Synergistic economy. Time and Change in Nonlinear Economics - V.B. Zang

Synergetic Economics

time and Change in Nonlinear Economics

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V.- B. Zang

Synergistic

ECONOMY

Time and change in the non-linear economic theory

Translation from English by N. V. Ostrovskaya, edited by

V. V. Lebedeva and V. N. Razzhevaikin

MOSCOW "MIR" 1999

BBC 16.22.9 З27

V.- B. Zang

Z27 Synergetic economy. Time and change in non-linear economic theory: Per. from English. - M.: Mir 1999. -335 p., ill.

ISBN 5-03-003304-1

The Chinese economist's book was written during his tenure at the Swedish Institute for Advanced Study and was published in 1991 in the famous Springer Literature Series on Synergetics, edited by Hermann Haken. The book uses the modern mathematical apparatus of nonlinear analysis for macro problems. economic dynamics.

It will be useful for specialists in the field of macroeconomics, applied mathematicians, graduate students and students economic universities.

BBC 16.22.9

The publication was supported by the Russian Foundation fundamental research project

Editorial Board of Literature in Mathematical Sciences

Originally published in English under the title:

"Synergetic Economics" by Wei-Bin Zhang.

Copyright © Springer-Verlag Berlin Heidelberg

1991. All Rights Reserved. © translation into Russian, "Mir",

ISBN 5-03003304-1 (Russian)

ISBN 0-387-52904

5.8 Optimal economic growth associated with endogenous fluctuations 142

5.10 Competitive business cycles in an economy with overlapping

7.5. Effect of noise on the trajectories of nonlinear stochastic systems near

7.6. Exposure to random external factors to a second order system

8.1 Spatially continuous economics and process description

8.3 Economic cycles in the spatial model "multiplier-

10.2 Connection of the synergetic economy with the traditional theory of economic dynamics | 297

10.3 Competitive and planned economy from the point of view of synergetic economy 303

10.4 Developed and developing models of the economy from the point of view of a synergistic economy 306

Foreword by the Translation Editors

All knowledge is only bringing the essence of life under the laws of reason.

Leo Tolstoy, War and Peace

characteristic feature modern stage development of economic science is its mathematization, which is manifested in the replacement of the studied economic process with an adequate mathematical model and the subsequent study of the properties of this model either by analytical methods or on the basis of computational experiments. Usage mathematical models in economics has more than a century of history. For example, one of the first models of market competition (O. Cournot) was published in 1838, and half a century later, L. Walras already used mathematical models when teaching a course political economy at the University of Lausanne. To date, various models of interaction between labor markets, goods and money markets, models of single-product and multi-product firms, a model of consumer behavior, a model of firm competition in the goods market, and others, which, in essence, are equilibrium models, have firmly entrenched in economic theory.

However, the vast majority of economic processes take place in time, as a result of which the corresponding mathematical models are, in principle, dynamic. One of the traditional approaches to predicting the development of economic processes is to study the shift in the equilibrium point of a dynamic system caused by a change in certain parameters of the model. This (quasi-stationary) approach relies on the key concept classical political economy- the "invisible hand" of Adam Smith. As is known, this concept is based on the hypothesis of the existence of an automatic equilibrium mechanism in competitive markets.

The use of a quasi-stationary approach to the analysis of the dynamic processes of the economy has led to the spread of the general

the idea that the development of any complex system can be viewed as a change from one stable state to another with a short period of transition between them. However, an analysis of real economic dynamics based on this approach may turn out to be erroneous, since the period of non-equilibrium development of many economic processes may turn out to be too long to be neglected. Perfectly understanding the importance of studying economic processes in dynamics, the classic of modern economic science A. Marshall justified the use of a quasi-stationary approach to assess changes in the market by the fact that "our analysis is still in its infancy."

Note that this approach is effective only for the time being, until, for some reason, the nature of the stationary state does not change radically. Such changes, called bifurcations, already belong to the field of application of methods of nonlinear dynamic analysis, the development of which leads to the growing spread of this point of view: "The world is a constant development, eternal instability, and periods of stabilization are only brief stops along the way."

Dynamic mathematical models, which have proven themselves in physics and then in biology, have much in common, although they retain the specific features of each of these sciences. Now models of this class are increasingly used in sociology and economics. To date, the modern methodology for the analysis of nonlinear dynamic systems has taken shape in a new scientific direction called synergy. This interdisciplinary science aims to identify general principles evolution and self-organization of complex systems in various fields of knowledge based on the construction and study of nonlinear dynamic mathematical models. Important concepts of synergetics are "catastrophe", "bifurcation", "limit cycle", "strange attractor", "dissipative structure", "traveling wave", etc. Arising from the use of relatively simple nonlinear models, these concepts allow us to penetrate deeper in the essence of many processes and phenomena. Physics, chemistry, and biology abound in examples of the successful application of this methodology. These include phase transitions between aggregate states of matter, turbulent fluid flows, structures in media in the presence of autocatalytic reactions, life waves and combustion waves, fluctuations in the number of natural populations, etc.

