|The academic community and the Russian Academy of Sciences have a long
and fine record of the study of prognostication problems. This paper looks
at the contribution of nonlinear dynamics to the analysis of an information
process like forecasting. Notably, it examines some fundamental restrictions
on the predictability of complex systems, which were established in the
last few decades, and discusses the risk management concept and the hypothesis
of human algorithms in making a prognosis. Examples illustrating the application
of these ideas to the forecasting of the behavior of complex social systems
are given, and new possibilities opening up in this field are described.
PREDICTABILITY AND THE ANALYSIS OF COMPLEX
A pendulum can serve as an illustration of some of the central ideas in
forecasting (Fig. 1). Observations of this pendulum show that, with a
95% probability, its oscillations will be nonperiodic, and with a 5% probability
we will see periodic movement. The result depends on the original momentum
we give the pendulum. Let us start the pendulum and see what happens.
Рис. 1. Простейший непериодический маятник, демонстрирующий
динамический хаос. Чтобы скомпенсировать трение, маятник снабжен
магнитиками, а в основание игрушки помещены катушка и батарейка,
создающие электромагнитное поле
Dynamic Chaos and Fundamental Restrictions in Prognostication
It would be more correct to say that for a given
accuracy (arbitrarily large but finite) it is always possible to
indicate a time interval for which predictions cannot be made. This
interval is not so large, which is the whole point.
Feynman Lectures in Physics
Before the 1960s, all processes were thought to fall into two broad classes.
The former was described by dynamics systems, where the future was uniquely
determined by the past. They were supposed to be fully predictable. The
great Laplace said with reference to such systems-if his words are translated
into our modern language-that, given sufficiently powerful computers,
we can look arbitrarily far into the future and arbitrarily far into the
past. The second class included processes where the future did not depend
on the past. We roll dice, generating a number in no way connected with
In the 1970s, it became clear that there was a third, very important class
of processes, formally described by dynamic systems, such as the pendulum
in Fig. 1, but their behavior could only be predicted for a short time
interval; later, the researchers would have to deal with statistics. A
simple linear model can be developed for our pendulum, which will permit
us to predict the position of, say, the small balls after five swings
of the larger ball below them. However, there is no way we could predict
their positions after sixty time intervals.
Ray Bradbury wrote a science fiction story in 1963 in which he in effect
formulated the idea of dynamic chaos. In this story, an electioneering
agent, after his candidate has won, sets out on a time travel. The firm
that organizes his trip proposes a hunt for dinosaurs which became extinct
long ago. One is expected to move along special paths so as not to disturb
the complex texture of causal relations and change the future. However,
the hero failed to meet this condition and accidentally squashed a golden
butterfly. Upon his return, he found changes in the composition of the
atmosphere, spelling rules, and election results. A barely noticeable
motion caused the smaller dominoes to fall, which in turn caused larger
dominoes to fall, and finally, the falling of giant dominoes led to a
catastrophe. There was a precipitous buildup of deviations from the initial
path, caused by the death of a butterfly (Fig. 2). Even small causes had
large effects. Mathematicians refer to this property as initial-data sensitivity.
In the same year (1963), Nobel prize winner R. Feynman suggested that
our ability to predict, even in a world ideally described by classical
mechanics, was fundamentally limited. If we are to have a forecast horizon, "God is not to play dice," adding some random terms to the equations
describing our reality. We must not come down to the microworld level,
where quantum mechanics describes the Universe in quantitative terms.
Objects whose behavior cannot be predicted for fairly large periods can
be quite simple, such as our pendulum.
Рис. 2. Расходимость фазовых траекторий в системах с динамическим
Любая динамическая система определяет в фазовом пространстве траекторию,
Динамический хаос обусловлен тем, что соседние
траектории удаляются от нее. Из-за этого малые причины могут иметь
The pendulum is equipped with small magnets to offset friction, and the
toy's base contains a coil and a battery, which produce an electromagnetic
American meteorologist E.N. Lorenz realized, again in 1963, that sensitivity
to initial data leads to chaos. He asked himself the following question:
why is it that the rapid development of computers, mathematical models,
and computational algorithms has failed to produce a method for making
reliable weather forecasts for the medium term of two to three weeks ahead
of time? Lorenz proposed a simple model of air convection, which plays
a significant role in atmospheric dynamics. The model is described by
seemingly simple equations :
where the variable x characterizes the field of velocities, and y and
z, the temperature field of liquids. Here, r = R/Rc, where R is Rayleigh
number, and Rc is its critical value; s is Prandtl number; b is a constant
related to the geometry of the problem.
Subjected to computer analysis, Lorenz's system yielded a fundamental
result: dynamic chaos, i.e., nonperiodic motion in deterministic systems,
where the future is uniquely determined by the past, has a finite forecast
In mathematical terms, any dynamic system, whatever its modeling object,
describes the motion of a point in phase space. A key characteristic of
this space is dimensionality or, put very simply, the number of quantities
that must be given to define the system's state. From the mathematical
and computational points of view, it matters very little what these quantities
are: the number of lynxes and hares in a territory, variables describing
solar activity or a cardiogram, or the percentage of voters supporting
a president. If we assume that a point moving in a phase space leaves
a trail, dynamic chaos will be represented by a tangle of paths, such
as the one in Fig. 3. Here, the dimensionality of the phase space is limited
to three (space x, y, z). In 1971, D. Ruel and F. Tuckens proposed a name
for steady-state oscillation: a strange attractor. J.H. Poin-care's prophecy
that it would be possible to predict new physical phenomena from the general
mathematical structure of the equations describing these phenomena was
made a reality by computer experiments.This computer-produced picture
(the calculations were made with r=28, s=10, and b=8/3) convinced E. Lorenz
that he discovered a new phenomenon, dynamic chaos. This tangle of paths,
which we now call the Lorenz attractor, describes a nonperiodic motion
with a finite forecast horizon.
The Lorenz system has a finite forecast horizon. The fact is that if we
take again the two close paths shown in Fig. 3, we find that they diverge
(as in Fig. 2). The rate of divergence is defined by the so-called Lyapunov
index, and the time interval for which a prediction can be made depends
on this quantity. Every system can be said to have its own forecast horizon
Рис. 3. Аттрактор Лоренца.
Такая картина, полученная на компьютере (расчет проводился при
r = 28, s = 10, b= 8/3), убедила Э. Лоренца, что он открыл новое
явление - динамический хаос.
Этот клубок траекторий, называемый сейчас аттрактором
Лоренца, описывает непериодическое движение с конечным горизонтом
The progress of science shows that every new fundamental theory not only
opens up new possibilities, but also strips us of illusions. Classical
mechanics divested us of the illusion of being able to devise a perpetual
motion of the first kind, thermodynamics of the second kind, quantum mechanics;
and that we can measure arbitrarily closely the coordinates of a microparti-cle
and its momentum and relativity theory, that information could be transmitted
in a vacuum at superlight velocity. Today, nonlinear dynamics shatter
the illusion of global predictability: starting with a time horizon, we
can no longer predict the behavior of many fairly simple systems, e.g.,
Lorenz's paper was published in a meteorological journal but. remained
unnoticed for ten years. Today's meteorologists believe that the weather
forecast horizon does not exceed three weeks. In other words, no matter
how closely we measure atmospheric parameters, it is generally impossible,
using available instruments, to predict the weather at a particular location
three weeks from now. Experts estimate, the forecast horizon for the ocean
to be one month.
Today, many experts in solar physics speculate that the same is true for
the Sun. For example, there is a phenomenon called the Maunder minimum,
which refers to a period of nearly 70 years during which there have been
no bursts of solar activity. The question arises; as to whether we can
predict the next similar minimum in ' solar activity. The work in progress,
the Lyapunov indices, and the forecast horizons being what they are, this
prediction cannot be made for decades ahead.
However, nonlinear dynamics brought out not only fundamental difficulties,
but also new and wonderful possibilities. Let us focus on one of them.
Let us try to find out how many quantities we need to describe the behavior
of our simple pendulum. Classical science maintains that an infinite number
of quantities is needed. A pendulum does obey the laws of mechanics, but
if this toy is to rotate and not stop because of friction, an electromagnetic
field has to be produced. Technically, our pendulum has an infinite number
of degrees of freedom.
Nonlinear dynamics, when applied to the analysis these kinds of systems,
helps us to establish how many variables are needed for their description,
how many descriptions are necessary for prognostication, and also the
kind of monitoring required. It turns out that such a system requires
ten variables at least. This opens up new possibilities. We have what
is technically a very complex system, and we need to isolate its essentials.
Whereas in the 1960s, systems analysis, which considered the general properties
of systems and appeared in them as entities, was all the rage, today systems
synthesis holds sway at the Keldysh Institute of Applied Mathematics.
This synthesis makes it possible to extract from a host of variables exactly
what is needed for decision making.
Having established that there are essential limitations in prognostication,
new generations of models and algorithms were developed, and forecasting
became an industry. What we witness today is a leap in forecasting not
unlike the one that occurred with the advent of personal computers. Before
the PC age, computers were immense and costly systems which only the very
large companies could afford to have. With the coming of PCs, computing
became affordable to very many. The same is happening in the field of
forecasting. Forecasting is ceasing to be a science and is becoming a
technology. In the past, it was the RAND Corporation and a few other teams
that made predictions for the US government and other entities, whereas
now even not: particularly large firms keep laboratories engaged in forecasting
or, to use a current phrase, "future design."
