Part 1: History of Chaos TheoryBe sure to check out Author's Homepage links in Footnotes

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Part 1: History of Chaos Theory


Be sure to check out Author's Homepage links in Footnotes




Introduction




The word "chaos" might have first appeared in
Hesiod
's
Theogeny

(@700 B.C.E.) in Par
t I: "At
the beginning there was chaos, nothing but void, formless matter, infinite space." Later in
Milton's
Paradise Lost
: "In the beginning, how the heav'ns and earth rose out of

chaos". Both
Shakespeare (
Othello
) and Henry Miller (
Black Spring
) refer to chaos. In these inst
ances one
inferred that chaos was an undesirable disordered quality. Historically our vernacular
incorporated this idea of disorder into chaos; dictionaries defined chaos as turmoil, turbulence,
primordial abyss, and biblical references to Tohu and Bohu ha
d the same referential character of
undesired randomness. Scientifically, Chaos implied the existence of the undesirable
randomness, but the self
-
organization concept at the edge of chaos denoted the order we get out
of chaos. The American essayist and his
torian
Henry Adams

(1858
-
1918) expressed the
scientific meaning of "chaos" succinctly: "Chaos often breeds life, when order breeds habit".
(1)



Li and Yorke
(2)

coined the word chaos to refer to the mathematical problem in chaos theory that
described a time evolution with sensitive dependence on initial conditions. Robert May
, a
mathematician
-
biologist whose research was well read, used the word and the theory from Li and
Yorke's paper, thus making them and the word famous. Chaos theory came in the back door, so
to speak, of the researcher's world. It was not a law like thermo
dynamics or quantum physics, but
it did enable the researcher to analyze events or areas with many problematic intricacies. Cambel
reported that it had even been proposed that we call chaos "divinamics"
(3)


after the ancient
Roman divinatio described by
Cicero
. (Because of the ubiquity of chaos found in nature, and
because my research is in the area of religion, I would certainly go
along with that name.)

Ilya Prigogine
,

the 1977 Nobel Prize winner in chemistry, pioneered the work in entropy of
open systems; this was the inflow and outflow of matter, energy, or information between

the
system and its environment. Prigogine used dissipative systems to show that more complex
structures can evolve from simpler ones, or order coming out of chaos.


What is Chaos/Complexity theory? Daniel Stein, in the Preface to the first volume of lect
ures
given at the 1988 Complex Systems Summer School for the Sante Fe Institute in New Mexico,
compares Chaos/Complexity to a "theological concept", because lots of people talked about it
but no one knew what it really was.
(4)


Several explanations for Chaos theory called for the
words synthesis, cross
-
discipline, edge of chaos, dynamical,
cellular automata
, or neural
networks, but all carry w
ith them the concept of complex systems. The implications of Chaos are
profound, for who could know the absolute conditions of any system for a complete prediction to
be made of the behavior of that system?

HOW IT ALL STARTED


For thousands of years human
s have noted that small causes could have large effects and that it
was hard to predict anything for certain. What had caused a stir among scientists was that in
some systems small changes of initial conditions could lead to predictions so different that
p
rediction itself becomes useless. At the end of the 19th century, French mathematician,
Jacques
Hadamard

proved a theorem on the sensitive dependence on initial conditions about the
fr
ictionless motion of a point on a surface or the geodesic flow on a surface of negative
curvature. All this was about billiard balls and why you can't predict what three of them will do
when they careened off each other on the table. French physicist
Pierre Duhem

understood the
significance of Hadamard’s theorem. He published a paper in 1906 that made it quite plain that
prediction was "forever unusable" because of the necessarily
present uncertain initial conditions
in Hadamard's theorem. These papers went unnoticed or rather unnoted by the man who was
recognized as the Father of Chaos theory,
Henri Poincare

(1854
-
1912).


In 1908 he published
SCIENCE ET METHODE

(5)
, that contained one sentence concerning the
idea of chance being the determining factor in dynamic systems because of some factor in the
beginning that we didn't know about. All three of these men and their ideas wen
t unnoted
because quantum mechanics had disrupted the whole physics world of ideas; and because there
were no tools such as ergodic theorems
(6)

about the mathematics of measure; and because there
were
no computers to simulate what these theorems prove.


In 1846, the planet Neptune was discovered, causing quite a celebration in the classical
Newtonian mechanical world, this revelation had been predicted from the observation of small
deviations in the or
bit of Uranus. Something unexpected happened in 1889, though, when King
Oscar II of Norway offered a prize for the solution to the problem of whether the solar system
was stable.
Henri Poincaré

submitted his solution and won the prize, but a colleague happened to
discover an error in the calculations. Poincaré was given six months to rectify the matter in order
to keep his prize. In consternation, Poincaré found there was no so
lution
.(7)


Poincaré had found
results that upset the accepted view of a purely deterministic universe that had reigned since
Sir
Isaac Newton

li
ned out linear mathematics. In his 1890 paper, he showed that Newton's laws did
not provide a solution to the "three
-
body problem", in other words, how one deals with
predictions about the earth, moon and sun. He had found that small differences in the ini
tial
conditions produce very great ones in the final phenomena, and the situation defied prediction.
Poincaré's discoveries were dismissed in lieu of Newton's linear model, one was to just ignore
the small changes that cropped up. The three
-
body problem wa
s what Poincare had to interpret
with a two
-
body system of mathematics. Why was it a problem? He was trying to discover order
in a system where none could be discerned.


Poincaré’s negative answer caused positive consequences in the creation of chaos theo
ry. About
eighty years later, as early as 1963,
Edward
Lorenz
8
, using Poincaré’s mathematics, described
a simple mathematical model of a weather s
ystem that was made up of three linked nonlinear
differential equations that showed rates of change in temperature and wind speed. Some
surprising results showed complex behavior from supposedly simple equations; also, the
behavior of the system of equatio
ns was
sensitively dependent on the initial conditions

of the
mathematical model. He spelled out the implications of his discovery, saying it implied that if
there were any errors in observing the initial state of the system, and this is inevitable in any
real
system, prediction as to a future state of the system was impossible.
(9)


Lorenz labeled these
systems that exhibited sensitive dependence on initial conditions as having the

"butterfly effect":

this unique name came from the proposition that a butterfly flapping its wings in Hong Kong can
effect the course of a tornado in Texas.


