The Part and the Whole

George Malinetsky

Ph.D . (Physics & Mathematics), Professor, Department
Head M. V. Keldysh Institute of Applied Mathematics)
gmalin@spp.keldysh.ru

Why is it that a whole can acquire properties that are not intrinsic to
any of its parts? Many of us have been confronted with a situation where
each member of a collective is "against" or hesitates, whereas
the overall vote is affirmative. The same problem can be noticed in chemical
kinetics. Chemical transformations take place at the molecular level, but
oscillatory reactions exist where almost all of the molecules start to run
synchronously, as if on a mutual agreement, or obeying the rule of an invisible
director. In this case, the solution in the test tube can periodically change
its color, becoming now blue, and now red.
One of the fundamental problems in biology is that of morphogenesis. As
believed by biologists, each cell has one and the same set of genes. Then
how is it that these cells, in the course of the development of the organism,
"know" which of them are to become brain cells, or which of them
are to become heart cells? In 1952, the attempt of Alan Turing, one of the
fathers of cybernetics, to construct an exceedingly simple model for explaining
this selection, marked the birth of synergetics.
Alan Turing made the simplest surmise: the "instruction" for each
cell is composed in the course of interaction, or in the process of collective
actions. He assumed that cells are capable of liberating chemical reagents
that can propagate throughout the tissue by means of diffusion. It soon
became clear that these suppositions, backed by certain mathematical constructions
created on their basis, are quite enough to explain the origination of peculiar
structures in an initially uniform tissue.
The origination of such structures is a property common to many open systems
that are capable of exchanging substance, energy, or information. Besides,
systems in which these structures can exist ought to be nonlinear (roughly
speaking, nonlinearity means that the increase of influence by one and
the same amount can lead to quite different results).
Nonlinearity is the basis of our existence. Without it, man would never
be able to sense blindingbright sunlight on some occasions, and react to
individual quanta on others.
Incidentally, here is another note concerning synergetics and nonlinearity.
It happens sometimes that everything in science connected with the origination
of structures is brought under the realm of synergetics. This might not
seem to be reasonable, because structures, ordering, and organization are,
in one or other form, the province of all sciences, and it would not be
appropriate to assign their credits to synerget
ics alone. It has enough of its own merits. Further, in order to describe
the formation of structures, of spiral arms of galaxies, of oscillatory
reactions, of the multitude of instabilities in the physics of plasma, of
rings surrounding the planets, and of many other phenomena, nonlinear equations
and computerized computations are not at all indispensable. A general conception
is in order: if we have a good grasp of the phenomenon, and we know our
end target, we can do without nonlinear equations. For example, we can
use linear equations, or even explain everything using simple physical models,
or just using sign language.
Ordinarily, the description of a whole  for example, our organism calls
for a huge amount of numbers, known as degrees of freedom. And one of nature's
secrets, embedded in the basis of synergetics, is that not all of these
numbers are equally vital. But a small number of degrees of freedom are
the "ruling class"; they govern the other degrees of freedom,
and have them duly ordered. These governing degrees of freedom are usually
called the order parameters. Selforganization is simply the isolation of
the order parameters in the course of development of a system.
For instance, we are not expected to think about how to govern all the degrees
of freedom of our organism. Various interlinks become established between
them in some definite way. The interlinks can be formed when a man learns
how to walk, to smile, to fix his attention. Physiologists call such links
synergies.
The recipe of synergetics, in its essence, is very simple. One has to identify
the order parameters describing the given complex system, find the interrelations
that link these parameters, analyze the resulting equations, reveal anything
of interest in them and, finally, confirm this experimentally.
However, it is not all that easy to identify the order parameters. Today,
the best achievements are made in the studies of continuous media: liquids
and gases. If the medium is complex, then, naturally, it involves complex
structures.
It is not unusual that, in the practice of synergetics, the media can be
quite simple, and have, for example, only two processes: combustion and
heat transfer. But even in these cases, we are confronted with a surprisingly
rich assortment of structures.
The analysis of remarkable examples of selforganization playing a vital
role in plasma physics, biology, and astrophysics has been conducted along
two lines. Inward  towards development of an absolutely new branch of mathematics,
and construction of methods that will, perhaps, become a basic tool of analysis
in the 21st century. And in breadth  towards generalizations, philosophical
interpretation, and establishing unexpected connections between oriental
cultural images and the ideas of synergetics.
At a side glance, the analysis of the most ordinary media that hide a huge
opulence of various forms and types of structures resembles the reflection
on a Japanese engraving. Just a few sparse lines designate a human image;
maybe here we have an image of new natural science.
The order parameters can be identified not only in hydrodynamics, chemistry
or ecology. The analysis of order parameters is, for example, the basis
of the concepts underlying the theory of macroeconomic models proposed by
Nobel Prize winner, V. Leontiev.
Nonlinear science is magnificent. It helps us to see the deep internal
integrity, universality, and harmony in the sequence of phenomena apparently
having no connection whatsoever. "The world is chaotically strewn with
ordered forms" the French poet Paul Valery once wrote. Synergetics
has assisted in the comprehension of the meaning of many forms. It seems
that such an audacious flight of thought could be born in the minds of artists,
quite distant from everyday troubles. However, the truth is quite the opposite.
Many fundamental ideas in synergetics have been born in large research centers
connected with the military industrial establishment, and in large engineering
research projects, such as nuclear arms, space flight, controlled thermonuclear
fusion, development of effective arms control systems, analysis of the dynamics
of processes in the atmosphere and oceans.
There are but a few such huge centers even in a superpower, equipped with
the most elaborate instrumentation, and they are very expensive. But it
is exactly these centers, rather than the reserves of oil, or the number
and rank of computers, that determine, to a large extent, the strategic
potential of a state.
The American research center at Los Alamos was founded to develop the atomic
bomb. The enormous potential of the scientific staff made it possible to
switch from huge applied tasks to fundamental problems. The Los Alamos scientists
have to their credit many basic works in several fields of nonlinear sciences.
A similar worldclass center founded for solving strategic scientific problems
was initiated in the USSR in the 1950s  the M. V. Keldysh Academic Institute
of Applied Mathematics.
In those years, the Institute cooperated most strongly with the developers
of Soviet nuclear arms that was its principal goal. Another key problem
was the mathematical provisions for space flights. The third problem was
the development of supercomputers. However, solving the last problem called
for more than brilliant scientific ideas; it required the creation of a
new industry.