Dynamic chaos made it possible, on several occasions, to diagnose grave
diseases from electrical activity data by using fairly simple algorithms,
and to propose new algorithms for information compression and protection.
Economic forecasts, relying on the ideas of chaos and strange attractors,
became a burgeoning activity. Mention must be made of the nonlinear journals:
Physica D, Chaos, Physical Review E, Izvestiya Vuzov: Applied Nonlinear
Dynamics, and Nonlinearity. It turned out that, viewed in the prognostication
perspective, there is more to link the objects of different disciplines
than to dissociate them.
Yes, a man is mortal, but this is only half the
trouble. What is worse, sometimes he is mortal all of a sudden:
that's the trick of it!
M.A. Bulgakov, Master and Margarita
Forecasting research is currently concentrating on the description and
prediction of the rate of catastrophic events. J. Van't Hoff, one of the
fathers of modem chemistry and the first Nobel Prize winner in chemistry,
in his day said, "I removed from my works everything that is difficult
to observe or happens rather rarely." Today's information technologies
provide us with capabilities that enable us to turn to the analysis and
forecasting of rare catastrophic events.
Here is an example showing that all kinds of catastrophic events must
obey the same laws. Curves of behavior of characteristics describing two
complex hierarchic structures, a stock exchange and a tectonic fracture,
just before a catastrophe, exhibit fast catastrophic growth, on which
accelerating oscillations are superimposed (Fig. 4). The smoothed curve
is finely described by the formula
that is to say, we have one and the same solution of equations that are
yet unknown. Note that the asymptotics of such processes prior to the
catastrophe is the so-called aggravating mode (where one or more quantities
describing a system grow to infinity within a finite time). This class
of modes has been studied by a scientific school that has formed, under
the guidance of one of the present authors, at the Keldysh Institute of
Applied Mathematics .
Рис. 4. Характерный вид зависимости, возникающей перед катастрофами
в сложных системах. а - зависимость от времени логарифма индекса
Доу-Джонса (этот индекс определяется ценой самого эффективного
пакета акций 30 ведущих компаний Соединенных Штатов) перед Великой
депрессией ; б - зависимость от времени логарифма концентрации
ионов хлора в родниках перед катастрофическим землетрясением в
Кобе (Япония) в 1995 г. . Точки - это точные данные, сплошная
кривая - сглаженная зависимость, построенная по ним
John von Neumann once said, "I do not believe that we can find genera!
laws in the behavior of complex systems. It is the same as building a
theory of non-elephants." The development of nonlinear dynamics refuted
this assertion. Nonlinear dynamics succeeded in establishing general scenarios
of the origination of chaos from an ordered state . Current developments
in science suggest that in some cases, we can speak of some general scenarios
of the inception of catastrophes.
Some thirty years ago, Feynman was asked, "If all the living physicists
were to die tomorrow, leaving a single phrase for posterity, what would
you say?" "The whole world consists of atoms and the void,"
said Feynman. "They will think out the rest." If this question
were to be asked of all scientists and not only physicists, the phrase
should probably be worded differently:
"Leam risk management." Risk management is one of the key technologies
of our civilization [7, 8]. It corresponds to the main road of progress:
to trade some threats and dangers for others. For instance, to trade the
danger of starving and freezing for the risk of reaping the fruits of
the pollution of the water, air, and earth owing to the operation of thermal
or nuclear stations.
According to the normal (Gaussian) distribution, large deviations are
negligibly rare. However, many disasters, accidents, and catastrophes
generate power-series distribution statistics, which decreases slower
than the normal distribution, therefore catastrophic events cannot be
neglected. In the logarithmic scale (below), power dependences acquire
the form of straights.
It should not be thought that tertium non datur and that we can only go
with the tide. There are alternatives. Sweden resolved to forgo nuclear
power engineering as an overly hazardous technology. In France, on the
other hand, where more than 70% of electricity is produced by nuclear
stations, the government is contemplating a boost to this industry as
a major way to conserve the environment. The stakes are high and the discretionary
power is quite great.
It was quite recently that the deep connection between nonlinear dynamic
notions and risk management became clear to us. The paradoxical statistics
of accidents helped us to realize this. Remember the Titanic, Challenger,
Chernobyl, Three mile, Bhopal... Each of these major catastrophes of the
20th century is associated with a long cause-effect chain, an "unfavorable
set of many unlikely incidentals," to use the usual wording of state
commission reports. As a matter of fact, an evildoer plotting something
along these lines would have a hard time of it. As one inspects these
disasters, one has a persistent feeling that we are simply having a long
run of bad luck.
What is the mathematical form of this "bad luck?"
The word random already been used once. In the early 19th century, K.F.
Gauss established that the sum of the independent, identically distributed,
random quantities obeys a certain law. The corresponding curve, obtained
after normalization, is shown in Fig. 5. It can be seen to rapidly decrease;
large deviations are very rare under this law. So rare, in fact, that
they can be ignored.
Рис. 5. Типичный вид нормального (1) и степенного (2) распределений.
В соответствии с нормальным, гауссовым, распределением большие
отклонения настолько редки, что ими можно пренебречь. Однако многие
бедствия, аварии, катастрофы порождают статистику со степенным
распределением, которое убывает медленнее, чем нормальное распределение,
поэтому катастрофическими событиями пренебречь нельзя.
В логарифмическом масштабе (внизу) степенные зависимости
приобретают вид прямых линий
The Gaussian distribution underlies a host of engineering calculations
and design codes. Every engineer knows the three sigma rule. It says that
the probability of a random quantity deviating from the mean by more than
three sigmas is less than 0.001 (see Fig. 5). The sigma here is mean-square
deviation. A simple example: people's heights are distributed according
to the Gauss law; hence, we can ignore with a light heart the likelihood
of coming across a three-meter-high giant.
But there is another class of laws called power laws (see Fig. 5). The
tail of this distribution decreases much slower, therefore such laws are
often referred to as "heavy-tail distributions." Here, large
deviations cannot be ignored. If people's heights were distributed according
to this law, it would be the world of oriental fairy tales, where ordinary
mortals could easily encounter thirty-meter jinn, ifrits, and divs. It
is in this world of oriental fairy tales that we usually find ourselves
when we face disasters, catastrophes, and accidents. This is according
to the statistics for earthquakes, hurricanes, nuclear arms stocking incidents,
market crashes, damage from confidential information leaks, and many other
To show that these are not mere assertions, here are the American statistics
for tornadoes, earthquakes, floods, and hurricanes in the past century
(Fig. 6). We can see that these observations fall, with a sufficiently
high accuracy, on the curves corresponding to the ideal power statistics.
Рис. 6. Распределение торнадо (7), наводнений (2), ураганов
(3) и землетрясений (4) по количеству погибших в них в США в XX
в. По оси абсцисс отложена фатальность F стихийного бедствия,
измеряемая логарифмом числа погибших, по оси ординат - логарифм
числа бедствий N, имеющих фатальность не меньше данной. Идеальным
степенным законам соответствуют прямые. Видно, что эти законы
являются хорошим приближением для реальной статистики бедствий
When we decide whether or not to undertake a particular engineering project,
we can use any of a variety of approaches. The first is one that was realized
and perfected already in Columbus' time: determine all possible outcomes,
N, multiply their probability pi by the corresponding rewards or losses,
xi and sum up:
Depending on what quantity results, we either undertake or forsake our
Note that Columbus' expedition was the only one to travel to the New World
at a treasury's expense. After him, business houses in Spain engaged in
the insurance and reinsurance of such projects, for the financial risks
were too high for any business house. However, the rewards were also high.
A historical anecdote: following his expedition to the New World, F. Drake
gave a present to the queen of England which equaled two annual budgets
of England. The queen paid off all her debts. There are indeed very many
dangerous, but also very profitable, projects in our world. That was the
foundation, laid even in Columbus' time, which provided the foundation
for the assessments of very many engineering initiatives up until the
However, a paradox was noticed as early as the 18th century. Let us picture
a game of heads or tails. If it is heads, you receive two gold ducats,
and the game ends. If it is heads again, you receive four gold ducats,
and the game ends; if a third heads occurs, your receive eight ducats.
The sum 5,, which is part of the Columbus algorithm, is infinite. The
question is how much one is prepared to pay to join the game.
Jacob Bernoulli, while watching such a game in St. Petersburg, was amazed
at the fact that people were not prepared to pay more than 20 ducats to
join. When one estimates the odds and decides if it is worth a try, one,
according to Bernoulli, does not act according to the Columbus algorithm.
What one estimates is not real winnings but the winnings utility:
where U(xi) is a utility function. If you have one ruble, 100 rubles
are great winnings for you. If you have 1000 rubles, you value 100 rubles
much less, their utility for you being much lower. In the mid-20th century,
von Neumann showed that the Bernoulli algorithm is good for economic behaviors
in a great many cases.
However, subsequent studies of economic behavior, notably the work of
M. Alle and his school, indicated that in many cases, people employ a
different, more complex decision making algorithm. A person deals not
with the Bernoulli formula, but with a formula including not just a utility
function, but also subjective probabilities f(pi), which reflect our notion
of danger :
The abscissa is the fatality F or a natural disaster, measured by the
logarithm of the number of disasters, N, with a fatality not less than
a given one. Ideal power laws are matched by straights. It can be seen
that these laws are a good approximation to the real-life statistics of
disasters and catastrophes.In the case of "Gaussian disasters," there are design, extradesign, and hypothetical accidents. The likelihood
of the first is determined by the area of the curvilinear trapezoid ABEF,
the extradesign, BCDE, and hypothetical accidents, the area of the path
behind the curve to the right of line DC. For visualization purposes,
the areas corresponding to extradesign and hypothetical accidents are
Psychologists contend that if one is told that the risk is Jess than 10У6
a yearУ1, one is bound to simply ignore this possibility. In other words,
in order to analyze projects, we need to have a certain system of estimates.
