010 Theory Validation   TR Young  1998

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A 4-Dimensional Bifurcation Map


                           In a Non-Linear World
Guest Speaker: Dr. Young 		     Topic: Theory Validation

A. Multiple Dynamic Regimes.  Modern science posits one and only
one dynamic regime for any given set of variables.  Given that set,
absent any change in the settings of any given variable and absent the
work of an intervening variable, one can predict to a nicety how that
set of variables will many teenage pregnancies; how
many bankruptcies; how many murders.
     Modern science, defined as that knowledge process set forth by
Newton in his "Principia Mathematica," (1687) uses measurement,
deductive logic, prediction, replication and statistical inferences about
dispersion and concentration in the effort to build theory.   The
immediate mission of research is to discover the place and point at
which any given system might be found at any given time as a direct
result of any particular set of variables.
Postmodern philosophy of science, grounded upon the new sciences
of chaos and complexity, posit an infinite set of dynamical states. 
For our purposes, we will think in terms of five such dynamical
regimes and think about theory validation for each.  But before we do
that, a word of advice is warranted.
Deterministic vs. Non-deterministic Regimes.  The non-linear regimes
of simple chaotic systems can be very, very complex but they all
share at least three things in common.  First, they are deterministic;
that is, there are very precise points at which they transform from one
dynamical regime to another.
     This set of transformations, discovered by Mitchell Feigenbaum 
and called a bifurcation map, is very, very similar to the bifurcation
maps of all chaotic systems and thus are styled, deterministic.  So
while the precise location of any given system in non-linear regimes
cannot be found, still there is sufficient order between regimes to
ground a very important knowledge process and to serve as basis for
theory validation of the sort favored by Newton and all those who
follow his mission and methods of discovery.
     Second, they can be controlled by finding and matching the
algorithm which produces the bifurcations.  The ability to control
chaotic regimes, even in deep chaos, is a remarkable finding for
simple chaotic dynamics.  It provides both a grounding for social
policy, and as we shall see later, a new mission for affirmation
postmodern social science.
     Third, the dynamical regimes created by non-linear dynamics
have a fractal geometry.  This mean that the tight correlations so
favored by modernist scientists are not to be found in any non-linear
regime, hence cannot be used to set truth values of any given
statement about any given social behavior.  Indeed, fractal truth
values are possible but not central to the knowledge process in chaotic
regimes.  Now on to non-deterministic regimes.
B. Non-deterministic Regimes. It is by no means demonstrated
that social systems are deterministic in the sense above; indeed, it is
very unlikely that any complex social system is deterministic in the
sense given by Feigenbaum in his work on bifurcation points, above. 
This means that some of the tools of theory validation available to
some researchers may not be available to those in social science. 
Any truth claims based upon bifurcation points may not be as solid a
grounds for theory, prediction and control as now they are in physics,
chemistry, physiology and biology.
However, the key element of non-linear dynamic regimes remains as
grounding to the knowledge process in social science.  It is the fractal
structure of non-deterministic chaotic regimes.  That structure, even
though it is fractal, still contains enough order to serve as base for a
postmodern philosophy of science.
It has been demonstrated that social fractals are found in complex
data sets not observable using the tools of modernist science.  Patti
Hamilton and her students at Texas Woman's University was first to
report such structures in a study of teenage birthing patterns
(Hamilton, 1994)
C. Missions of Modern Science:  Since the time of Newton, the task
of the research scientist has been to build theory.  Theory validation
has been focussed upon the stable patterns of behavior of simple
deterministic systems.  Deterministic systems of the sort with which
Newton worked produce a very structured set of causal relationships.
     These linear dynamics are elegant and eternal....once a
researcher has found the dynamical pattern produced by linear
dynamics, s/he can build an axiomatic theory with which to predict
the behavior of all such dynamical systems.  Validation is a simple
task; imagine the variables which drive a system to a specific pattern;
measure them before and after a small change in one; test whether the