During 1970
-
71, interest in turbulence, strange

attractors and sensitive dependence on initial
conditions arose in the world of physics.
(10)


E. N. Lorenz published a paper, called
"Deterministic nonperiodic flow" in 1963 that proved that meteorol
ogists could not predict the
weather.
Jim Yorke
, an applied mathematician from the University of Maryland was the first to
use the name Chaos, but actually it was not even a chaos situation, but the name cau
ght on.
(11)



A chaotic system is sensitive to initial conditions and causes the system to become unstable.
Cambel identifies chaos as inherent in both the complexity in nature and the complexity in
k
nowledge. The nature side of chaos entails all the physical sciences. The knowledge side of
chaos deals with the human sciences. Chaos may manifest itself in either form or function or in
both. Chaos studies the interdependence of things in a far
-
from
-
equi
librium state. Every open
nonlinear dissipative
(12)

system has some relationship to another open system and their
operations will intersect, overlap and converge. If the systems are sensitive to the i
nitial
conditions, in other words, you don’t know exactly in detail every little piece of information, and
then you have a potentially chaotic system. Not all systems will be chaotic, but those where a
lack of infinite detail is unknown, then these systems

have an indeterminate quality about them.
You can’t tell what’s going to happen next. They are unpredictable. If these systems are
perturbed either internally or externally, they will display chaotic behavior and this behavior will
be amplified microscopi
cally and macroscopically.


Further research in non
-
linear dynamical systems
(13)

that displayed a
sensitive dependence on
initial conditions
came from
Ilya Prigogine
, a Nobel
-
prize winning chemist, w
ho first began
work with far
-
from
-
equilibrium systems in thermodynamic research.
(14)


Ilya Prigogines'
research in non
-
linear dissipative structures led to the concept of equilibrium and far
-
from
equi
librium to categorize the
state

of a system. In the physical studies of thermodynamics,
Prigogines' research revealed far
-
from
-
equilibrium conditions that led to systemic behavior
different from what was expected by the customary interpretation of the
Second Law of
Thermodynamics.

Phenomena of bifurcation and self
-
organization emerged from systems in
equilibrium if there was disruption or interference. This disruption or interference became the
next step to
Chaos Theory; it became Chaos/Complexity Theory. Prigogine talked about his
theory as if he were
Aristotle
: a far
-
from
-
equilibrium system can go ‘from being to
becoming’.
(15)


These ‘becoming’ phenomena showed order coming out of chaos in heat
systems, chemical systems, and living systems.


From Lorenz simulation, René Thom, mathematician, proposed ‘catastrophe theory’, or a
mat
hematical description of how a chaos system bifurcates or branches. Out of these bifurcations
came pattern, coherence, stable dynamic structures, networks, coupling, synchronization and
synergy. From the study of complex adaptive systems used by Poincaré,
Lorenz and Prigogine,
Norman Packard and Chris Langton developed theories about the ‘edge of chaos’ in their
research with cellular automata.
(16)


The energy flowing through the system, and the
fluctu
ations, cause endless change which may either dampen or amplify the effects. In a phase
transition of chaotic flux, (when a system changes from one state to another), it may completely
reorganize the whole system in an unpredictable manner.
(17)


Two scientists, physicist Mitchell Feigenbaum
(18)

and computer scientist Oscar

Lanford
(19)

came up with a picture of chaos in hydrodynamics using Renormalization ideas. They were
studying non
-
linear systems and their transformations
.(20)


Sin
ce then, chaos theory or
Nonlinear Science has taken the scientific world by a storm, with papers coming in from all
fields of science and the humanities.
Strange attracto
rs

were showing up in biology, statistics,
psychology and economics and in every field of endeavor.

Properties of complexity






Complexity or the edge of chaos yielded self
-
organizing, self
-
maintaining dynamic structure that
occurred spontaneously in a

far
-
from
-
equilibrium system. Complexity had no agreed upon
definition, but it could manifest itself in our everyday lives. Intense work is being done on the
implications of complexity at the
Santa Fe Institute

in Ne
w Mexico. Here Ph. D.’s from many
fields use cross
-
disciplinary methods to show how complexity in one area might link to another.
Erwin Laszlo, from the Vienna International Academy, has the most interesting statement about
Complexity:

In fact, of all the

terms that form the lingua franca of chaos theory and the general
theory of systems, bifurcation may turn out to be the most important, first because
it aptly describes the single most important kind of experience shared by nearly
all people in today’s wo
rld, and second because it accurately describes the single
most decisive event shaping the future of contemporary societies.
(21)

Bifurcation once meant splitting into two or more forks. In chaos theor
y it means: When a
complex dynamical chaotic system becomes unstable in its environment because of
perturbations, disturbances or ‘stress’, an
attractor

draws the trajectories of the stress, and at the
point of phase transition, the system bifurcates and i
t is propelled either to a new order through
self
-
organization or to disintegration.


The phase transition of a system at the edge of chaos began with the studies of John Von
Neumann
(22)

and Steve Wo
lfram
(23)

in their research on cellular automata.
(24)

Their research
revealed the edge of chaos was the place where the parallel processing of t
he whole system was
maximized. The system performed at its greatest potential and was able to carry out the most
complex computations. At the bifurcation stage, the system was in a virtual area
(25)

wh
ere
choices are made
--
the system could choose whatever attractor was most compelling, could jump
from one attractor to another
--
but it was here at this stage that forward futuristic choices were
made: this was deep chaos. The system self
-
organized itself t
o a higher level of complexity or it
disintegrated. The phase transition stage may be called the
transeunt

stage, the place where
transitory events happen. Transeunt is a philosophical term meaning that there is an effect on the
system as a whole produced
from the inside of the system having a transitory effect; and, a
scientific term in that it is a nonperiodic signal of sudden pulse or impulse.