Fig. 1. Any dynamic system describes a trajectory in the phase
space, for example, such as that is shown in red. Dynamic chaos
is conditioned by the fact that the adjacent trajectories (in
green) are moving away from it. That is why some unimportant causes
can have farreaching consequences.

It is far from easy to hold back from the temptation to write about the
scientific schools, about the scientists engaged in synergetic studies.
But the life of ideas in these fields is even more compelling when compared
to the life of people. This is why we shall further deal with ideas  to
be precise, with one idea that is closest to the sphere of high technologies
dynamic chaos.
The Faces of Dynamic Chaos

In 1963, Ray Bradbury published a science fiction story in which he formulated
the concept of dynamic chaos. In this story, one of the organizers of
an election campaign, after his candidate's victory, sets off on a voyage
in time. The firm organizing these voyages offered hunts of dinosaurs
that were destined to die in the near future. In order not to upset the
complex fabric of genetic relationships, and not to alter the future,
one had to follow specially identified paths. The hero, however, failed
to stick to this rule, and accidentally crushed a golden butterfly. On
returning, he found changes in the composition of the atmosphere, the
rules of orthography, and the results of the elections. A tiny movement
upset the smallest domino, and that upset a larger piece, until, finally,
the fall of a gigantic piece resulted in catastrophe. The deviations from
the initial trajectory caused by the crushed butterfly increased abruptly
(See Figure 1). Small causes can lead to large consequences. Mathematics
calls this property, sensitivity to initial data.
In the same year, Richard Feynman, also a Nobel Prize winner, formulated
an idea on the limitation, in principal, of our ability to predict (or,
as it is known today, on the existence of a prediction horizon or limit)
even in our world, which is ideally described by classical mechanics.
The existence of the prediction horizon does not necessitate "a God
playing dice", adding some random members into the equations describing
our reality. There is no need to sink to the level of the microworld,
in which quantum mechanics offers probabilistic descriptions of the Universe.
Objects whose behavior we are unable to forecast for sufficiently large
periods of time can be quite simple: for example, the far from intricate
systems of pendulums with small magnets that are on sale today in many
shops, and referred to as works of "dynamic art".
The American meteorologist Edward Lorentz came to an understanding  also
in 1963!  that sensitivity to initial data leads to chaos. He was bewildered
by the question: why the skyrocketing developments in computers did not
lead to the realization of the wish of all meteorologists  reliably medium
term (23 week) weather forecasts? Edward Lorentz proposed the simplest
model for describing air convection (which plays an important role in
atmosphere dynamics), ran it through his computer, and was bold enough
to take the result seriously. The result was dynamic chaos: that is, nonperiodic
movements in determinate systems (meaning in systems in which the future
is unambiguously defined by the past), possessing a finite prediction
horizon.
From the point of view of mathematics, one can accept that any dynamic
system, irrespective of the object it simulates, describes the path of
a point in the space known as the phase space. The most critical characteristic
of this space is its dimensionality or, to simplify this, the number of
figures that are to be prefixed in order to determine the state of the
system. For a mathematician or computer scientist, it is not at all important
whether these numbers represent hares or lynxes in a certain territory,
or variables describing solar activity, or a cardiogram, or the percentage
of voters still supporting the president. If we accept that the point,
on passing through the phase space, leaves a trace, then dynamic chaos
would be represented by a tangle of trajectories, such as exemplified
in Figure 2. In this case, the dimensionality of the phase space is only
three. Is it not wonderful that such amazing objects exist even in just
a threedimensional world? Two other classic thinkers in nonlinear science
 D. Ruel and F. Tackens  in 1971 christened these tangles with a beautiful
name strange attractors.