In the 1950s, it was presumed that if people have sufficient training
and are paid regularly, they can ensure the absolute safety of any installation.
But the 'State Scientific and Technological Program "Safety" (managed by RAS Corresponding Member N.A. Makhutov) demonstrated that
the course prevailing the world over is more preferable for isolating
design, extradesign, and hypothetical accidents (Fig. 7). The consequences
of design accidents (for which there is a certain likelihood) are to be
removed by the company itself. The consequences of extradesign accidents
(which have their own likelihood) are to be liquidated by the Ministry
of Emergencies and the appropriate organizations best suited for accomplishing
it. As for hypothetical accidents, their likelihood was thought, until
recently, to be negligible.
Рис. 7. Типичная схема оценки аварий. В случае "гауссовых
бедствий" выделяют проектные, запроектные и гипотетические
аварии. Вероятность первых определяется площадью криволинейной
трапеции ABEF, запроектных - BCDE, гипотетических - площадью участка
под кривой, лежащим справа от линии DC. Для наглядности площади,
соответствующие запроектным и гипотетическим авариям, на рисунке
Much that was designed in this country was based on this supposition,
from weapon systems to nukes. It turned out that the assumption of Gaussian
statistics leads to the inference that the probability of a nuclear power
station accident is 10У7 a yearУ1, that is, one accident in 10 million
years. However, recent studies demonstrated that in each of these cases,
we deal with power statistics. Therefore, our estimates must be quite
different. In the event of power disasters, we should count on the worst.
To give you an idea of the scale of rare catastrophic events, here are
some episodes from the 20th century history. During the 1931 flood of
the river Yangtze in China, 1.3 million people died, and during the Tian
Shan earthquake of 1976, about 650000. The 1970 flood in Bangladesh cost
more than 500000 people their lives and left 28 million homeless .
The essence of risk management is connected not only with the description,
statistics, and understanding of mechanisms, but also what in some cases
can be termed precursors. This kind of behavior is exemplified by a curious
phenomenon called hard turbulence. It was discovered in plasma physics
in the 1970s and more recently, in a variety of reaction-diffusion type
systems. Let there be a quantity that changes chaotically but sometimes
makes gigantic leaps (Fig. 8).
Рис. 8. Типичная картина при возникновении жесткой турбулентности.
На "хаотическом фоне" изредка возникают гигантские пики
For such model problems we can identify precursors that signal danger.
Nothing has occurred yet, and disaster is very remote, but a certain slowly
changing variable already indicates that we have entered a danger zone
Рис. 9. Изменение медленных переменных Р, М и logE - перед гигантскими
пиками. Наиболее важна с точки зрения предупреждения катастрофических
событий переменная М
Today, such things arc being sought for many real-life systems.
A number of steps in the development and application of risk management
theory is being taken within" the framework of a federal goal-oriented
program, initiated by the Russian Ministry of Emergencies, to prevent
and mitigate the aftereffects of emergencies in natural and anthropogenic
environments. The program focuses on forecasting and preventing disasters
and catastrophes because, in economic terms, forecasts and preventive
measures cost dozens, sometimes hundreds, of times less than the liquidation
of the consequences of past calamities. However, the scale of these efforts
in Russia does not seem to match their importance. A broad multidisciplinary
approach is in order here, as well as a much more active contribution
from the Academy of Sciences. Many things must be reviewed and reappraised.
The complexity paradigm and the theory of self-organized
The more general a regularity, the easier it is to formulate.
Where do power statistics come from? This question is answered by a new
paradigm of power dynamics, the complexity paradigm, and the theory of
self-organized criticality which was developed within its framework [10,
Power dependences are characteristic of many complex systems: earth crust
fractures (the famous Richter-Gutenberg law), stock exchanges, or the
biosphere during the evolutionary time periods. They are typical of highway
traffic, computer network traffic, and many other systems. What all of
them have in common is the emergence of long cause-effect relations. One
event may lead to another, a third one, and then an avalanche of change
affecting the whole system. For example, mutation, which with time changes
the appearance of a biological species, affects its ecological niche.
A change in the ecological niche of this species will, naturally, affect
those of other species. They will have to adapt. The end of this avalanche
of change, leading to a new equilibrium, can be long in the coming.
A sand hill is a simple physical model demonstrating this kind of behavior.
Imagine the following picture: We drop a grain of sand upon the top of
the sand hill. It either stays there or slides down, causing an avalanche,
The avalanche may have one or two grains of sand, or it may have very
many. The statistics for a sand hill proves to be power-type, as is the
case for a number of disasters and catastrophes. It is like the statistics
that we have for, say, earthquakes; in other words, the danger is at the
boundary between determinist and stochastic behavior or, to use a current
phrase, at the edge of chaos.
Studies of complex systems demonstrating self-organized criticality showed
that such systems, on their own, seek a critical state, which is possible
with avalanches of any scale. Because this kind of system includes the
biosphere, society, various infrastructures, the military-industrial complex,
and a host of other hierarchic systems, the findings of the theory of
self-organized criticality are very important for the analysis of control
action and the development of methods for their prevention and destruction.
Extensive work on the complexity paradigm and its forecasting applications
is in progress the world over. The newly established Institute of Complexity
in Santa Fe, New Mexico, is an example. It is headed by M. Gell-Mann,
a recipient of the Nobel prize for physics, and has on its staff B. Arthur,
a winner of the Nobel prize for economics. The institute engages itself
in a variety of tasks, from disaster prediction and computer simulation
of economic processes to scenarios of the destabilization of political
regimes and artificial life . In Russia, the work conducted on the
complexity paradigm is at our institute and other RAS institutes. However,
its scale is a far cry from what is needed.
How can we predict?
Nature, whatever it should be, Was coauthored by the devil-
This is the whole point.
If prediction is so difficult, even when based on the use of state-of-the-art
computer techniques, how can we successfully reason in this complex and
changing world of ours? How do we manage to act in a reasonable way despite
a very modest time horizon? A theory of riverbeds and jokers, which is
under development now, attempts to give answers to these questions, and
hence, forecast algorithms.
George Soros, the well-known financier, is credited (rightfully) to be
one of its authors. In his Alchemy of Finance he put forward the idea
of an "informational" or "reflexive" economy. According
to this idea, variables such as credibility level, expected profit, and
many others, which describe our virtual reality, play a , key role in
today's economy. It is thanks to these variables that grand financial
pyramids can be built and afterwards destroyed. But these variables can
change quickly, something which is quite alien to mathematical models
built in the natural sciences.
In other words, in the phase space of many entities with which we deal
in our daily life, there are places called joker fields, in which chance,
a game element, or a factor of no consequence in any other situation can
turn out to be decisive and not only affect the future of the system but
even shift it in a stepwise fashion to another point in the phase space.
Joker refers to the rule by which this step is made. The name was borrowed
from card games. The joker is a card that a player can substitute for
any other card in the pack. Obviously, this greatly increases the number
of variants and the degree of uncertainty.
Рис. 10. Система с руслами и джокерами.
Картинка, возникающая в задаче в разорением банка. Небольшая область
внутри окружности соответствует области джокера, в которой надо
принимать серьезные меры
Picture appearing in a problem with a destroyed bank. The small area
inside the circle corresponds to the joker area, where serious action
has to be taken.
Consider a simple example. Let us assume that we own a small bank. Business
is going from bad to worse, and how can it be otherwise in an age of crisis?
A decision must be made. The first, and most natural decision (which is
taken with the probability p1, whereby the system abruptly goes over to
a phase space point a1 Fig. 10) is to stage a presentation at the Hilton:
publicity, journalists, new clients, and opportunities. The second is
to act like all honest men and declare ourselves bankrupt (probability
p2 and correspondingly, point a2). Finally, we can consider our nearest
and dearest and get away with the remaining cash, to preach to local reformers
from across the Atlantic (probability p3, and point a3). We can see that
we have time and again a symbiosis of dynamics, predetermination, and
We can translate the following into the language of medicine. Away from
the joker area, therapy must produce an effect, whereas only surgery can
be effective within the area itself. The situation in this case can change
rapidly and radically.
If we have bad luck in our forecasts in the joker area, there must be
some area where we have good luck. Let us see what having good luck in
a forecast means exactly. It means that the behavior of the system is
defined, with an acceptable accuracy, by only a few variables, and the
rest can be disregarded in the first approximation. Besides, we should
be able to make predictions for a reasonably long time horizon. The phase
space areas where these conditions are fulfilled were termed riverbeds
It was probably the ability to effectively isolate riverbeds and to learn
not only by trial and error and perfecting its predictive system, but
also by relying on the commonsense that gave mankind the decisive advantage
in its evolutionary development. We can also take a broader view: different
theories, approaches, and sciences prove to be useful and necessary if
they succeed in finding the right riverbeds. After all, science is an
art of simplification, and it is particularly easy to simplify when dealing
with riverbeds. Of course, on average, we cannot glimpse at what is beyond
the forecast horizon, but in particular, having found ourselves within
the riverbed parameters and became aware of the fact, we can act intelligently
and with caution.