After the bifurcation, the system may settle into a new dynamic regime of a set of more complex
and chaotic at
tractors, thus becoming an even more complex system that it was initially. Three
kinds of bifurcations happen:
1
. Subtle, the transition is smooth.
2
. Catastrophic, the transition is
abrupt and the result of excessive perturbation.
3
. Explosive, the transi
tion is sudden and has
discontinuous factors that wrench the system out of one order and into another.
(26)


Per Bak
(27)
,
with his co
-
researchers

Chao Tang and
Kurt Wiesenfeld

reckons nature abiding on the edge of
chaos or what they call ‘self
-
organized criticality’
.


Our daily encounter with Chaos/Complexity is seen i
n traffic flow, weather changes, population
dynamics, organizational behavior, shifts in public opinion, urban development and decay,
cardiological arrhythmias, epidemics. It might be found in the operation of the communications
and computer technologies o
n which we rely, the combustion processes in our automobiles, cell
differentiation, immunology, decision making, the fracture structures, and turbulence.

Here are a few of the statements that Cambel makes about the ubiquity of chaos:

1. Complexity can oc
cur in natural and man
-
made systems, as well as in social structures and
human beings.

2. Complex dynamical systems may be very large or very small, indeed, in some complex
systems, large and small components live cooperatively.

3. The system is neither
completely deterministic nor completely random, and exhibits both
characteristics.

4. The causes and effects of the events that the system experiences are not proportional.

5. The different parts of complex systems are linked and affect one another in a
synergistic
manner.

6. There is positive and negative feedback.


The level of complexity depends on the character of
the system, its environment, and the nature of the interactions between
them.28


W
HERE ITS ALL GOING


If we lived in a completely deterministic world there would be no surprises and no decision
making because an event would be caused by certain conditions that could lead to no other
outcome. Nor could we consider living in a completely
random world for there would be, as
Cambel says, "no rational way of reaching a well
-
reasoned decision".
(29)


What kind of answers
do we get when we recognize that a system is indeed unstable and tha
t it is indeed an example of
chaos at work. The American Association for the
Advancement of Science
published nineteen
papers presented at their 1989 meeting that was devoted entirely to chaos theory usage on s
uch
ideas as chaos in dynamical systems, biological systems, turbulence, quantized systems, global
affairs, economics, the arms race, and celestial systems. Stambler
(30)

reported that the Electric
Pow
er Research Institute was considering the applications of chaos control in voltage collapses,
electromechanical oscillations, and unpredictable behavior in electric grids. Peng, Petrov and
Showalter
(31
)

were studying the usefulness of chaos control in chemical processing and
combustion. Ott, Grebogi, and Yorke cited the many purposes of chaos and said it might even be
necessary in higher life forms for brain functioning. Freeman studied just such brain

functions
related to the olfactory system and concluded that indeed chaos "affords an opportunity to
exploit further these manifestations of brain activities"
(32).



Not only are research papers prol
ific, but an array of books are being published monthly on
chaos applications. Bergé, Pomeau, and Vidal assert that chaos theory has "great predictive
power"
(33)

that allows an understanding of the ov
erall behavior of a system. Kauffman
(34)

uses
the self
-
organization end of chaos to assert that nature itself is spontaneous; Cramer claimed that
by overcoming the objections to mysticism and scientis
m
(35)
, that the "theory of fundamental
complexity is valid" (this will most likely turn into a book
--
so many researchers refer to it). This
perhaps gives some idea as to far reaching applications of c
haos theory in the scientific areas.


A few last words about the edge of chaos will be added here because they will allow you to see
how research has gone from linear science to nonlinear applications. Wentworth d'Arcy
Thompson, in his book
On Growth and
Form
(
36)
,
used transformations of coordinates to
compare species of animals. Comparing one form of a fish, as an example, with another could be
shown on a coordinate map and used to show how they differ and how they were alike. The same
kind of transformation coordinate map could compare chimpanzee skulls to human skulls. Where
Thompson used order to compare the workings of nature, Stuart Kauffman
, in his book
The
Origins of Order: Self
-
Organization and Selection in Evolution
,
(37)
took the next step in
studying nature. He was seeking the origins of order in complex systems that were chaotic. Hi
s
research is rife with examples of the interconnectedness of selection and self
-
organization. The
essence of his findings are that much of the order seen in organisms stems from spontaneous
generation from systems operating at the edge of chaos, or in oth
er words, systems that are
unstable purposely. Thompson applied physics to biology, and now Kauffman is applying chaos
/complexity theory to biology. Cramer sees the interaction of order and disorder as a necessity in
nature. "In nature, then, forms are no
t independent and arbitrary, they are interrelated in a regular
way...And even organs arising to serve new functions develop according to the principle of
transformation. At the branch points where something new emerges, disruptions of order are in
fact ne
cessary; abrupt phase changes occur. Indeed, the interplay of order and chaos constitutes
the creative potential of nature."
(38)



The great French mathematician Henri Poincaré first noticed the idea
that many simple nonlinear
deterministic systems can behave in an apparently unpredictable and chaotic manner. Other early
pioneering work in the field of chaotic dynamics were found in the mathematical literature by
such luminaries as Birkhoff, Cartwright
, Littlewood, Levinson, Smale, and Kolmogorov and his
students, among others. In spite of this, the importance of chaos was not fully appreciated until
the widespread availability of digital computers for numerical simulations and the demonstration
of chao
s in various physical systems. This realization has had broad implications for many fields
of science, and it has been only within the past decade or so that the field has undergone
explosive growth. The ideas of chaos have been very fruitful in such diver
se disciplines as
biology, economics, chemistry, engineering, fluid mechanics, physics, just to name a few. As
you can see, Chaos Complexity theory can become a real research tool for many fields.
Metaphorically it can be used outside the scientific field.

This author plans to apply this theory to
religious research.


Part 2: Order and Instability in Chaos




ORDER


There's order in music: click the symbol below.