Fig. 2. This computersimulated image convinced E. Lorentz that
he had discovered a new phenomenon dynamic chaos. This tangle
of trajectories, now called Lorentz's attractor, describes a nonperiodic
motion. Motion in this case does not become periodic, no matter
how long we wait.

The individual models, computer experiments and observations, once put
together, produce a fascinating picture. One can read, among the fragments
of this mosaic, the prophecy of Henry Poincare  it will be possible some
day to predict new physical phenomena on the basis of the general mathematical
structure of the equations describing these phenomena. Well, computer
experiments have transformed this dream into reality.
Another fragment  the endeav isors of theorists that substantiated
statistical physics and 35discussed the question: why is it possible
to use probabilistic language for describing not only movement, but dynamics
as well. An important element of the mosaic is offered 15 by synergetics
or nonlinear dynamics, which came to life in the 1980s as an interdisciplinary
approach.
Dynamic chaos made it possible, in a number of cases, to diagnose serious
illness from electrical activity data using relatively simple computer
software. Economic forecasts based on such notions as chaos and strange
attractors even became a branch of industry!
A tremendous achievement  the development of a scenario for the transition
from order to chaos. Irrespective of the equations describing a system,
there are but few universal scenarios in our world. They do not depend
on whether we open a water cock and observe the ordered smooth flow being
transformed into a chaotic turbulent flow, or we pour a solution into
a test tube in which a chaotic chemical reaction is under way, which gives
us the pleasure of observing the play of colors. Behind all this multiplicity
there is an intrinsic unity.
What Comes After Postmodern Science?