This raises the following questions: Where does the riverbed start and
end? What is the structure of our ignorance? How can we go from one information
field and notions adequate to this riverbed to others when this riverbed
is at an end? As one comes to know different economic, psychological,
or biological theories, one has a persistent feeling that their originators
deal, without realizing it, with different realities, or different riverbeds.
This is akin to the principle in quantum mechanics, where the answer to
the question of whether an electron is a wave or a particle depends on
THE FORECASTING AND DYNAMICS OF COMPLEX
Having realized the existence of a forecast horizon, understood how complex
the systems with which we deal can be, clarified the questions that can
be asked and the data we need to be able to answer these questions, we
obtained a tool for the description of a great variety of phenomena and
processes. It is particularly useful when we predict the behavior of socio-technological
systems, for which quantitative patterns determining their dynamics are
Modeling the development of higher education
In 1994, we were approached by the Russian Ministry of Education and the
International Bank for Reconstruction and Development. The matter at hand
was the granting of a two billion credit for the reconstruction of Russia's
higher education; it was a more trouble-free time than the one we are
living in today. The following question arose: If the World Bank's wishes
were realized, what would it lead to in a five-, ten-, and twenty-year
perspective at a macrolevel (the macroeconomic level), middle level, and
a microlevel. Let us dwell on the macromodel.
We analyzed United Nations statistics within a nonlinear dynamics framework.
It was found that industrial development and the role of science and education
can be determined (if we aim at a crude, qualitative picture) by the computer
analysis of a discrete mapping of three variables . One describes
the resources; another, output (gross domestic product); and the third
one, science plus education (Fig. 11). There are two key quantities in
this system. The first is the time lag. If science and education improve
their performance tomorrow, the economy is not likely to see the results
until three to five years later. The second is receptivity to innovation.
According to available statistics, if we take the receptivity of the Japanese
economy as 10, then that of the United States economy will be 8, that
of Western Europe will be 6, and that of the Soviet Union will be 1.
Now let us assume a model situation. A country rich in resources initiates
industrialization and invests in science. However, its economy has a receptivity
factor that equals zero. Science is making great progress in this country,
but because its economy is not receptive to any research findings, we
eventually find ourselves at the renewable resources level (Fig. 1la).
The role of science in this situation is quite different: we need it in
order to find new sources of development. To illustrate this point, uranium
salts were known to be fine dyes in the 1930s. Later, it was discovered
that uranium had some other useful applications.
Рис. 11. Макроэкономические траектории экономики, невосприимчивой
к нововведениям (а), восприимчивой к инновациям (б), восприимчивой
к инновациям при урезании финансирования (в).
Кривые показывают, как меняются выраженные в условных единицах
ресурсы (1), объем производства (2) и научно-технический потенциал
(3) в некоторой стране с течением времени;
а - соответствует "банановой республике",
б - ситуация, когда общество достигает некоторого
уровня развития, после чего происходит смена основных ресурсов развития
и дальнейший рост обеспечивается интеллектуальной сферой,
в - ситуация, когда в результате сокращения вдвое
финансирования интеллектуальной сферы к критическому моменту начала
спада производства развитие этой сферы не достигло необходимого
уровня и не смогло оказать заметного влияния на развитие общества
Now, let us imagine that we have managed, through some reforms, to raise
the receptivity of our economy. Our postreform situation is close to what
happened in Japan, where an accelerated growth was in evidence (Fig. 11b).
If, during this rapid growth, we reduce the support of education and science
by half, the country will find itself in the same situation it was in
from the beginning (Fig. 11c). We are in a trap: science is not supported
because the economy is poor; the economy is poor because there are no
projects or effective technologies.
The IBRD models with which we compared our results yielded roughly the
same picture. The bank's experts believe that creating a sustainable low-productivity
operation would be normal for Russia. We think otherwise.
Toward a "direct-action sociology"
Totally new opportunities are opening up in societal management. We shall
use the terms "social barometer" or "direct-action sociology" to describe them. What do they mean?
Let us assume that we are measuring some parameters of our society. The
question is how many variables, in reality, characterize it. Sociological
survey data and the capacity available in many Russian regions make it
possible to monitor public opinion, yielding dozens and hundreds of indicators.
If computer networks are used, this kind of monitoring can be carried
out at daily or hourly intervals. However, what use is this vast and,
evidently, important information to us? Decision makers can keep in their
field of view only a handful of factors and qualitative indicators, not
more than seven, if we are to believe psychologists. How do we select
these indicators and help make intelligent and balanced decisions?
The fact that help is possible is shown by a simple device like the barometer.
If we cannot effectively solve equations describing atmospheric dynamics,
from which we could predict the weather, our barometers still warn us
before a storm that problems may await us.
For social systems, computer technologies can serve as a barometer of
sorts: they reduce the information available to a few indicators that
help us in decision making. Techniques tested in earthquake prediction
furnished the basis for these approaches . We do not know the equations
that we can solve to forecast a disaster, but we have a vast file of data
we can use to teach appropriate computer systems to forecast. We have
conducted work on the sociological applications of these approaches jointly
with I.V. Kuznetsov and his colleagues at the RAS International Institute
of Mathematical Geophysics and Earthquake Forecasting Theory and also
with S.A. Kashchenko and researchers at Yaroslavl State University.
A word of caution against excessive expectations typical of a society
pinning too many hopes upon computer technologies. Initially, it was supposed
that computerized control systems would be instrumental in raising the
efficiency of the economy, but the economy proved unprepared for this.
Great expectations were entertained for an experiment in a computer-aided
solution of various equations. However, it was found that we lacked suitable
equations for the description of many important entities, and even if
we had these equations, finding the coefficients and adjusting the model
was in itself a challenging problem.
Data is the Achilles' heel of prediction algorithms for socioeconomic
systems and risk management problems. To teach a computer system, we need
long arrays of valid and reasonably accurate data describing the different
aspects of the concerned object. So far, this has been lacking practically
everywhere. If this gap is filled,' the quality of our forecasts can be
When we took up sociological data, we found many curious things. It transpired
that the reaction of Moscow and St. Petersburg to many events was the
direct opposite to that of the rest of the country (Fig. 12). Obviously,
this behavior is connected with the socio-economic structure of our society
and its terms of reference. Using these approaches, many of the conclusions
made by researchers at the RAS Institute of Social and Political Studies
 can be corroborated and rationalized in quantitative terms at another,
Рис. 12. Разность между позитивными (и нейтральными) и негативными
ответами на вопросы ВЦИОМ в Москве и Санкт-Петербурге и в остальной
России. а - "Что бы вы могли сказать о своем настроении в
последние дни?"; б - "Как бы вы оценили в настоящее
время материальное положение вашей семьи?"
These methodologies, like most research findings, cut both ways. By relying
on them we can, on the one hand, manipulate the behavior of our electorate
even more successfully than we do today. On the other hand, they show
key variables and order parameters in the social conscience. It is they
that determine the main problems of the future and opportunities connected
with Russia's revival after the crisis.The curves show changes through
time in a nation's resources, expressed in conventional units (1), output
(2), and scientific and technological potential (3), with (a) standing
for a "banana republic," (b) standing for a situation where
society arrives at a certain level of development, followed by a change
in the main development resources, with the further growth supported by
the knowledge sphere; and (c) standing for a situation where, due to the
support of the knowledge sphere being cut by half. the development of
this sphere has not reached the necessary level, by the critical point
of the start of the decline in production, and could not have a pronounced
effect upon societal development.
Innovation development. Scenarios for Russia
Today, many hopes are being centered on what is called the "innovation
economy." We' at the Institute of Applied Mathematics, together with
colleagues at other RAS institutes, are conducting a study, commissioned
by the RF Ministry of Industry, Science, and Technologies, into the possibilities
available to Russia for embarking on a sustainable development path and
shifting to an innovation economy.
Our analysis has shown that, from a ten-year perspective, the complex
socioeconomic system of Russia is threatened by collapse. Its systemic
crisis has brought the nation to a line where the supercritical depreciation
of the main assets leads to a series of anthropogenic and social disasters,
the growth of energy prices leads to the ultimate destruction of the manufacturing
industry and the raising of transportation rates, and to the irreversible
breakup of the nation. Given the present trends, the nation will completely
lose its sovereignty and disintegrate, and the Russian people will disappear
from the historical arena.
Because of its geographic and geoeconomic position, and by virtue of the
high energy intensity of the industry and living in this cold country,
which has four-fifths of its territory located in the permafrost area,
Russia cannot for any length of time be a raw-materials appendage of the "golden billion" nations . Therefore, the question of the
new development of resources became a vitally important one . One
possibility is to redirect some of the sectors of the economy to high-technology
production. Russia's government has announced a strategy of transfer from
a "tube economy" to innovative development.
The official view of innovation focuses on the neoliberal conception and
the imitation of foreign models. It treats an innovation as something
that has found its place in the market, lays an emphasis on the development
of venture businesses, and sees the state's role as that of an arbitrator
providing the conditions and infrastructure for the application of innovations.
Studies made at the Institute of Applied Mathematics and other RAS institutes
have shown that this is a dead end path for Russia.