Finding the order of something is necessary for scientists, historians, artists, waitresses,
musicians, theologists, cooks, and Daddys putting together Christmas toys. Ordering can be done
mathematically or wit
h pictures. Order in a straight line is easily understood for it can be
constructed by just a series of equal segments; here order is defined by a single, similar
difference. To find the order in curves you need to know what the starting point is and the
c
ommon difference in successive line segments. Here again, order is defined by a single similar
difference. By noting the similar differences between successive segments of a curve or other
geometric figure, you can determine their order.
(1)


Order is also seen in randomness, as
Bohm

and Peat explain:

...whatever happens must take place in some order so that the notion of a ‘total
lack of
order’ has no real meaning. Indeed, even what are called random events do
happen to take place in a definable and describable sequence and can be
distinguished from other random events. In this elementary sense they obviously
have an order.
(2)

Order in language, art, music, games, architecture, social structures, and rituals is very subtle
because it is context dependent
--
the participator must understand all its complexities for a
meaningful and satisfy
ing appreciation of it. Order in nature, inanimate objects and physical
systems also have an infinite, but subtle order. Flowing water can have a smooth flow in
unobstructed areas, or complex eddies and whirlpools can develop when obstructions are there;
e
ven
chaotic order

can erupt in extreme agitation. Randomness can result also, but only when it
is "understood as the result of the action of the very small elements in an overall context that is
set by the boundaries and the initial agitation of the water.
"(3)


This is where Chaos theory fits
into the idea of order, the flowing water is a dynamic system that uses
non
-
linear systems
theory
.


The arena of a system is called the state space or the phase s
pace. Mathematically this phase
space would be the "space where each dimension corresponds to one variable of the system.
Thus, every point in state space represents a full description of the system in one of its possible
states, and the evolution of the s
ystem manifests itself as the tracing out of a path, or trajectory,
in state space."
(4)


When you investigate the behavior area, or phase space, in a dynamical
system, there might be tiny perturbations

or disturbances external to the system that can cause
the whole system to change.
(5)


Sally Goerner gives a good definition of non
-
linearity as ‘any
system in which input is not proportional to output
’.
(6)


Non
-
linear systems in chaos theory
display aberrant, illogical behavior; they can give either positive or negative feedback; they can
produce stability or instability; they can produce coherence

through convergence, coupling or
entrainment, or produce divergence or even explosion. For chaos to happen, you have to have a
system that is
sensitive to the initial conditions

and that is interdependent with its environment.
What seems obvious when one
begins to look at non
-
linear systems, is that they look like what is
going on around us in the everyday world.


INSTABILITY


Listen to the sense of perturbation this music evokes. Click icon.



Chaos/Complexity involves dynamics, or what Lorenz called "far
-
from
-
equilibrium" states. The
word equilibrium may remind you of a tranquil lakeside scene; a state of rest is o
ne of its
definitions, but it also entails the idea of balance. For complex dynamical systems, equilibrium is
a rarity, or as Çambel calls it "a temporary weigh station".
(7)


For a dynamical process to

take
place, the system deviates from equilibrium. Prigogine and Stengers tell us that the more
complex a system is, the more numerous are the perturbations, disturbances, or fluctuations that
threaten its stability
.(8)


As the system becomes more vulnerable to these disturbances, its energy
requirements escalate as it tries to maintain coherence. Instability can occur in all kinds of
structures from solids to gases, from animate to inanimate, from orga
nic to inorganic, and from
constitution to institution. External and internal disturbances can cause stable systems to become
unstable, but this instability does not necessarily happen from just some ordinary perturbation.
Çambel says it depends upon the "
type and magnitude of the perturbation as well as the
susceptibility of the system"
(9)


that must be considered before the system is rendered unstable.
He adds that sometimes it takes more than one kin
d of disturbance for the system to transform
into an unstable state. Prigogine and Stengers speak of the "competition between stabilization
through communication and instability through fluctuations. The outcome of that competition
determines the threshold

of stability."
(10)


In other words, the conditions must be ripe for
upheaval to take place. We could reckon this to many observable situations in areas such as
disease, political unrest, family and c
ommunity dysfunction. Cambel used the old adage that it
might be the straw that broke the camel’s back that finally allows the system to go haywire.


Stephen Kellert says, "Chaos theory investigates a system by asking about the general character
of its lo
ng
-
term behavior."
(11)


Chaotic solutions seek a qualitative account of the behavior of a
system at some future time. Quantitative closed solutions might tell you
when

three elliptically
orbiting plan
ets will line up. Qualitative solutions will tell you
how

the elliptical orbits may have
formed as opposed to circular or parabolic orbits. What will be the characteristics of all solutions
of this system? How does the system change behavior? A system like

a marble at the bottom of a
bowl can be jostled and will exhibit some behavioral antics but will eventually settle down to the
bottom of the bowl. A system like a watch will stop momentarily if given a jar but will continue
ticking reliably soon after. Th
ese systems are said to be "stable". Unstable or aperiodic systems
are unable to resist small disturbances and will display complex behavior making prediction
impossible and measurements will appear random. Human history is an excellent example of
aperiodi
c behavior. Civilization may appear to rise and fall, but things never happen in the same
way. Small events or single personalities may change the world around them.
(12)


Kellert goes
on to say, "The
standard examples of unstable aperiodic behavior have always involved huge
conglomerations of interacting units. The systems may be composed of competing human agents
or colliding gas molecules".
(13)


An unstable aperiodic system is deterministic because it is
usually composed of less than five variables in a differential equation, and because " the
equations make no explicit reference to chance mechanisms."
(14)



In far
-
from
-
equilibrium complex systems changes can frequently occur that upset the fine
-
tuning
between the internal forces structuring the system and the external forces that make up their
environment. Most of the time, the fine tuning a
llows the system to operate smoothly, but when
the perturbations escalate and the system is "stressed" beyond certain threshold limits, subtle
indications of unrest crop up, sometimes sudden non
-
linear "chaos" takes place. The subtle forms
of chaos begin b
y aberrant behavior. The shift to an attractor or the swing from one attractor to
another causes the system to behave differently. What can be done when so many problems
come upon us? Uri Merry says a human system will have to give "more awareness, attenti
on, and
care ...to maintaining its internal ties and communication networks.
(15)


Human decision making
has the unmistakable imprint of chaos on it. There are always so many assumptions and
implicatio
ns to consider that it sometimes is quite overwhelming. It is here that a strange
attractor aids in the decision making process. The strange attractor might take the form of one’s
belief system. This has been considered by this writer in her research.



