2001. V. 71, №2, pp 94-113

Nonlinear Dynamics and the Problems of Prediction

G.G. Malinetskii and S.P. Kurdyumov

Discussion at the RAS Presidium

This 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.
Everyone takes an interest in forecasts, both the specialist and the man on the street. Both wish to know the extent of reliability of complex engineering structures like nuclear power stations, aircraft, and ocean liners, or to receive timely information about an approaching tsunami or an impending earthquake. In recent years, thanks to advances in nonlinear dynamics, risk management theory, and self-organized criticality theory, some fundamental limitations of forecasts were identified. These limitations were discussed at a RAS presidium meeting held at the end of 2000. The meeting's deliberations formed the basis of the paper and discussion below.
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.



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 previous results.
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 хаосом.
Любая динамическая система определяет в фазовом пространстве траекторию, например X(t).
Динамический хаос обусловлен тем, что соседние траектории удаляются от нее. Из-за этого малые причины могут иметь большие следствия

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 field.
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 [1]:

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 horizon.
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 [2, 3].

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 прогноза

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., our pendulum.
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 [6].

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 [4]; б - зависимость от времени логарифма концентрации ионов хлора в родниках перед катастрофическим землетрясением в Кобе (Япония) в 1995 г. [5]. 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 [3]. 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 tribulations.
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 и катастроф

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 project.
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 1950s.
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 [9]:

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 greatly enlarged.
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 значительно увеличены

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 [8].
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 (Fig. 9).

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 criticality
The more general a regularity, the easier it is to formulate.
Petr Kapitsa



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, 11].
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 [12]. 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.
Goethe, Faust

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 chance.
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 [13].
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 experiment.


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 yet unknown.
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 [14]. 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 [8]. 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 greatly improved.
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 [15] can be corroborated and rationalized in quantitative terms at another, deeper level.

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 [16]. Therefore, the question of the new development of resources became a vitally important one [14]. 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 strategies; and
- 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 [18]. The development of such a course should be the central task for the nation's research community and leadership alike.
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" [19]. 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 stability.
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 foresaw.

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. 130-141.
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, 2000.
14. Kapitsa, S.P, Kurdyumov, S.P., and Malinetskii, G.G., Sinergetika i prognozy biidushchego (Synergetics and Forecasts of the Future), Moscow: Nauka, 1997.
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, no. 3.

Discussion at the RAS Presidium

This 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 forecast horizon.
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 for Russia.
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 of this.
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 events?
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 to forecasts.
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 prognostication.
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 seriously.
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 chaos.
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 like.
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 prevention.
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 Kurdyumov.
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.

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