Innovation in today's Russia should ensure the solution of strategic tasks
in the sustenance of its population and its gradual transition to a progressive,
sustainable development path, not the "filling up of the market,"
"assuring macroeconomic stabilization," etc. Most of the innovations
of vital importance for Russia are non-market ones. They include the production
of high-quality and affordable foodstuffs and medicines, the building
of housing and roads, the provision of communications, alternative technologies,
and innovations increasing the safety of the technosphere. Many of the
innovations being publicly discussed today [16, 17] are not needed for
the economy's harmonization but for the nation's survival. Reliability,
endurance, and maintainability are characteristics of new technologies
at a premium for Russia today.
The state can and must be the only customer for such innovations. It must
assume the key function of goal setting in the fields of economic and
social development. This calls for a fundamentally different level of
coordination compared to the present one and much higher demands on the
prediction and monitoring of the socioeconomic system. This presupposes
the reestablishment, on the basis of new methods of social management,
forecasting, and modern information technologies, of something along the
lines of the State Planning Committee of Russia.
Its primary tasks should be:
- To raise the reliability and quality of forecasts;
- To make use of Russia's available resources;
- To define the nation's scope of opportunities, given alternative development
- To detail the policy chosen (not only in cost but also physical indicators).
We must realize that the country is in an emergency. a historical dead
end. To lead our country out of this dead end, we need programs on the
scale of ED. Roosevelt's New Deal . The development of such a course
should be the central task for the nation's research community and leadership
Returning to innovation, we shall note that the variables that the Ministry
of Industry and Science regarded as the key ones and the mechanisms it
acknowledged as important-innovation/production complexes, their accelerated
development, market penetration, etc.-are actually secondary. When we
analyzed the items on which hopes were pinned, these hopes proved to be
unjustified. What matters is not innovation/production complexes but their
symbiosis. The Zelenograd Innovation and Production Complex is a case
in point. It includes the Proton plant, which is a donor for a host of
smaller enterprises. Each of them receives money from the government.
However, if we cast the total (how much such an enterprise receives and
how much it contributes to GDP), it turns out that they give about ten
times more than receive. Therefore, as we encourage innovation in this
particular case, we should think not only about small businesses, but
more importantly, about the Proton plant. As can be seen, when one analyzes
seemingly obvious things from the standpoint of nonlinear dynamics and
information processes, the results can be rather unexpected.
Theoretical history, or, a search for alternatives
Arnold Toynbee, one of the greatest historians of our time, wrote a very
short work, a "historical heresy" as he termed it later in his
memoirs, "If Philip and Artax-erxes Had Survived" . It is
on record that Alexander the Great came to power as a result of a plot
allegedly engineered by his mother. It was for this reason that his mother
was to die very soon thereafter. According to Toynbee, history would have
taken a radically different course if there had been no Alexander and,
correspondingly, his opponent. There would have been no Rome, the era
of great European empires would never have come, and city-states would
have long retained very good development prospects. At the same time,
Oriental despotisms would have slowly transformed while retaining their
The techniques, methods, and formalisms offered by nonlinear dynamics
and undergoing active development make it possible to consider historical
development alternatives for some simple model situations [14, 20, 21].
Here is an example relating to the situation examined by Toynbee. Computer
calculations of the Mediterranean population densities yield two variant
developments (Fig. 13). According to the first, there is Rome, and history
has developed precisely as it has developed. In 96% of the cases, computations
do indeed yield this variant. But there is still the 4%, when history
takes quite a different course: if there is no Rome, there is no Roman
civilization, whereas Greece is developing at an accelerated pace. In
other words, computer analysis admits to both the possibilities that Toynbee
Рис. 13. Результаты компьютерного расчета плотности населения
в Средиземноморье Слева - вариант, реализовавшийся в истории,
справа - альтернативный, когда нет Рима и Римской империи
Of course, these simple models are rather conditional. They only recognize
elementary links between natural, social, and demographic factors-a very
limited set in comparison with the vast file of data that professional
historians deal with. However, even the recognition of these few relationships
allows one to see historical alternatives. It is to be hoped that more
complex models of this kind will be useful in strategic planning, and
in due course, history will increasingly pose as an applied science, a
kind of whetstone on which to sharpen global dynamics models, whose importance
is growing in the context of the sustainable development concept.
To summarize, researchers working in different scientific disciplines
have reached a common understanding of essential problems in forecasting
and fundamental limitations connected with prediction. In order to pursue
a sensible policy (technological, innovation, or economic), it is critically
important in some instances that we have both a forecast and a team capable
of making it.
1. Lorenz, E.N., Deterministic Nonperiodic Flow, J. Atmosph. ScL, 1963.vol.20.pp.
2. Predely predskawemosti (Prediction Limits), Moscow: Tsentrkom, 1997.
3. Malinetskii, G.G., Khaos. Sfruktury. Vychislitel'nyi eks-periment.
Vvedeme v nelineinuyu dinamiku (Chaos. Structures. Computer Experimentation:
Introduction of Nonlinear Dynamics), Moscow: Editorial URSS, 2000.
4. Sornette, D. and Johansen, A., Large Financial Crashes, Phys. A, 1997,
vol. 245, nos. 3-4.
5. Johansen, A., Sornette, D., et al.. Discrete Scaling in Earthquake
Precursory Phenomena: Evidence in the Kobe Earthquake, J. Phys. France,
1996, vol. 6.
6. Rezhimy s obostreniem. Evolyutsiya idei: Zakony koevolyutsii slozhnykh
struktur (Aggravation Modes. The Evolution of an Idea: The Laws of Coevolution
of Complex Structure), Moscow: Nauka, 1998.
7. Proceedings of the Workshop "Reduction and Predictability of Natural
Disasters" held Jan. 5-9, 1994, in Santa Fe, Rundle, J.B., Turcotte,
D.L., and Klein, W., Eds., New Mexico, 1995.
8. Vladimirov, V.A., Vorob'ev, Yu.L., Malinetskii. G.G., et al., Upravlenie
riskom. Risk, ustoichivoe razvilie, sinergetika (Risk Management. Risk,
Sustainable Development, and Synergetics), Moscow: Nauka, 2000.
9. Larichev, O.I., Teoriya i melody prinyatiya reslienii (Theory and Methods
of Decision Making), Moscow: Logos, 2000.
10. Bak, P. How Nature Works: The Science of Self-organized Criticality,
New York: Springer, 1996.
11. Malinetskii, G.G. and Podlazov, A.V., The Self-organized Criticality
Paradigm: The Hierarchy of Models and the Limits of Predictability, Izv.
Viiw. Prikl. Nelineinaya Dinam., 1997, vol. 5, no. 5.
12. Waldrop, M.M., Complexity: The Emerging Science at the Edge of Order
and Chaos, New York: Touchstone, 1993.
13. Malinetskii, G.G. and Potapov, A.B., Sovremeimye problemy nelineinoi
dinamiki (Problems of Nonlinear Dynamics Today), Moscow: Editorial URSS,
14. Kapitsa, S.P, Kurdyumov, S.P., and Malinetskii, G.G., Sinergetika
i prognozy biidushchego (Synergetics and Forecasts of the Future), Moscow:
15. Rossiya u kriticheskoi cherty: vozrozhdenie ili katastrofa. Sotsial'nay
a i sotsial'no-politicheskaya sitiiatsiva v Rossii v 1996 godii: analiz
i prognoz (Russia at the Critical Line: Revival or Catastrophe. The Social
and Socio-Political Situation in Russia in 1996: An Analysis and Forecast),
Osipov, G.V., Levashov, V.K., and Loko-sov.V.V., Eds., Moscow: Respublika,
1997. Why is Russia Not America, Moscow:
16. Parshev, A.P., Forum, 2000.
17. Weizsecker, E., Lovince, E., and Lovince, L. Factor Four, Moscow:
Academia, 2000. Translated under the title Faktor chetyre.
18. Roosevelt, FD., Fireside Chat, Moscow: Gos. Duma R.F, 1995. Translated
under the title Besedy u kamina.
19. Toynbee, A.J., If Philip and Artaxerxes Had Survived, Znanie-sila,
1994, no. 8. Translated from English.
20. Malkov, S.Yu., Kovalev, V.I., and Malkov, A.S., Mankind's History
and Stability: A Mathematical Modeling Experiment, Strateg. Stabil'nost',
2000, no. 3.
21. Chernavskii, D.S., Pirogov, G.G., et at. The Dynamics of the Economic
Structure of Society, /zu Vuzov. Prikl. Nelinein. Dinam., 1996, vol. 4,
Discussion at the RAS Presidium
scientific communication was discussed by RAS academicians, R.F.
Ganiev, Yu.A. Izrael', N.A. Kuznetsov, D.S. L'vov, G.A. Mesyats,
R.T. Nigmatulin, N.A. Plate, D.V. Rundkvist, V.I. Subbotin, and
S.Yu. Malkov, Dr. Sci. (Eng.) of the Center for Strategic Nuclear
Forces at the Academy of Military Science.
G.G. Malinetskii, having made a scientific
communication on "Nonlinear Dynamics and Prediction Problems" at the RAS Presidium, answered questions.
Academician Yu.A. Izrael': You have showered us with a tremendous
amount of information, which seems to have a fair share of emotion thrown
in. There are different kinds of forecasts but you pretend to use all
of them, natural, economic, and political alike. I wish to center my question
on natural processes.
Early in your report, you mentioned the prediction limit. What is your
view of the prediction limit: is it lack of information, lack of theory,
or a matter of principle? If it is a matter of principle, i.e., there
is a prediction limit, how can it be determined?