Pa
rt 3: Strange Attractor in Chaos Theory



Listen to the odd reoccurring sound.


Click the icon.


To define an attractor
is not simple. Tsonis gives the definition of attractors as "a limit set that
collects trajectories".
(1)



A strange attractor is simply the pattern of the pathway, in visual form,
produced by graphing

the behavior of a system. Since many, if not most, nonlinear systems are
unpredictable and yet patterned, it is called
strange

and since it tends to produce a fractal
geometric shape, it is said to be attracted to that shape. A system confines a particula
r entity and
its related objects or processes to an imaginary or real frame as the subject of study, this is its
"state space" or phase space. The behavior in this state space tends to contract in certain areas,
this contraction is called "the attractor".
The attractor is actually "a set of points such that all
trajectories nearby converge to it". Now tell me what an attractor is. You can't and neither can I,
even with Tsonis's definition. Scientists, mathematicians, and computer specialists can show you
pi
ctures of how they operate, but they cannot tell you what they are. Maybe, that is why Daniel
Stein, compares Chaos/Complexity to a "theological concept", because lots of people talk about
it but no one knows what it really is.
(2)


(Found in the Preface to the first volume of lectures
given at the 1988 Complex Systems Summer School for the Santa Fe Institute in New Mexico)


Several researchers have defined and studied strange attractors. The first was
Lorenz in
"Deterministic nonperiodic flow" in 1963
(3)
, and Later Ruelle in "Sensitive dependence on
initial condition and turbulent behavior of dynamical systems" in 1979
(4)

When computer
simulation came along, the first fractal shape identified took the form of a butterfly; it arose from
graphing the changes in weather systems modeled by Lorenz. Lorenz's attractor shows just how
and why weather p
rognostication is so involved and notoriously wrong because of the butterfly
effect. This amusing name reflects the possibility that a "butterfly in the Amazon might, in
principle, ultimately alter the weather in Kansas."
(5)



For an in depth story of how this butterfly
effect developed into the science of Chaos and Complexity, see James Gleick's
CHAOS

and
Mitchell Waldrop's
COMPLEXITY
(6).
Kauffman e
xplains that the tiny differences in initial
conditions make "vast differences in the subsequent behavior of the system"
(7)


as Lorenz
illustrated in his weather prognosticator.


Chaos is centered on
the concept of the
strange attractor.
Watch the flow of water from you
faucet as you turn the water on to give faster and faster outpour; you will see activity from
smooth delivery to gushing states. These various kinds of flow represent different patterns

to
which the flow is attracted. The feedback process is the feedback displayed in most natural
systems in nature. There are
four
basic kinds of feedback or cycles a system can display: these
are the
attractors.
(
8)
Tsonis gives the definition of attractors as "a limit set that collects
trajectories"
(9)

The four kinds of attractors
(10)

are:


1. Point attractor, such as a pendulum swinging back and forth and eventually stopping at a
point. The Attractor may come as a point, in which case, it gives a steady state where no change
is made.

2. Pe
riodic attractor, just add a mainspring to the pendulum to compensate for friction and the
pendulum now has a limited cycle in its phase space. The periodic attractor portrays processes
that repeat themselves.

3. Torus attractor, picture walking on a larg
e doughnut, going over, under and around its outside
surface area, circling, but never repeating exactly the same path you went before.
(11)


The torus
attractor depicts processes that stay in a confin
ed area but wander from place to place in that
area. (These first three attractors are not associated with Chaos theory because they are fixed
attractors.
(12)


4.
Strange attractor
, this attractor dea
ls with the three
-
body problem of stability. The strange
attractor shows processes that are stable, confined and yet never do the same thing twice.

Three non
-
linear equation solutions exhibit a fractal structure in computer simulations of the
strange attr
actor.
(13)


In other words, each solution curve tended to the same area, the attractor
area, and cycled around randomly without any particular set number of times, never crossing
itself, staying in th
e same phase space, and displaying self
-
similarity at any scale.
(14)


The
operative term here is self
-
similarity. Each event, each process, each period, each end
-
state in
phase
-
space is never precise
ly identical to another; it is similar but not identical.The attractor
acts on the system as a whole and collects the trajectories of perturbation in the environment.
(These trajectories of perturbation are the positive and negative events going on in and
around
the system.) Though these systems are unstable, they have patterned order and boundary.

FRACTALS




French mathematician
Georges Julia

studied these chaotic orbits in complex analyt
ical systems
back in the 1920s, but Benoit Mandelbrot, in the early 1970s, gave some rules for computation.
His work on noise interference problems revealed distinct ratios between order and disorder on
any scale he used. The seemingly chaotic behavior of
noise displayed a fractal structure.
(15)

Mandelbrot recognized a self
-
similar pattern that the fractals formed. He then cross
-
linked this
new geometrical idea with hundreds of examples, from cotton pr
ices to the regularity of the
flooding of the Nile River.

Mitchell Feigenbaum found the constants or ratios that are responsible for the
phase transition

state when order turns to chaos. These Feigenbaum numbers helped to predict the onset of
turbulence (
chaos) in systems: applications in the real world began. Optics, economics,
electronics, chemistry, biology, and psychology quickly used this new analytic tool. Fractal
geometry is now being used to graphically show change and evolution in technology, soci
ology,
economics, psychotherapy, medicine, psychology, astronomy, evolutionary theory, and the
metaphorical application is spreading to art, humanities, philosophy and theology.
(16)






METAPHORICAL
APPLICATIONS

Kauffman explains that the tiny differences in initial conditions make "vast differences in the
subsequent behavior of the system."
(17)


Unstable or aperiodic systems are unable to resist

small
disturbances and will display complex behavior, making prediction impossible and
measurements will appear random. Human history is an excellent example of aperiodic behavior.
Civilization may appear to rise and fall, but things never happen in the s
ame way. Small events
or single personalities may change the world around them.
(18)


The symbolic use of chaos to delineate the interactions of a system and its
environment can be more enlightening wi
th chaos theory as the tool, especially
when explicating an historical personage or situation. Here the human being or set
of human beings creates a pattern in the time
-
space, this pattern is the basin of
attraction within which the attractor or multiple a
ttractors form. The Newtonian
paradigm of linear mechanics does not reveal all the ramifications that effect the
event or person. Newtonian expectations proposed smooth transformations that
can be plotted by linear actions or reactions; but chaos/complexit
y will allow the
researcher to see the symbolic interaction of the person or event with their
environment.