Since we find ourselves in a culdesac and are striving to find a way
out, it is natural to take a look at the landscape from above. All talk
about stable evolutions indicate that mankind is exactly there. Well,
let us ask ourselves: what does society expect of fundamental science
in the 20th century?
It may be that the sad prophesy of E. Wiener, one of the fathers of quantum
mechanics, has come true: science had a beginning, and it will also have
an end. And the end will come soon. First, the next generation of fundamental
theories will be of no interest to an increasing number of people. Secondly,
because we failed to design a fundamental "building" for science,
in which all the stories, secluded spots, cellars, and lofts are somehow
connected with one another. It would be good to know that when one is
in one part of the building, one could if necessary reach any other part.
But this is still not possible.
Evolution offers many advantages over revolution. Therefore, we should
predict the forthcoming changes, and be ready for them. Marie Antoinette
was reproached by her mother because she did not expect any abrupt turns
in society, such as revolution. It is important to say that the present
generation of scientists, including those involved in the "nonlinear
approach" to the development of nature, does not deserve such a rebuke.
Computer technologies have enabled huge databases to be created. The only
problem is what should be done further with the majority of them. The
most important thing is that these databases could be involved not only
in the presentday computer business, but also in the principal game that
our civilization is now playing.
In the 21st century, scientists will face a key problem, which can be
called neuroscience. Neuroscience deals with consciousness, perception,
and the intellection of the human being. This branch of science is at
the interface between computer science, cognitive psychology, neurobiology,
and nonlinear dynamics. Chaos plays an extremely important role here.
The brain, as well as many other systems of the human organism, works
in a random mode. The recent theory of control over chaos suggests that
it provides great scope for new approaches. In addition, specialists in
nonlinear dynamics try to involve encephalograms. But this is only a
shadow of the successes that will be required in the future.
Another key problem to be solved by 21st century science is, conditionally
speaking, the theory of risk and safety. Scientists faced this superproblem
fifteen years ago, although it was forecasted to appear in the 1960s by
Polish writer Stanislav Eem, in his book titled "The Sum of Technology".
We are living in a technological civilization, based on technology, not
on theology as it was in the Middle Ages. All we need is to learn how
we should handle this instrument with care and efficiency.
About thirty years ago, Richard Feynman was asked: "If all the living
physicists died tomorrow, and only one phrase describing the Universe
could be left for future generations, what should it say?"  "The
Universe consists of atoms and a vacuum," Feynman answered.  "This
description is sufficient and exhaustive. They will hit upon all the rest."
If presentday scientists, not only physicists but from any branch, were
asked the same question, the phrase would have to be different: "Eearn
to control risks." Controlling risks  this is one of the major technologies
of our civilization. In general, the main path of the world's development
is to exchange one threat or danger for another. It was obvious only recently
that ideas of nonlinear dynamics were intimately associated with the
concept of risk control. This link became clear when paradoxical statistics
on accidents were analyzed in detail. Let's recall the "Titanic",
Chernobyl, Trimail, Bhopal... Each of these great disasters of the 20th
century is related to a set of reasons and results, which are called "unfavorable
coincidence of many improbable and rare circumstances". How can these
random circumstances be described via a mathematical model? Karl Gauss,
who was called the king of mathematicians by his contemporaries, established
that the sum of independent, equally distributed random parameters obey
certain laws. The corresponding normalized curve is shown in Figure 3.
As can be seen, the curve descends very fast. According to this law, large
deviations occur very rarely. Hence, they can be ignored.
There is another class of laws that are called exponential laws (red curve
in the same figure). In this case, the "tail" descends much
slower  for this reason, these laws are called "distributions with
heavy tails". Here, great deviations cannot be ignored.


Fig. 3. The Gaussian Distribution (shown in red) is considered
to be classic and traditional. Following it, considerable deviations
are so rare in occurrence that they can be neglected. However,
many disasters, accidents, or debacles form statistics with exponential
distributions (shown in green). In this case, infrequent disastrous
events cannot be neglected.

What is the origin of exponential laws? In 1978, the American explorers
P. Buck, C. Chang, and K. Weisenfeld put forward a simple hypothesis:
an occasional impact in interactive dynamic systems can cause an avalanche
to appear, similar to the case when dominos upset each other.
The source of danger lies between dynamics, which we mentioned at the
beginning of this article, and an occasional event introduced from the
outside world: as is said now, at the edge of chaos. This laid the foundation
for a new star in nonlinear dynamics: the theory of selforganized criticality.
This theory can be applied to the behavior of stock markets, biological
evolution, earthquakes, traffic highways, computer networks, and many
others.
When scientists specializing in the study of chaos, computer simulation,
and operation with large databases worked on the theory of controlling
risks, they faced another interesting problem, which can conditionally
be called the analysis of long relationships of reasons and results.
One can approach the problem of risk control based on work on information
with the use of computer technology and global telecommunications. First,
one should make a conclusion after every disaster, taking it as a lesson.
All the disasters in the 20th century had forerunners, i.e. similar emergencies
of he same type, but with less catastrophic effects.

Fig. 5. The picture evolved from problems related to the ruin
of a bank. The small red area corresponds to the joker area, in
which much care should be taken to avoid troubles.