system behaves in the expected way when the variables are thus
increased or decreased in value.  If the pattern is stable, every other
researcher should find the same results...replication is, for simple
systems, a powerful validating tool.
     Since the work of Lorenz (1964) and, later, Mandelbrot (1977),
concern with the formal structure of systems has switched from
discovery of ordered behavior...and the building of a formal theory
with which to predict the behavior of such clarification of
the process by which given systems come to be behave in different
D. Missions of Postmodern Sociology:  The missions of postmodern
sociology are manifold.  First there is the venerable task of serving as
the 'sociological eye' of the country and world (Collins, 1998).  Joey
Sprague (1998) emphasizes a role which emphasizes social action and
to facilitate social policy formulation.  Marx and his intellectual heirs
call for the relentless criticism of all things existent, including
sociology itself.  This latter requires both immanent and transcendent
critique; one criticizes a society and a science both upon its own
claims and self-stated tasks as well as upon external grounds.  Finding
external grounds not contaminated by special interests remains a task
of postmodern philosophy and religion.
The most general task then becomes to unravel the changing ratios of
order/disorder.  This serves society as its 'sociological eye.'  Then
there is the twinned task of discovering:
     a.   which ratios of certainty/uncertainty are most conducive to
          the human tasks of symbolic interaction, to creativity in
          politics, economics, religion and other important social
     b.   which ratios of uncertainty/certainty trigger great increases
          in bankruptcy, crime, divorce, ethnic/racial violence as
          well as suicide and other forms of self-destructive
This second task serves as grounds for social policy in ways quite
different from a modernist sociology oriented to the quest for order or
a social policy task oriented to stability.
From Structure to Process.    As one can see, in this postmodern
specification of mission, theory building is not primal.  Yet there is
need for some set of truth claims upon which an affirmative and
empirical postmodern sociology may ground the knowledge process. 
In this lecture, I propose a sub-set of missions which presume the
relevance of non-linear social dynamics to the sociological
imagination.  We need to know:
     1.   just which dynamical regime is at work and just what kind
          of fractal is involved.  Just how loose and how tightly
          dynamical regimes behave in terms of its overall structure
          in cartesian time-space mappings.
     2.   just when one dynamical regime is about to transform into
          a more (or less) complex regime...into a more predictable
          and controllable regime or, alternatively, a much less
          predictable regime.
     3.   just what algorithm is driving the system and just what set
          of parameters are involved in that algorithm.  Fortunately,
          there are analytic techniques to discover these.  Hamilton
          and colleagues have led the way in their discovery to two
          fractals; one driven by a four part algorithm and one by a
          seven part algorithm.
     4.   just which dynamical regimes are most congenial to basic
          social values.  The nature of those values seem outside the
          purview of both modern and postmodern sociology but, I
          think, there are some few guides to be found.  I have tried
          to suggest some in prior papers on symbolic interaction,
          social change, social control and criminology.
     5.   just what algorithm stabilizes a given system at a given
          dynamical regime.  The choice of when and what to
          stabilize is, of course, a political question to be settled, I
          claim in democratic discourse.
     6.   just what kind of feedback is at hand: both negative and
          positive feedback tend to cripple a system; the first by
          extinguishing it, the second by exploding it beyond all
          human comprehension...and control.  Indeed, in non-linear
          dynamics, the concept of feedback displaces the concept of
E. Fractals and Theory Validation.  There are an infinite number of
dynamical regimes in any given bifurcation of map of non-linear
dynamics.  As I said, we will artificially cut this range down to five
most interesting regimes.  Each dynamical regime carries its own
private number of outcome basins called Attractors.  Of these
attractors, the first two serve as base for certainty about truth claims.  
They may be seen in Figure 1, below.  The first one, mapped by the
straight line, is a called a point attractor since, if one were to look at
it at any given point in time, its geometry would take the shape of a

