often are nonlinear; that geometries may
be and usually are fractal; that the scientist preshapes outcomes in the
research endeavor by choice of scale and parameters to isolate and study. 
Then too, such knowledge often reenters the world from which it came to
change the reality quotient of the dynamics it studies as part of a self
fulfilling prophecy.  Vide political, economic or gender theory.

Parameter, key:     Any factor which affects the behavior of a system. 
Food supply is a key parameter which affects the population growth and
decline of animals.  Climate, predator population, trace minerals, micro-
wave radiation or competition from other species also serve.  In social
dynamics, class, race, ethnic, kin and gender status are key variables which
preshape, nonlinearly, the destinies of people.  It is most important to
note that a slight change in a given key parameter can produce widely
differing outcomes; not the effect of new and different variables as
presumed in modern science.  

Phase-space:  A Cartesian map of the dynamics of a system (or a set of
systems) usually created in 3-dimensional space by turning time-series
data into a picture (above called an attractor) showing its topology. 
String Theory suggests that there are up to 26 naturally occurring
dimensions.

Postmodern:  The term, postmodern, came into use in the 1950s in reference
to architectural styles as a protest to the cubes, circles, pyramids and
cones that marked modern architecture.  The term was extended to literary
criticism of all the pretensions of universal standards for novels, poetry,
and essays.  Its use spread to art criticism; to critique of music and
theatre.  Now it is used to denote an era in which all absolutes; all
universals; all claims to objective Truth and all pretensions of perfection
are challenged as political expressions of some human interest.  Such a
perspective fits excellently well into the findings of the new science of
complexity since there are no centers from which absolute judgments of
time and velocity can be made nor are there final states to which systems
'naturally' evolve.

Soliton:  In 1834, John Russell noticed that a wave on a canal did not fade
away: did not degenerate into its most probable state as the second law of
thermodynamics predicted.  It continued intact down the length of the canal
as far as he could see.  The concept of the soliton refers to such
phenomena.  This capacity of a system to move through its environment and
to maintain its integrity grounds the field of irreversible thermodynamics
of which Von Bertalanffy was an early Nobel Laureate for his work in 1928. 
In 1977, Ilya Prigogine took a Nobel prize for his work on the emergence of
order out of dissipative systems.  In brief, nonlinear feedback sustains
the structure of a complex system.  Much as two waves are able to pass
through each other in deep water, two social systems are able to co-exist
in the same eco-system provided that the feedback loops between them are
nonlinear.

Self-SimilarityIn complex system dynamics, nothing ever happens exactly
the same way twice.  Similarity replaces sameness.  Even when conditions
are precisely alike, a second iteration of a word, a tool or a work is
slightly different from the one going just before.  No one electron is ever
precisely identical to another; no atom, no molecule, no wave, no leaf, no
snowflake, no bacterium, no angleworm or human being is identical to any
other such entity; nor does it maintain precisely the same 'structure' for
long.  A day in the life of a business, a school, a church or a factory is
never precisely identical to the previous day nor will it be exactly the
same as the next day.  Every thing is change and thus everything is changed
in both the philosophy and the practice of science.  If there is sufficient
similarity across a set of iterations of a system, we can speak of
structure even if the process[es] which produces structure is loose,
fuzzy, riddled with gaps, skips from point to point and back again, and
untrackable by the closest observation.

Structure:In abstract terms, the pattern made in phase space by the
regularities made by its dynamics in the space through which it passes.  At
some point, a graph of the dynamics of a system stabilizes enough to take
the form of an attractor.  A system's dynamics, as process, turns into
'structure' for other systems if other systems in a field adjust to the
regularity/disorder of those dynamics to maintain their own integrity.  The
structure comes to have a life of its own when the response of other
systems are such that, in turn, the first system comes to depend upon
those responses.  Brian Arthur, Berkeley, calls this emerging mutual
dependency, 'lock-in.'  He lists all sorts of sub-optimal dynamics which have
become structured by such inter- and counter-dependency; clocks, cars,
gas engines, light-water reactors, the  QWERTY keyboard layout and more.

     One should note that, in both physical and abstract terms, the
dynamics of electrons constitute the structure of atoms; the dynamics of
atoms, the structure of molecules, the dynamics of molecules, the
structures of tissues and so on.