Malinetskii: There is indeed a prediction limit; this is the point
I wanted to make. It appears that nature being what it is, near paths
diverge in many systems, even fairly simple, low-dimensionality ones.
That is to say small causes lead to great effects. The rate at which these
effects grow with time determines the forecast horizon. When Edward N.
Lorenz became aware of this fundamental limitation, he gave the following
striking example: If the earth's atmosphere is what we think it is, a
butterfly's wingbeat - a very small action at the right place at the right
time - can change the weather in a vast region in, say, two to three weeks
time. In other words, the formulated limit is as much a matter of principle
in meteorology as it is in quantum mechanics or thermodynamics.
There are different ways to determine our time horizon. In particular,
we can maintain monitoring and recording, every tenth second, the position
of a specific ball in the pendulum I have demonstrated. Furthermore, we
use computer techniques to measure quantitative characteristics of the
ball's path. Many a time, our ball finds itself in the vicinity of one
and same point in the phase space. Let us have a single path. A second
path, starting from an adjacent point, can be considered a disturbed first
path. From these two paths we can determine the mean rate of their divergence,
and hence, the forecast horizon.
Izrael': You estimate the forecast horizon in meteorology at two
to four weeks. Can you give us a more definite figure?
Malinetskii: We once settled on three weeks. Visiting American
specialists maintained that three weeks was, indeed, the magnitude.
I wish to be understood correctly; therefore, I will return to my pendulum.
After I start it, there is a five-percent chance that it will go over
to a simple periodic mode, which is perfectly predictable. In other words,
there are strange spots in the phase space where predictability is anomalously
good. In meteorology, there is a well known phenomenon called blocking.
If the atmosphere is in a certain special state, we find ourselves in
the neighborhood of a quite definite point in the phase space, in which
the forecast horizon can be rather distant. On average, however, the system
has a particular finite time horizon.
Academician G.I. Marchuk: Mikhail Alekseevich Lavrent'ev at the
RAS Siberian Division made some experiments. There is a wave, a rather
big one, perhaps even a tsunami wave. Then it begins to rain, and suddenly
the wave's energy dissipates, the wave grows smaller and smaller, and
finally disappears. How does this experiment fit the theory you are developing?
Malinetskii: To be frank, I have anticipated this question. Here
is a demonstration I have prepared. Look at this toy (see picture). When
in equilibrium, it has a steady form which is unchanged irrespective of
the action it is exposed to. If we start to slowly alter a parameter,
at some point there will be an abrupt change, and this form of equilibrium
disappears. The change is followed by a bifurcation, with the system becoming
very sensitive to small actions. This is a typical picture in many complex
systems, from social to economic. It would appear that our Novosibirsk
colleagues observed this kind of phenomenon.
Игрушка, иллюстрирующая аномальную чувствительность системы
вблизи точки бифуркации. Эта игрушка имеет два устойчивых состояния
равновесия (а, б). Меняя число витков пружины, зажатых в руке,
мы изменяем параметр. Вблизи точки бифуркации (в), где исчезает
одно из состояний равновесия, пружина обладает аномальной чувствительностью
к малым возмущениям. Последние скачком могут привести пружину
в состояние равновесия а
Academician N.A. Shilo: I once noticed that the time distribution
of the half-lives of both stable and radioactive isotopes of chemical
elements fit in the Fibonacci series. What is the relation between the
Fibonacci series and the tremendous process of decay of radioactive elements,
which can be said to embrace the whole Universe?
Malinetskii: We did not address this problem; we simply did not
meet people who asked these kinds of questions.A toy demonstrating the
anomalous sensitivity of a system near a bifurcation point.
The toy has two steady states of equilibrium, (a, b). By varying the number
of coils of the spring clutched in the hand we change a parameter Near
the bifurcation point (c), where one of the states of equilibrium disappears,
the spring has an anomalous sensitivity to small disturbances. The latter
can bung the spring by a rapid change to the state of equilibrium.
Academician R.I. Nigmatulin: It seems to me that thresholds are
one of the reasons for the appearance of various uncertainties. Another
reason is that most processes are described by numerous parameters, and
since we cannot cover all of them, we have to reduce their number. Instead
of billions, we have seven or eight equations. For instance, in classical
mechanics, uncertainty is the price paid for there being thresholds or
the reduction of the number of variables or other values.
Malinetskii: What you are speaking about are indeed important causes
of uncertainty. However, along with these causes, there is an even deeper-seated
reason. An elementary system like the Lorenz system has no thresholds,
none of the factors you listed, but it does have uncertainty. Nature made
it so that there would be a fundamental limitation associated with the
Academician V.A. Kabanov: A few years ago, you read a report at
the chemical faculty of the Moscow University in which you analyzed the
possibility of predicting the state of education in Russia depending on
the amount of funding received. If 1 remember correctly, your model led
to the following conclusion: if we take a country with a fairly high level
of development of science and education, which are funded in one way or
another, and then reduce the funding, this high level will persist for
the time being, followed by a collapse. Today, the percentage of the GDP
appropriated for science and education is decreasing in this country.
Is it possible to use your computations to predict in how many years science
and education will collapse in Russia?
Malinetskii: The study you mentioned was concerned primarily with
education. We did establish a funding threshold, after which "science
plus education" cease to have any effect on the macroeconomy. This
is not to say that education has no effect on a microlevel; people satisfy
their curiosity, raise their social status, etc. We also traced the future
development of the teaching community. This was in 1995.
When we submitted our forecasts, we were highly praised, but told that
our prediction was too gloomy, that we should be realistic, and that there
was no way the funding of science and education could be increased, not
just by a matter of percent, but several times over, as we advised. Unfortunately,
the reality proved to be close to our predictions, if not gloomier still.
According to our computations, the reform of higher and secondary education
being conceived today is criminal, for it will lead to a rapid degradation
of our whole system.
I believe that it is, in principle, possible to analyze the scientific
sector and the innovation sector at a macrolevel, but there are two problems.
One is social need: there has to be people who are really interested in
forecasts; the other is that a large body of data is required. We work
with the Yaroslavl region and with the Moscow government. We have found
that all data are privatized and every item has to be paid for. These
two circumstances prevent our work team from developing the same kinds
of model and speak about science as seriously as we once spoke about education.
Academician V.I. Subbotin: You used the term extradesign accidents.
It is not of your invention, but it carries very dangerous undertones.
To be sure, nothing is absolutely safe. A system created by humans has
a right to accidents but not to disasters. If this cannot be achieved,
this area must be simply closed, and other approaches to the final goal
should be sought. The idea of an extradesign accident creates the possibility
of a collapse.
Malinetskii: I will give an example which is related not to nuclear
power, but to oil extraction. Drilling platforms are operating both in
the North Sea and the Gulf of Mexico. They represent more than one million
tons of metal and concrete, and their total cost is upwards of two billion
dollars. The platforms are built to be extra safe. When they were started,
the general feeling was that no accidents at all could occur. The risk
estimates made at the time said that a breakdown could occur, not once
in one million years as in the case of an atomic reactor, but in 20 million
years; that is to say, they were designed with an order of magnitude more
reliable than an atomic reactor. Nevertheless, heavy accidents have occurred
at 15 platforms.
We must face the fact that disasters can occur in complex engineering
systems. We should count our money, but we should also build when it pays
to risk. It is simply not possible to rule out the likelihood of a disaster,
as we realize now, therefore, during the design process we should have
in mind the worst of possible scenarios too.
There is one last point to make in this connection. Individuals, with
their skills, psychological state, etc., are also a part of a technological
system. When human factors sharply deteriorate, what happens is something
that we have always warned about: through human fault, the technosphere
starts to break down. Roughly speaking, given particular human skills
and pay level, we can use particular technologies; when the skills and
pay level are decreased, the use of sophisticated technologies is pregnant
with disaster and catastrophe. This aspect seems to be very important
Academician G.S. Golitsyn: I should like to remind you that the
problem of prediction was first formulated by the astronomer and meteorologist
Philip Thompson as early as the mid-1950s. Lorenz further developed all
You never mentioned-perhaps for lack of time- the fact that we can predict
the statistics of events or the weather. A weather forecast is made for
a particular time period and an averaged area. As a rule, the longer the
time period, and the larger the area for which we make a prediction, the
greater the forecast horizon. We already know the prediction limit in
the study of climates. Are there examples of the extension of forecast
idomains in time, space, etc., to other fields of science?
Malinetskii: In the technosphere, we faced what is called the planner
paradox. Let us assume that we have very good models, a fine strategy,
and very good solutions designed for five years. The question is, "What
happens after 10 years?" These strategies may prove to be ineffective
in 10 years, and simply criminal after 20 years. This raises the question:
"How long are we going to live, and how are we going to average?" If we are going to live in the Principality of Muskovy and average for
Moscow oblast, we will have particular models and solutions. If, on the
other hand, we are going to act throughout the Russian territory, there
should be a different strategy. Russia borrowed much money in the belief
that everything would be fine after ten years. This never came to pass.
Moreover, this money was borrowed from the condition of the domestic market
and not the global dynamics. This money was borrowed on our good intentions.
Therefore, let us set our task straight-what we want to have. Further,
depending on the formulation of our problem, we will arrive at different
equations and different models. Here, in my opinion, the situation is
the same as in meteorology. True, it is easy to predict the climate, but
it is extremely difficult to predict the weather.