The second regime is called a Limit attractor.  As process, it embodies a geometry with two
lines; and as structure, it has two outcome basins which may be seen in a time-frozen cross
section in Figure 1, below

Figure 1.  A Bifurcation Map showing five dynamical regimes in the transition from order to disorder.

Note that Regions A and B are very ordered; that Regions C and D have a mix of order and disorder but that order prevails and that Region E, shown in the body of the Bifurcation Map has regions of order mixed with regions of great disorder.

The white regions in E indicate new forms of order emerging from deep chaos.

These first two attractors are not found in social dynamics; all
distinctly social dynamics are too complex to give up such neat and
tidy attractors.  Yet both pre-modern social philosophy and modern
social science assume the relevance of the point attractor for all
'normal' behavior' and the limit attractor for such social forms as,
say, gender.
The Torus. All subsequent dynamical regimes are non-linear and, in
the case of social dynamics, they are, most probably, non-
deterministic as well.  The third attractor which is of great and
central interest to the postmodern researcher in social science.  It is
called a Torus since its dynamics take a roughly donut shaped
geometry.  It is found in Region A, of the Bifurcation Map in Fig. 1.
Theory validation in linear regimes is easy; Newton and Bacon gave
the formula, mentioned earlier.  The 'scientific method' set forth in
all graduate programs in the social sciences embody the method;
define a problem, make hypothesis about how changes in two or more
variables affect a system, control all other variables, gather data,
analyze the data using inferential statistics and report findings to the
scientific community.  Subsequent researchers can use these findings
to replicate the study, to improve the definitions, to improve the
research design, to improve the observational techniques, to improve
the inferential tactics and, with these new and improved techniques,
help build formal axiomatic theory.
This whole process, the scientific method, the replications and the
improvements in data gathering and data analysis constitute a larger
modernist project often called the 'method of successive
approximations' by which at the end, a superb theory is produced
with which one can know the behavior of all dynamical systems for
all time.  This later accomplishment defines a new epoch in the
knowledge process at which all is known; at which science meets 'the
end of history.'
F. Theory Validation in a Non-linear World.  History never, never
ends in postmodern philosophy of science.  This knowledge process,
accepting the non-linearity of system dynamics, assume a never-
ending and ever-changing configuration of attractors for a given
dynamical system; for a given interacting set of variables.  
Yet a scientifically grounded knowledge process is available.  Far
from complete disorder and hopeless confusion, there is enough order
in chaotic regimes to serve as both the sociological eye of a people
and the impartial advisor to the social policy of an era.
The torus is central to this knowledge process for several reasons:
     1.   It embodies First Order Uncertainty from the linearity of
          the point and limit attractor.  One knows that a system will
          be somewhere within the boundaries of the doughnut
          shaped form created by non-linear dynamics even if one
          cannot, as with the limit and point attractors, know
          precisely where the system might be.
          A case in point for the social researcher is seen when a
          given family will have a given number of children at a
          given point in time.  One cannot predict the precise
          number but one can be confident the number will be
          confined within upper and lower limits.  Or the number of
          bankruptcies or the number of murders or the incidence of
     2.   The torus is the most ordered of all non-linear regimes. 
          Indeed fact that most social research reports only in terms
          of .10, .05, or 01. levels of certainty rather than the
          .000000000000001 level of certainty found in the behavior
          of point and limit attractors is significant.  It is not
          necessarily the case that the research design is faulty or
          that the researcher is biased or that there is a mysterious
          intervening variable which if we could define and measure
          it, would give greater levels of is that all non-
          linear behavior is by definition less than certain.
          It is true that there is bad theory, poor research design,
          observer bias and all the other flaws of which nihilistic
          postmodern critics make much.  But the larger truth is
          that, in non-linear dynamics, there is a mix of order and
          disorder...the torus is the first complex structure in which
          the assumptions of modernistic, linear sociology fail.
     3.   The good news is, in terms of truth-laden scientific
          statements, that all subsequent non-linear dynamical
          regimes, no matter how complex, how chaotic, may be
          seen as a set of linked tori.
          The bad news is that, between linkages, there may be great
          disorder.  Disorder great enough to defeat the knowledge
          process; defeat the effort to make truth claims.
The Butterfly Attractor.  As I mentioned earlier, a torus is the basic
unit of non-linear dynamics.  When a torus expands to occupy more
of the social space available to it, it becomes a butterfly attractor. 
Some people call each wing of a butterfly attractor, a tori.  Some
refer to the entire outcome field as an attractor.  It is helpful to clear
communication to call each wing of any complex attractor, a tori.
The bifurcation process which transforms a single torus into a
butterfly attractor may be seen in Figure 2, just below.  In it one may note the emergence

of a 'tongue' which, if even a slight change occurs in a key parameter, will

lead to an entirely new outcome basin. That new basin is linked to the existing

one and forms what is known as a 'butterfly' attractor since it has two wings.
Figure 2.  Note the tongue which is shown in the cross-section of a torus.  Even a very small change can trigger the emergence of an entirely new outcome basin. 