Part 4: PHASE TRANSITION


Listen to how the music bifurcates.

The edge of chaos seems to be the phase transition state of the system or the p
lace where choices
are made and take place. There has always been turbulence in the universe, it has been
recognized in the scientific world since Poincare and Lorenz researched the motions of the
atmosphere and their relevance to weather prediction. David

Ruelle and
Floris Takens

opened up
a new way to look at turbulence in their paper "On the nature of turbulence."
(1)


Most of the time
if turbu
lence showed up in an experiment, it was ignored, accounted for by factoring it out, or
declared a failed experiment. But, Ruelle and Takens used ideas of
Rene Thom

and
Steve Smale
,
who were mathematicians working with "differentiable dynamical systems."
(2)


They proved
that not only could the onset of turbu
lence be mathematically formulated by use of nonlinear
equations, but also it showed that turbulence was directly related to the sensitive dependence on
initial conditions, and that the turbulence was described by strange attractors. Chaos theory
developed

from open dynamical systems with a time evolution with sensitive dependence on
initial conditions.
(3)


It has also been called
deterministic noise,

which describes irregular
oscillations that appear n
oisy, but the mechanism that produces them is deterministic.
Mitchell
Feigenbaum

proved that there was a mathematical relationship in all open, dynamical
systems.
(4)


This relationship became the universal number ratios now called Feigenbaum
numbers. Use of these Feigenbaum numbers, continued development in studies of nonlinear
mathematics, experiments with chaotic systems an
d now the discovery of
strange attractors

took
chaos theory to a new development horizon.


Phase transition studies came about from the work begun by John von Neumann
(5)
and carried
on by Steven Wolfr
am
(6)

in their research of cellular automata. M. Mitchell Waldrop in his book
Complexity
: The Emerging Science at the Edge of Order and Chaos

gives a lively account of the
discoveries made at the begin
ning of the research with cellular automata.
(7)


They found there
were two kinds of phase transitions first order and second order. The First order we are familiar
with when ice melts to water
--
(molecu
les are forced by a rise in temperature to choose between
order and chaos right at 32° F, this is a deterministic choice.) Second order phase transitions
combine chaos and order, there is a balance of ordered structures that fill up the phase space in a
so
rt of "dance of submicroscopic arms and fractalfilaments."
(8)



In the material world, phase transitions are so intimately intertwined at the molecular level that
there is no way of predicting what sta
te a system will take. Also the real astonishing discovery
was that at the edge of chaos, not only would you encounter complexity at its most mysterious,
but maybe life itself. Complex adaptive systems, like individuals, families, organizations, and
nation
s, "are able to survive and adapt more effectively in turbulent environments, when they are
functioning in a mode that is described as ‘the edge of chaos’".
(9)


Stuart Kauffman, theoretical
biologist,
is involved with phase transition at the Santa Fe Institute. His studies show that
dynamical systems are at their optimum fitness at phase transition states. These systems seem to
reach the boundary between order and chaos by themselves and adapt to that s
tate of transition at
peak fitness. Kauffman’s studies, in his book
The Origins of Order: Self
-
Organization and
selection in Evolution
, reveal that complex systems carry out and coordinate the most complex
behavior, adapt most readily and can build the mos
t useful models of their environment.


Kauffman and Christopher Langdon speak of the edge of chaos as the place where systems are at
their optimum performance potential.
(10)


This edge of chaos seems

to be the phase transition
state of the system or the place where choices are made and bifurcations take place. It is the time
and place when there are many options, many positive and negative influences from these
options, and a time of great mental turm
oil if the system is a human being. The
strange attractor

boxes behavior into a small, easily handled package, allows coherence to the many positive and
negative influences, and self
-
organizes the system into something new without causing any
damage to cas
cade throughout the system.
(11)


When a system is operating on the border of
chaos, a self
-
organized critical state produces a weak form of chaos that will allow long term
predictions to be made about

the system. Per Bak explained this in relation to his earthquake
model.
(12)



Per Bak describes the perturbances related to chaos as ‘self
-
organized criticality’
(13)


Bak and
his co
-
researchers use the metaphor of a sand pile with someone steadily dribbling new grains of
sand onto it. Because the whole system of the sand pile is so interwoven and interlocked, the pile
grows higher until at some
point either a large avalanche or a small cascade will happen. They
assert that this ‘power law’ (the average frequency of a given size of avalanche is inversely
proportional to some power of its size) is common in nature. Examples are critical mass of
plu
tonium, activity of the sun, light from galaxies, flow of water through a river, and
earthquakes. This vivid metaphor allows one to see how disturbances from the outside can take a
system to the edge of chaos and then cascade into a new order. The small ch
anges and large
upheavals, or cascades and avalanches in the case of the sand pile, are the signaling methods that
a system is operating at the edge of chaos.

What you have at the edge of chaos is a sublime balance, between stability and instability. This

sublimely balanced area is the place where creativity evinces itself, the place where decisions are
roughly wrought, and the place where mental turmoil is at its most torturous. We all have heard
the stories of great thinkers, writers, and artists who liv
e on the edge, so to speak, as they retch
forth their
magnum opi
; the stories of their near and real insanities are the lure to millions of
readers. It is this poised state between stability and instability that propagates the perturbations,
that allows so
me parts of the sand pile to remain the same, while other parts are changing. This
poised state is a system far
-
from
-
equilibrium.

PART 5: DEEP CHAOS


Can you hear the sense of impending doom that this Mars.midi evokes?