G.S. Golitsyn: It is very important that we realize what we can
and what we cannot do, and what is dangerous. Science is undergoing commercialization,
which in itself poses a number of important mathematical problems.
Academician N.P. Laverov: I am baffled by the State Duma having
adopted, in the first reading, a new prediction law, which covers the
prediction of both processes and phenomena. Considering the great influence
that various external contingencies exert upon an operating system, have
we matured enough to pass in the Duma a law forecasting processes and
Izrael': Isn't it a wonder what the Duma is doing! How can we enact
a prognostication law?
Laverov: The Duma will be the Duma, but we should be kept advised
of what is being done there.
Malinetskii: I can explain why this kind of law is being passed
because I happened to talk with the experts. Their comment is this: today,
no one in this country bears any responsibility for any forecast.
How is a forecast made in a normal situation in a normal country? Assume
that a forecast for the development of the economy has been made. There
are verifications and models, which are discussed, there are competent
people who can state: "Yes, our scientific community realizes that,
at present, we have no better model, therefore, given the present standard
of our technology, we shall rely on it in our predictions of the economy.
In the course of time, we shall see how well we have predicted the future
and correct our models."
What happens in this country? The government team is changing rather often.
It sets up its analytical center and recruits forecasters who make decisions.
When the government is asked, "What has happened?," or "What
have you done?," the usual answer is this: "You know, that was
the forecast we had, and we relied on it." I believe that it was
in order to avoid this kind of talk, and partly succumbing to emotion,
that the Duma is passing a prediction law.
Laverov: I will continue. If this law is enacted, we shall act
in the framework defined by the law, and shift the blame for failed forecasts
to the fact that a law has been passed, and we need no change in our models.
Have you seen what the law says at all?
Malinetskii: Yes, I have. Its attitude to prognostication is as
if we lived in Laplace's times. Notably, it ignores the existence of an
objective forecast horizon. Viewed in this perspective, the law is, I
believe, ill-judged. We should be more sober in our appraisal of the capabilities
of contemporary science. Also, it fails to acknowledge one circumstance.
Conceptually, a forecast is a process. There is a commission. You submit
a forecast to it, and you find out whether or not your methodology works.
You find it out primarily from whether or not your forecast has held up.
However, this kind of mechanism is not found in either the State Committee
for Engineering Supervision, or a host of other vitally important departments,
notably the Ministry of Defense. If the general attitude were the same
as in earthquake prediction, namely, that a forecast is a nonrecurrent
act, on the one hand, and a process and ongoing work that must be perfected,
on the other, we would face no problems.
I feel that if the Russian Academy of Sciences does not make a move to
introduce amendments, the prediction law will be passed unamended.
Academician D.S. L'vov: In contemporary economics, there are so-called
alternative approaches to socioeconomic forecasting, which can be presented
by cones expanding with time. In our present situation, the overlapping
area of these cones proves so short in duration as to render the different
options in the development of, say, Russia's macroeconomic parameters
practically indistinguishable. In this connection, I have two questions
to ask. Have you investigated Gref's famous forecast, if only as a rough
estimate? Have you estimated how much it will cost to expand our forecast
horizon and to see somewhat further than we can today?
Malinetskii: Unfortunately, we did not find a client for this work,
although we wanted very much to undertake it.
When Gref's program was discussed at our institute, the first thought
that came to mind was, where are the models that underlie, e.g., that
dreadful pension alternative visualized by Gref? True, the workforce will
decrease but so will the number of pensionable-aged peoples! Thus, based
on ambiguous models, quite awful things are adopted.
Now, when we at the Institute of Applied Mathematics requested from the
Gref center the models on which the Gref program relied, they responded
with silence. I believe it says something about our culture if people
regard it as normal that someone puts forward a program without backing
it by any forecasts and serious models.
Also note the following fact. Seismologists have learned to predict earthquakes
because they have vast data files, which every forecaster can analyze.
Nothing like this is found in economic statistics. It is bad enough that
every department of any size that maintains some inhouse statistics is
not liable to make them available, and often seeks to sell them. Many
important data are simply not collected or discarded.
To my mind, the Duma should pass'a law, not on forecasting, but on statistical
data, which are a strategically important resource.
Academician G.A. Mesyats: I should like to note that three RAS
institutes gave their opinion on the Gref program. The main question was
whether or not i( was possible, within the framework of the proposed concepts
or models, to assure a 5-percent growth fot the GDP, as envisioned in
the Gref program. One institute gave one-percent growth, another gave
zero growth, and the third, one-percent growth. You are absolutely right;
nobody shows their models. The models that our economic theorists have
are, of course, more realistic.
Academician A.F. Andreev: The word prediction was repeatedly used
here. However, prediction is what science is always concerned with: given
particular starting conditions, to determine what will happen to the system
afterwards. Therefore, our talk about prediction is actually a talk about
the destiny of science in contemporary society. When the Duma enacts a
prediction law, it thereby enacts a science law. Prediction cannot be
separated from science, nor science from prediction. They are one and
the same thing.
Recently, the prediction problem has been used in reference to something
vitally serious for the economy and for life; therefore, the general attitude
to prediction is entirely human, being somewhat different from the attitude
to science. I totally disagree with this view.
I liked the report very much. It shows that when we take up a forecast
problem, i.e., a prediction about what will happen to a structure, or
a system, we must have in mind some technicalities, which are very many.
A system can be described with high accuracy by simple equations, and
they will be a model of the system. No simple model ever completely describes
a system. Something is always left out. A simple model has a certain degree
of accuracy, and its accuracy may happen to be quite high. But as a system
develops, it may enter an unstable region. It can be the model's own instability,
which is apparent to all, or instability with respect to some parameters
not recognized by the model. Then, however long you analyze the model
you are not likely to see any instability. The rapporteur gave an example
of an electromagnetic field in a pendulum, whose action is not visible
at first sight.
I do not agree that, until the 1960s, people did not understand any of
these things. What happened is that, starting in the 1960s, the science
of prediction has experienced rapid growth. This very important research
area is finding ever new applications in all fields of science and society.
These predictions are essentially not different from all those problems
that have been solved by science long ago, and we must accord them the
same esteem that we generally accord science.
Izrael': I shall make some observations. As far as meteorological
forecasts are concerned, their prediction limit is two to three weeks.
In climatology, as Academician Golitzyn said, scientists even gave up
the word forecast, using the word projection instead. They make these
forecasts for 50 or 100 years ahead, without computing every single path.
We should realize, therefore, that there are two different approaches
Now to the extradesign accidents that Academician Subbotin mentioned.
These accidents are, as a rule, considered in projects. They are accepted
as they were. But there are other accidents, which are, as the rapporteur
said, so rare as to be ignored. However, they are precisely our main interest.
The Chemobyl disaster was not included in the extradesign category. All
the fantasy that designers incorporated in their nuclear power plants
never lived up to the situation that occurred in the Chemobyl disaster;
it is a well known fact. It would seem, therefore, that where we refer
to power statistics, to rare events, we should examine once again what
extradesign accidents these data refer to.
The last observation. I did not read the prediction bill being discussed
in the Duma. If it is about the order of the use of forecasts, it is quite
natural, but if it preaches some scientific truths, then, I believe, in
principle, there must not be such a law.
Kuznetsov: I shall name yet another reason for prediction limitation.
It is connected with the fundamental limitation of computers.
If a system's path fills some area of the phase space, this path cannot
be predicted by computer "path by path." The fact is that computers
have their capacity. Every equation is converted into digits. For example,
we use discretization to reduce a set of partial differential equations
to a set of ordinary derivative equations, then computation follows. However,
when we wrote our set of ordinary derivative equations, we assumed that
some of the coordinates are continuous; we discretized some coordinates
leaving others continuous. A computer has no continuous coordinates, all
of its coordinates being discrete. We have at every time slice a finite
set of points where the system may find itself, in other words, there
can, in principle, be no nonperiodic motion, and we must be fully aware
of this. We can predict distributions but not every individual path.
Methods have been developed in the last two to three years, which make
it possible to judge, from the resulting semihyperbolicity, conditions
of whether or not we can in principle simulate a particular path on a
discrete computer, and if we cannot, what must be done to compute the
path distribution. In other words, we cannot say what course a path in
a phase space will take, but we can say in what neighborhood of this point
it will remain one time, and in the neighborhood of another at another
time. This is a prediction too, albeit a peculiar one.
Subbotin: I have a very short remark about Chemobyl. Today, it
is generally admitted that its design was faulty. Besides, it is important
that the control system be self protected: If someone initiates operations
that are contrary to logic, they must not be executed. Unfortunately,
this was not implemented. I call your attention to the fact that the terms "extradesign accident" or "hypothetical accident" often hide a faulty design.
L'vov: I found the report extremely interesting because of its
multidimensionality, which also includes the study of economic processes.
In my view, such studies could be of great applied value. Therefore, it
was with a purpose that I asked to speak.
It grieves me, as an economic expert, to say what I must say, but the
information we manipulate is thoroughly individualized, with a great many
overruns. For this reason any experimenter wishing to use my model to
replicate my result will arrive at a totally different one. We allow ourselves
wonderful levity to handle key statistical indicators. The problem of
statistics is typical everywhere, but it is only in this country that
the measuring system we use has a prediction period close to zero is from
the start from the start. However, we believe such predictions and create
heightened expectations of economic theory. We believe that it ostensibly
knows how to do something, and is doing something, instead of analyzing
the models underlying this or that construction. This is my first point.