It is important to note that the very same set of variables can produce two entirely different outcomes.  Modern science permits one and only one outcome basin for any given set of variables and all changes in dynamics must be proportional to change in a variable. 

Not so in non-linear social dynamics.

tongue.gif (2466 bytes)

The doubling of Torii goes on with each new bifurcation.  There may be two, four, six, eight, nine, ten, twenty...n number of Torii each with links to  adjacent tori.  In any given cartesian map of a set linked Torii, one may see the great mix of order and disorder so typical of non-linear dynamics.  If one branch of a complex attractor bifurcates and another wing does not bifurcation, the complex attractor may have an odd number of tori.

Let's now take a look at pointers on research design, truth claims and
theory validation for attractors with two or more tori.
Theory Validation in Complex Bifurcation Maps.
Theory validation becomes local and temporal in postmodern
knowledge processes.  However, reliable knowledge is still possible
even in greatly disordered, i.e., chaotic regimes.  
Several points are helpful to theory validation in complex bifurcations
maps...defined as any bifurcation may with two or more linked tori.
     1.   Beginning with the Butterfly Attractor, one meets Second
          Order Uncertainty.
     2.   Although there is a great deal of disorder in such
          attractors, still there is enough order for truth claims and,
          thus for theory...and theory validation.
     3.   Modern science research designs tend to select one Torii;
          tend to define all other linked Torii as outside the research
          frame and thus lose other, equally valid, research results.
     4.   To remedy this, one must first find out how many Torii
          there are in any given outcome field; find out where the
          linkages are and then make conclusions about the whole
          outcome field rather than just part of it.
     5.   Replication becomes a dangerous, misleading validation
          tactic since, in a truly dynamical social field, new Torii are
          arising all the time.  
     6.   Falsification is equally harmful to the knowledge process
          since, if one were to sample any other region of an
          outcome field, one would of a matter of course, find
          different outcomes.
     7.   Thus controls, also, are hostile to theory validation.  One
          must have a research design broad enough to encompass all
          the linked Torii which define that particular outcome field.
     8.   Measures of significant association remain helpful to the
          knowledge process in postmodern empirical sociology.
          However, the measures of association hold only for:
               a. for one Torii
               b. for the time period between bifurcation
     9.   As more and more Torii emerge, measures of association
          reveal ever weaker association...since they include Torii
          with conflicting results from the same set of variables.
     10.  However, weak associations should not be discarded; they
          may well indicate the approach of a new bifurcation and
          thus are most valuable to both the knowledge process and
          to social policy.
     11.  After the third bifurcation, i.e., in an outcome field with
          16 or more attractors, prediction rapidly becomes
     12.  Truth claims become local...confined to the limits of a
          given Torii/attractor.
     13.  One cannot predict behavior at bifurcation points; one can
          only predict within those fragments of the whole field here
          called tori.
G. The Fifth Attractor--Deep Chaos.  Deep chaos, defined here,
refers to any dynamical regime with more than 16 attractors...or
outcome basins if you prefer.  Deep chaos is the region of Third
Order Uncertainty.  Entirely new, surprising, unpredictive and often
very useful forms of natural and social life arise in deep chaos.  For
sociologists, theories of change must be confined to the third and
fourth regimes of our bifurcation map even though most of the more
interesting forms of social change occur in this region of non-linear
social dynamics.
     Theory validation is impossible in the fifth regime.  Truth
statements are very, very fractal as are the entire set of attractors
which present themselves as alternative ways to do gender, family,
economics, politics, crime and other social activities.  
     However, social policy and social action are not impossible.  It
is in deep chaos that non-linear sociologists are most helpful since
they can advise persons, firms, groups and whole societies how to
respond.  Some sorts of helpful advice include:
     1.   the point that deep chaos is not helpful to most social life-
          world activities.  Symbolic interaction, productive labor,
          educational routines, art, music and science itself require a
          conservative mix of order and disorder.  Outcome basins
          having 4 to 12 options are, I should think, the best mix for
          most social activity.
          A society needs disorder since it provides surprise,
          innovation, creativity and flexibility.
          A society needs order since it permits planning, goal-
          oriented behavior, dependability, co-ordination and
          intersubjective agreement.
     2.   identification the key parameter(s) which drives a given
          form of behavior into deep chaos are helpful to the
          restoration of first and second order uncertainty.
          A small change in unemployment may drive otherwise
          productive citizens into new forms of crime.  A small
          change in tax rates may produce new forms of corporate
          crime.  Multiple uncertainties for a given group may drive
          suicide, domestic violence or drug use.
     3.   Adoption of any one, two or three emergent attractors as a
          solution to the problem of third order uncertainty may be
          helpful to a community, to a people or to a nation.
          It takes wisdom, insight, intuition and just plain dumb luck
          to select the more helpful of the new forms of social
          behavior as ways to cope but the most general point to
          make is that:
          Only chaos can cope with chaos.
          One must be ready to accept new ways of doing medicine,
          school, church and state if one is to help move a society to
          that ultra-stability necessary to cope with third order
H. CONCLUSION.  There are five dynamical regimes.  The first
two regimes permit that superb theory of which Newton brought to
the knowledge process.  The third and fourth regimes; that of the
torus and the multiple linked Torii sometimes called butterfly attractors,
are the region of limited theory and short term truth statements.  The
usual tools of social science continue to be helpful here except that
one should not give preference to tight correlations and, conversely,
dismiss low correlations. 
The tasks of a researcher working in non-linear social dynamics includes:
     1.  Discovery of the parameters which produce non-linear
          patterns of behavior of complex systems.
     2.  Discovery of the point at which bifurcations in one or more
          of these parameters increase the number of outcome basins
          possible for a given set of complex systems.
     3.  Discovery of the algorithm which describes the feedback
          between parameters.
     4.   Estimating the fractal value of the geometrical structure
          produced by a given algorithm.
     Low correlations may, just may, predict the onset of deep
chaos.  The same is true for contradictory findings; they may reveal
the existence of multiple outcome basins rather than provide grounds
for dismissing research as 'inconclusive.'
     Finally, the postmodern social science must be prepared to
defend disorder in both method and theory for complex adaptive
systems.  Too much order is death to the social process; too much
disorder is death to social structure.  Chaos/complexity theory teaches
us that the best mix of order to disorder is found, most probably, in
the fourth dynamical regime of the bifurcation map above.