Thus, to prevent the "premiere", one should make conclusions
based on a smallscale "rehearsals," and in keeping with this
approach, change standards, plans, and rules of the game applied to social
life and the technosphere. The investment of one thousand dollars in the
forecast and prevention of an accident is better than one million in liquidation
of the consequences. Secondly, information and forecasting reduce the
time required to respond to events, and hence allow many thousands of
lives to be saved. Third...
However, it would be better to stop at this point. The Institute is engaged
in very promising work, in which Chaos plays a key role.
Let's return to dynamic chaos, and ask ourselves: if the task of prevention
of disasters is a considerable challenge, even with the use of modern
computer technologies, how can we orient ourselves in this complex and
rapidly changing world? How can we solve problems that can hardly be forecasted?
A new theory of channels and jokers tries to answer this question, and
also to obtain the algorithms for forecasting.
This theory was developed, among others, by the wellknown financier G.
Soros. In his book "The Alchemy of Finances," he put forward
the idea of an "informational" or "reflexive" economy.
According to the theory, a set of variables such as "level of confidence",
"expected profit", and many other parameters characterizing
our "virtual reality" play a key role in the modern economy.
These parameters allow one to create and destroy great financial pyramids.
But exactly these parameters can vary stepwise, which is not typical of
mathematical models simulated in the natural sciences.
In other words, in life we deal with a number of items composing a phase
space, in which there are other spots called joker's areas. Any occasion
or element of a game, or any factor that is absolutely unimportant in
any other situation, can play a key role in the joker's areas. This not
only can considerably affect the destiny of the system, but also transfer
it in a stepwise manner to another point of the phase space. The rule,
according to which the jump is made, is called the joker. This title comes
from gambling the Joker is a card that can be assigned the value of any
other card at the player's wish. Undoubtedly, this rule drastically increases
the number of variants and the degree of uncertainty.
Here's a simple example. Let's say we have a small bank. The financial
situation worsens every day; an epoch of crisis presents no other alternative.
It is now time to make a decision. First, the most natural (probability
pi, See Figure 5): to organize a presentation at the Hilton hotel. This
can cause a stir  an influx of journalists, and then new clients and
opportunities. The second possibility: be honest enough to declare bankruptcy
(probability p2). The third way: escape from the country with the rest
of the bank's capital in cash, taking care of only family and close friends,
and living abroad (this is a lesson from the Russian reformers  probability
p3). So we face again a symbiosis of dynamics, predestination, and occasional
coincidence.
This case can be reworded in terms of medicine. When one is located a
long distance from the joker, the best results can be obtained from therapy.
But location in the area of the joker calls for a surgeon's attention.
Thus, the situation can be changed rapidly and considerably.
Dynamic systems can also be considered using the same approach. The picture
shown in Figure 6 was designed by I. Feldshtein, a research fellow of
the M. V. Keldysh Institute of Applied Mathematics of the Russian Academy
of Sciences. This is again the Lorentz system. The colored arrows in the
figure depict the speeds of divergence (the area over zerolevel) or convergence
(the area under zerolevel) of the trajectory. As can be seen, the divergence
area that naturally should be compared to the joker is rather small.

Fig. 6. Local speeds of divergence (convergence) for the Lorentz
attractor. The areas above zerolevel correspond to divergence,
and under the zerolevel, convergence. As can be seen, the first
areas occupy a relatively smal part of the image.

If we fail to forecast the situation relative to the joker area, it could
be that we will be fortunate enough to forecast something else. Let's
consider: what does it mean that we are fortunate to forecast something?
It implies that the behavior of the system is determined by only a few
variables with the accuracy we need, and any other parameters in the first
approximation can be neglected. In addition, we are able to predict the
situation for a long enough period. These areas, belonging to the phase
space, are called channels. Most probably, our ability to separate the
channels effectively in the phase space and perfect the forecasting system,
and common sense, gave us an advantage over other animals, rather than
use of the trialanderror method. We may approach the problem more widely:
different theories, approaches and sciences can find a wide utility and
demand, when they have successfully found the channels. Science is the
art of simplification, and simplification can be achieved when one deals
with channels. Certainly, "on the average", "generally",
we cannot take a look beyond the horizon of the forecast. But "in
particular", when we find ourselves in the range of parameters corresponding
to the channel, and realize this fact, we can act reasonably and with
caution. The problem is: where is the origin and the end of a channel?
What is the structure of our ignorance? When we reach the end of a channel,
how can we transfer from one information field and point of view, which
corresponds to the previous channel, to others? Facing various economic,
psychological, or biological theories, we imply, without necessarily keeping
it in mind, that the developers of these theories dealt with different
channels. The same situation exists in quantum mechanics, where the answer
to the question, is the electron a wave or a particle, depends on the
particular experiment.
At a conference on artificial intelligence, the following definition of
the problem was presented. Simple problems are those that can be easily
solved, or prove to be irresolvable; all other problems are complex ones.
The development of our notion of chaos, and application of this approach
to various problems, proves that we were lucky. Designing the future,
comprehending a new reality, human nature, and algorithms of development
and control has turned out to be a complex problem.