My second point is that it seems to be quite obvious that our academy-and
the report we have heard is vivid proof-has a fairly large scientific
backlog, which can be used to analyze economic information and assess
economic development parameters in order to make a significant step forward.
Today, the academy stands aloof of these efforts (I beg to be understood
correctly), demonstrating the destitution of all principles. We know this:
what is incorporated in the development of our economy has no scientific
verification. And what do we do? We raise our hands and give our support.
In conclusion, a few words about the expectations we create in our students.
Any textbook on economics describes a model proposed by a winner of the
Nobel Prize in economics. But the model is not confirmed by reality, because
if you take different time intervals, or different countries, the model
will not work. These are the facts, and something must be done about them.
Malkov: We are concerned with the matters of provision and prediction
of the strategic stability of this country in the military field. Recently,
we expanded our scope to include research into information stability,
and social-psychological, economic, and other kinds of safety. In cooperation
with the Keldysh Institute of
Applied Mathematics and other organizations we are modeling social processes,
specifically historical processes.
We established that a closed society, bounded by its territory and resources,
has a steady-state condition, being a closed-loop system. If it is an
open-loop system, i.e., it has neighbors and no clear boundaries, it is
inherently unstable, as feudalism was in its time. A steady-state condition
does exist for a capitalist society, but the domain of attraction is continuously
changing, and certain correlations of parameters give rise to several
attractors, which can, for the same level of productive forces, lead to
a feudal-type society with an uneven distribution of earnings. This social
structure is also stable.
In other words, transitions to local chaos and the multiplicity of attractors
are characteristic of social systems with the same macroparameters. When
we speak about forecasts, we should realize that accurate predictions
for any reasonably long period are not possible, and it is self-deception
to believe otherwise. We can only speak about short-term forecasts, about
the presence or absence of steady-state conditions, there generally being
several steady-state conditions. In conclusion, our prognostication strategy
should be as follows:
we need to ascertain what parameters and in what combination (because
the combination of parametric changes is also a very important thing)
must be changed in order to arrive at a needed limiting condition. Our
country is currently located in an attractor with a low-productivity state
of economy, characterized by a feudal societal structure.
It would be a good thing if the Academy of Sciences paid more attention
to interdisciplinary research, and colleges and universities trained students
who are at home in both soft sciences and nonlinear analysis techniques.
If this does not happen, we must not expect any significant progress in
Nigmatulin: I think that today's meeting is a rare example of a
case when every word said by the rapporteur and the speakers is extremely
interesting. I will not speak on the gist of the matter but, as a State
Duma deputy, I deem it necessary to touch on the prediction law now on
the floor at the Duma. I, too, have given it some thought.
It is common knowledge that state budget figures are based on forecasts.
The most fundamental forecast says that the 2001 GDP will be 7.5 trillion
rubles and the state budget, nearly 1.2 trillion rubles. The tax code
is being adopted, which also allocates rates. The government states that
it will keep the state budget. But what is the basis for this statement?
Figures change depending on the methodology used in the computation of
the GDP. There must be certain standard methodologies. They are not partial-derivative
equations, they are a group of simple (perhaps ten to twenty) arithmetic
operations. We do not need dozens of figures, what we wish to know is
whether or not we shall collect one or two trillion rubles worth of taxes.
In this connection, I believe that government should be put within the
prediction law framework. We need limitations on the particular state
budget figures that are confirmed every year. However, the method of their
acquisition-especially as some of them are fundamental-should, I feel,
be regulated by the prediction law.
Ganiev: Nonlinear mechanics is a field our team has worked on all
its life, especially in recent years in connection with the need for development
of science-, intensive processes. There will be people aspiring to create
a new science. They will coin a name, take an example from the biology
of something like a self-oscillating system: many hares, fewer wolves;
many wolves, fewer hares. This is what the new science of synergetics
is all about. No sir, a science becomes a science when it has common mathematical
models, common mechanisms, and common methods. And there are none in synergetics!
There are various applications, there are fine analogies that physicists
can take from mechanics and biologists can take from physics. But models
in biology are very complex, and they should be studied specifically and
Incidentally, I read an article by Mathematician V.I. Arnol'd the other
day. He attempted to predict some social processes using simple mathematical
systems. This is all very interesting, but with all due respect to this
distinguished mathematician, these questions should be the concern of
students, maybe sociologists, in order to get the feel of the trends.
The models that were considered today are very simple, therefore, professionals
in the fields of sociology and economics ought to regard them with great
caution. Much work remains to be done on the development of mathematical
models themselves. I should like to hear more about attractor models and
When this trendy theory first appeared, we mechanics knew perfectly well
that in deterministic systems, along with regular processes, there are
always unstable processes in evidence. Let us take a glass filled with
water and begin to sway it. At certain frequencies the surface of the
liquid in the glass will execute plane oscillations, at other frequencies
it will gyrate, i.e., spatial motions will oscillate in two distinct planes.
Between these two states there is a nonstable domain, where the whole
liquid is in chaotic motion. This phenomenon is described by simple mathematical
techniques, without recourse to terms like attractors, chaos. and the
In conclusion, I want to emphasize once again that models related to social
phenomena or economic phenomena are very complex. Therefore, we must be
very cautious in basing far-reaching conclusions on them.
Plate: I am glad that the report provoked such a good reaction,
because I was among those who had initiated its presentation at a RAS
presidium meeting. In truth, there is food for thought here, and for the
interaction of specialists in different fields.
A few short comments. Of course, extradesign, ill-predictable disasters
should perhaps be our primary concern. Many of those present will remember
that epoch-making presidium meeting two months after Chemobyl in this
room, where Valerii Alekseevich Legasov spoke; Anatolii Petrovich Aleksandrov
was also here. At the time, the likelihood of that disaster was estimated
at less than 10У7 a year. It did occur nonetheless. The design, which
might have been faulty, had, of course, contributed to the disaster, but
there was also a crazy conjunction of many other things. It seems to me
that we must be in a position to estimate the likelihood of such events
by means of models, to predict them, and to give some advice for their
As for the direct application of the modeling approach in economics, I
think that this possibility should be discussed. The nation has for decades
invested heavily in its agribusiness. The Soviet Union could not be fed,
and now we cannot feed Russia either. Every year, we have a battle for
the crop, a battle for the seedage, procurement, etc. Maybe it is simply
an imperfect policy. I set apart all manner of ideological and political
things, which are abundant here, but these things must be looked into.
Georgii Gennad'evich Malinetskii showed us a drawing, which demonstrates
how the funding of science and education affects the economy. A very important
conclusion follows from it. We are now rejoicing that the Duma and government
promise to increase the 2001 science budget by 30%. But if our models
and computations are correct, it may turn out that this 30% will not affect
any qualitative change in the present situation, and then disappointment
and disaffection will follow; we are investing in science, but it has
no effect on the economy. We must assess the critical magnitudes of appropriations
for science in fractions of a percent of the GDP, which are purposeless
in terms of our future strategy. Perhaps what we need is to demonstrate
in quantitative terms that an increase in appropriations by 300, not 30%,
will bring about economic growth in two years' time.
This is the kind of educative work with government officials, the Duma,
and others that should be carried out by our economics and spokesmen for
the group headed by Georgii Gennad'evich Malinetskii and Ser-gei Pavlovich
Rundkvist: It was with great expectations that I
came to hear this report, because prediction is the number one question
for geology and seismics. Regrettably, I am leaving without the clarity
that I had hoped to gain.
The rapporteur set a singularly important task: to consider the prediction
of all processes and phenomena in the social sphere, in nature, and in
the technogenic sphere. It would seem that, having posed such a super-general
problem, we should have made supergeneral conclusions, to be detailed
by us in particular areas. Unfortunately, his conclusions (which I tried
to faithfully record) will not be easy to use.
I am not at all objecting to those who highly appreciated the report.
The aim of my intervention is simply to state that when such reports are
made, one wishes more clearly formulated conclusions on fundamental issues,
which we can be applied in the future. A super-generalization such as
the we one just heard is from a small perspective.
On the whole, I am glad that this report took place. This is an exceptionally
important field, and I hope that future contacts will help us, including
myself, to better understand what precisely is of use for geologists today.
Mesyats: Most likely, it not easy to deliver a report from which
something new could be derived by both nuclear scientists, mechanics,
and economists. I think that this is quite a natural reaction.
Malinetskii: I wish to thank you for a very interesting discussion,
and above all else, for your understanding. It was only once that the
speaker's understanding of my words was the direct opposite to my meaning.
When I spoke about accidents, I said that there are engineering systems
where extradesign, or hypothetical accidents, can be ignored. This can
be done, in particular, with regard to car breakdowns. But there are complex
systems where we cannot act in this manner and where we must count on
the worst. Our institute, jointly with the Institute of Control Problems,
is currently engaged in the scenario modeling of precisely the worst possible
situation we can face.
As far as the conclusions from the report are concerned, I should like
you to-realize the following: There are general fundamental limitations
in prognostication. Science has established essential restrictions on
predictions for most diverse systems. Prediction is becoming major technology
in many fields. As for the fields where it is most effective, I think
that this is a subject for a general study. Thank you very much for a
very exciting discussion.
Mesyats: I believe that many of us have gleaned from this report
new and necessary information. At any rate, a feeling has emerged that
there is much common ground between the different fields of knowledge.