Click on the Morph




Deep chaos is the fractal dimension where patterns of self
-
similarity reveal themselves in
descending scales of order. Uri Merry likens

them to "a set of wooden Russian dolls, each
containing a smaller replica of itself within."
(1)


This complexity can occur in natural and
man
-
made systems, as well as in social structure; therefore be
cause it is so ubiquitous to
nature, it has no agreed
-
upon definition. Çambel describes complex systems from 15
categories, but to be succinct, complex systems have size, purpose, and are dynamic.
(2)



Cramer
(3)

gives his definition in the form of a logarithm taken from information theory
which in essence means that "the more complex a system, the more information it is
capable of carrying.
(4)


In deep chaos, there is a displacement of being, the chthonic realm
of turmoil; it is the dimension between states. It is here in the deep chaotic state that the
system becomes complex, and hence the ter
m Complexity enters in. Kauffman and
Christopher Langdon speak of the edge of chaos as the place where systems are at their
optimum performance potential.
(5)


When the constraints on a system are suffi
ciently strong, (many positive and negative
perturbations), the system can adjust to its environment in several different ways. There
may be several solutions possible from the whole basin of attraction, and chance alone
cannot decide which of these soluti
ons will be realized. It is the
attractor
that will help
determine the solution. The fact that one solution among many does occur gives the system
a historical dimension, a sort of memory of a past event that took place at a critical moment
and which will
affect its further evolution. This is the phase transition of the system, the
place where the system is isotropic; it has no preferred direction to go in, it is an either/or
decision, the past old ways or the future new ways. You might visualize the phase
transition
as the coin tossed into the air; while it is in the air there is only probability, no actual choice
has been made until it lands. There is no observance of transilience (leaping from one state
to another) in the system, but there is a phase tran
sition that takes place at the edge of
chaos before an actual self
-
organization into another state.



There might be some similarity of phase transition to the sense of NOW. The sense of NOW
worries all philosophers and theologians and even scientists. Wha
t is NOW?
Karl Popper

reckons NOW as a single frame in a filmstrip
--
the future and past are all known within the
whole of the strip.
(6)


Einste
in worried about NOW as a physics question
--
NOW was
special for humans, but did not have a meaning in physics.
(7)


The NOW in chaos theory is
the phase transition state where all choices are open. Pau
l Tillich
(8)


spoke eloquently
about living one’s life in the
Eternal Now
. Perhaps that is precisely what we do, each
moment is a phase transition to the next, our choices moment by moment determine th
e life
we live.



COMPLEXITY


SELF ORGANIZATION IN CHAOS



Complexity is the most difficult area of chaos and the cu
tting edge of field study at the
present time. To better under stand what is happening, try a few of the clickable spots on
the internet. Then read about complexity below.


For your enjoyment see:
The Game

of Life.


Emergence games
, you can make your own.


Your brain as a
cellular automata.


Visit Capow and s
ee how a power company is using
cellular automata.


Minifloys

are artificial flies, see them swarm online.


And for another fun time go to
Cellular Automata.








Physicist, Ilya Prigogine has shown how classical open "dissipative structures" held far
from thermal equilibrium by matter
-
energy flow can be self
-
organizing. He defines
complexity as ‘the abilit
y to switch between different modes of behavior as the
environmental conditions are varied.’
(1)


Out of Chaos has come the self
-
organizational
properties that have genuinely surprised and delighted sci
entists in the last decade.
Prigogine states:


We know now that nonequilibrium, the flow of matter and
energy, may be a source of order. We have a feeling of great
intellectual excitement: we begin to have a glimpse of the road
that leads from being to bec
oming.
(2)

In the
Instability

section we discussed order and disorder, but in this section we observe
another aspect of these two terms. Ord
er suggests that there is symmetry in the model, or
an invariance of a pattern under a group of transformations. One part of the pattern is
sufficient to reconstruct the whole: in order to reconstruct a mirror
-
symmetric pattern,
like the human face, you ne
ed to know one half and then simply add its mirror image. A
crystal structure
is typically invariant under a discrete group of translations and rotations
therefore, the smaller t
he part needed to reconstruct its whole, because it has a more
redundant or "ordered" pattern.


Disorder also contains symmetry of the probabilities that a component will be found at a
particular position. A gas is statistically homogeneous in that any pos
ition is as likely to
contain a gas molecule as any other position, though the individual molecules will not be
evenly spread. The law of large numbers says the actual spread will be symmetric or
homogeneous. Even a random process can be defined by the fac
t that all possible
transitions or movements are equally probable.


Complexity may then be characterized by a lack of symmetry or "
symmetry breaking"
. No
part or aspect of
a complex item can provide sufficient information to actually or
statistically predict the properties of the others parts. This again connects to the difficulty
of modeling associated with complex systems. Prigogine speaks of the symmetry breaking
in natur
e:


Here another interesting question arises: In the world around
us, some basic simple symmetries seem to be broken.
Everybody has observed that shells often have a preferential
chirality.
Pasteur

wen
t so far as to see dissymmetry, in the
breaking of symmetry, the very characteristic of life. We know
today that DNA the most basic nucleic acid, takes the form of a
left
-
handed helix. How did this dissymmetry arise? One
common answer is that it comes from

a unique event that has
by chance favoured one of the two possible outcomes [but] ....
To speak of unique events is not satisfactory; we need a more
"systematic" explanation.
(3)

Alan Turing
hypothesized a mechanism based on the process of chemical reactions and
diffusion to explain how living organisms develop. But the idea was too small to encompass
all the complexities involved with

biological
morphogenesis
. His work did give birth to
more work in the theory and experiment with spatial dissipative structures. From that
grew the work with oscillations, prop
agating waves, pattern formation on catalytic
surfaces, mulltistability and chaos. Kondepudi and Prigogine give many examples of what
is happening in the materials science using instability and self
-
organization occurring in
far
-
from
-
equilibrium systems. T
hey also cite biological, geological and social investigations
of the same processes.
(4)


We find the mechanisms of instability and self
-
organization in complexity. Complexity
implies two or more compo
nents joined together in such a way that it would be difficult to
separate them. Methods of analysis or decomposition into independent modules cannot be
used to develop or simplify models of complexity. These complex entities will be difficult to
model, th
e models will be difficult to use for prediction or control, and the problems will be
difficult to solve. Chaos theory has enabled the analysis of such systems in diverse academic
research of both science and humanities.