Note: This lecture is dedicated to the members of the Class in Theory
Validation taught by Dr. Patti Hamilton in the Winter Semester at
Texas Woman's School of Nursing.  In particular, I am indebted
those who took notes and shared them with me after this lecture.  To
Patti Hamilton, I remain indebted for all the support she has given
over the years.  Her demands that I help answer her questions push
me far beyond my own expectations and intentions.
                                                  TR Young,
                                                  Feb, 1998

Other articles in the Non-linear Social Dynamics Series at:
Postmodern Theories of Crime 

A Constitutive Theory of Justice: Architecture and Content 

Chaos and Causality in Complex Social Dynamics 

Chaos Theory and Postmodern Philosophy of Science 

Symbolic Interactional Theory and Nonlinear Dynamics 

Chaos Theory and the Knowledge Process 

Chaos Theory and Human Agency 

Chaos and Management Science 

Chaos and the Drama of Social Change 

Chaos and Crime 

Managing Chaos 

Paradigm Theory 

Reinventing Socialism 

Chaos and the Concept of Structure 

Class Structure and Non-Linear Social Dynamics

			Some Basic References

Feigenbaum, Mitchell     1978 .  Quantitative Universality for
a Class of Nonlinear transformations, in the Journal of Statistical
Physics, 19:25-52.  Cited in Gleick, p. 157.
Hamilton, Patti, Bruce West, Mona Cherri, Jim Mackey, and Paul
Fisher.  1994. Preliminary Evidence of Nonlinear Dynamics in
Births to Adolescents in Texas, 1964 to 1990.  Theoretic and
Applied Chaos in Nursing.  Summer, 1994. 1:1.
Lorenz, E.N. 1963. Deterministic Nonperiodic Flow. Journal of the
Atmospheric Sciences 20:130-141.
Mandelbrot, Benoit  1977  (rev. 1983)  The Fractal Geometry of
Nature.  New York:  Freeman.

Newton, Isaac, 1946 (1687)    Mathematical Principles of Natural