Two aspects of complexity concern
distinction and connection. Distinction denotes variety
and heterogeneity, and to the fact that different parts of the complex behave differently.
Connection signifies constraint, redundancy, and the fact that different parts are not
independent, but that
the knowledge of one part allows the determination of features of the
other parts. In a gas, where the position of any gas molecule is completely independent of
the position of the other molecules is an example of distinction leading to disorder or chaos
o
r entropy. A perfect
crystal,
where the position of a molecule is completely determined by
the positions of the neighboring molecules to which it is bound is an example of connection
leading to order o
r negentropy. Complexity can only exist if both aspects are present.
Complexity is therefore situated in between order and disorder, or, using a recently
fashionable expression, "on the edge of chaos".


It has been noted that the
strange attractor

boxes behavior into a small, easily handled
package, allowing coherence to the many positive and negative influences, and self
-
organizes the system into something new without causing any damage to cascade
throughout the system.
(5)


Kauffman and Christopher Langdon speak of the edge of chaos
as the place where systems are at their optimum performance potential
(6)
.


Computer
simulations of randomly generated boolean networks are used to explore: the dynamics of
evolution on rugged fitness landscapes; the tendency to react to perturbations by returning
to a stable cycle or "attractor" that was active w
hen the perturbation occurred; and the
relationship among the different attractor loops within such networks. This experimental
work is tied in with knowledge of biology and chemistry to explain the emergence of life,
autocatalytic systems of chemicals, ce
ll development, and natural selection. Kauffman's
work is relevant to all complex systems and offers lasting insight into the mechanisms
underlying cells, societies, and even thought.


This edge of chaos seems to be the
phase transition

state of the system or the place where
choices are made and bifurcation take place. It is the time and place when there are many
options, many positive and negative influences from these options, and a time of
great
mental turmoil if the system is a human being. What is it that stabilizes the system? What
allows the system to clearly weigh the options and select an option that is fittest for that
system to reorganize itself and go on to the next level? There hav
e been many theories
popping up recently to explain how this self
-
organization in Complexity works.


David Bohm was one of the leading quantum physicists of our age. In recent years, Bohm
attempted to explain an ontological basis for quantum theory. Bohm's

theory is that
elementary particles are actually systems of extremely complicated internal structure that
act essentially to amplify information contained in a quantum wave. His new theory of the
universe is a new model of reality that he called the "Impl
icate Order." This entails a
holistic cosmic view. It connects all things through a sort of enfoldment. In principle, any
individual entity could reveal detailed information about every other entity in the universe.
Bohm's theory states that there is an "u
nbroken wholeness of the totality of existence as an
undivided flowing movement without borders." The layers of the
Implicate Order

can go
deeper and deeper to ultimately the "unknown and undescribable
totality" that Bohm calls
the holomovement. The holomovement is the "fundamental ground of all matter." Does
this sound like
Tillich'
s 'ground of all being"? Bohm's implicate order implies a sort of
c
omplexity of being where order and disorder join. The Explicate order is what is
manifested as the universe. Bohm's theory of the Implicate Order emphasizes that the
cosmos is in a state of process: it is a feedback
-
universe that continuously recycles forw
ard
into a greater mode of being and consciousness.

This is precisely what Chaos/complexity
entails.
(7)


Another similar theory is the Unifying Theory
-

which is called
Tripartite Essentialism
. The
implications of non
-
linearity are that atoms belong to observed groups with similarly
classified properties and that no two similarly classified 'atoms' are absolutely iden
tical due
to their Chaos Ontology. It is more fully explained at:
http://easyweb.easynet.co.uk/~pegasus/

where it continues to make progress in Philosophy
of Science, Mind, 5
th

Generation Artificial In
telligence, and Cosmology.


Stuart Kauffman's
At Home in the Universe, The Search for the Laws of Self
-
Organization
and Complexity


(N.Y.: Oxford Univ. Press, 1995) contains the new idea that Darwinian
natural selection from random variations while necessa
ry is not sufficient to explain
evolution. There is also a spontaneous self
-
organizing mechanism. Kauffman states:


Darwin devastated this world. .... Evolution left us stuck on the earth with no
ladder to climb, contemplating our fate as nature's Rube Gol
dberg machines.
Random variation, selection
-
sifting. Here is the core, the root. Here lies the
brooding sense of accident, of historical contingency, of design by
elimination. ... We human, a trumped
-
up, tricked
-
out, horn
-
blowing, self
-
important presence o
n the globe, need never have occurred. So much for our
pretensions; we are lucky to have our hour. So much, too, for paradise.
(8)

Research into cellular automata is being done by Chris Langdon who form
erly worked at
the Santa Fe Institute, but is now with Artificial Life, see:
http://www.santafe.edu/~cgl/

.


Certainly the basic process of self organization is iteration. A good example is provided by
cellular

automata. Automaton consists of cells, which perceive their surroundings and
perform a decision to change their state according to some rule. The rules need not be
deterministic, but the dynamics dictated by it is irreversible. See above for links to
illu
strations of cellular automata online.


Autopoiesis

is another theory that has come out of the work of Ilya Prigogine.


Heinz von
Foerster's work in the Biological Computer Laboratory (Universit
y of Illinois) emphasized
the self
-
organizing features of living systems.


He even suggested that we call ourselves
'human becomings'.


Erich Jantsch in 1980 studied self
-
organizing systems and
hypothesized the integration of a variety of theories of self
-
regulation and self
-
organization
within the framework of the phenomenon of dissipative self
-
organization. He hypothesizes
the unification of Prigogine's theory of dissipative structures (order out of chaos),
Maturana's concept of autopoiesis (self
-
producti
on) and Eigen's (1971) theory of self
-
reproducing hypercycles.


Maturana defines an autopoietic system as a unified system in
which one is unable to distinguish product, producer or production since it is self
-
producing.


Jantsch viewed autopoiesis as a wa
y that the self organization of non
-
equilibrium systems manifests themselves.


Characteristically, living systems continuously
renew themselves and regulate regulate their processes so that the integrity of their
structure is maintained.