It is not surprising that this universal methodology, which has arisen relatively recently and has proven itself in the natural sciences, began to penetrate into the traditional humanities, and

primarily to the economy. Without fear of making a mistake, it can be argued that any branch of economic science can be attributed to the field of applications of synergetics, since when considering any dynamic economic process, some active, i.e., feedback element is always present as an acting factor. Therefore, if we want to look beyond the horizon of a narrow world in which everything seems to be stable and in which there is no place for catastrophes and restructuring, we cannot do without using a synergistic approach.

In the book offered to the attention of readers by V.-B. Zang "Synergetic Economy" an attempt is made to give a general idea of ​​the possibilities of a synergistic approach in the economy. At the same time, the main attention is paid to the consideration of relatively simple mathematical models of small dimensions, which, as a rule, can be investigated by analytical methods. The use of synergetic methods in the economy is not a tribute to fashion, but an urgent need to move forward beyond the limits outlined by the quasi-stationary approach, to look for new ways to use powerful modern computing tools to solve serious problems. practical tasks.

The mathematical toolkit of the book is a fairly compact set of methods that make it possible to conduct a very effective analysis of nonlinear models of real economic processes. The undoubted advantage of the approach used is that the analysis of the low-dimensional models discussed in the book is easy to comprehend, since the set of properties that are the most striking consequences of nonlinearity is rather limited. Therefore, the mathematical apparatus used in the book should become not only the alphabet for a new generation of economists, but at the same time a beacon to which the mathematical training programs of economic universities should be tuned. Apparently, it is in connection with this that V.-B. Zang recommends his book not only to specialists, but also to students. economic specialties.

The scale of the task that the author set himself did not allow him to avoid some shortcomings. This concerns, first of all, the excessive conciseness of the presentation of fundamental hypotheses in the formulation of mathematical models, which, unfortunately, is inherent not only in this, but also in many other books on mathematical economics. It can be noted that; what economic models often serve as illustrations of well-known mathematical results in the book. This places the models under consideration in a subordinate position in relation to the mathematical apparatus, which, of course, cannot but cause some feeling of dissatisfaction among economist readers. However

in As a result of this approach of the author to the presentation of the material, the reader discovers, for example, that economic cycles are also natural, like fluctuations in the number of populations, and "jumps" in society, i.e., changes of a revolutionary type, are like phase transitions for matter. So, this can be regarded as a deliberate methodological approach in the presentation of material, which forces readers to delve more carefully into those stingy lines that set out the main hypotheses and mathematical constructions of the models, and to show maximum independence in understanding not only the results presented, but also mathematical formulation of the problem.

To some subjective assessments (and self-assessments) of the author should be treated quite critically by the reader. For example, speaking about Haken's subordination principle, it is impossible not to mention another formulation of this principle - Tikhonov's theorem for systems of equations with singular perturbations. And in general, speaking of synergetics, it should be remembered that many of its results are directly related to the development of mathematical modeling, at the origins of which in our country were A. A. Dorodnitsyn, N. N. Moiseev, A. A. Samarsky and others (for For the convenience of readers, we provide at the end of this preface a small list of literature in Russian on this topic).

At the same time, we would like to draw the attention of readers to the main advantage of the book: on the whole, the author managed to give a broad panorama of the state of affairs

in today's synergetics on the example of the analysis of relatively simple models of dynamic economic processes. Moreover, the book is aimed at developing a non-linear style of thinking among readers, which is important in any field of knowledge, including, of course, in modern economy.

When working on the translation manuscript, we corrected the observed inaccuracies of the original without any special reservations, and where necessary, made footnotes. It should be especially noted that the publication of the book in Russian was carried out thanks to the initiative of the translator of the book, N.V. Ostrovskaya, who supported her initiative. Russian fund fundamental research (head of the publishing department V. D. Novikov), employees of the Mir publishing house, as well as A. V. Fedotov, who took part in the translation of chapters 5 and 9.

We would also like to express our gratitude to the author of the book, Prof. V.-B. Zang for his attention to the Russian edition - he kindly sent, at our request, a list of typographical errors, which was taken into account in the Russian edition, and also answered a number of questions regarding the clarification of certain places in the text. In conclusion, we express the hope that the book will be useful to all readers interested in applications of nonlinear analysis methods in economics. Who knows, maybe among them will be those who, with its help, will find the very thread, unraveling which, it will be possible to get to a clear synergistic picture of the economic problems that we are all experiencing today and, having this picture in front of us, find real ways to decent economic development.

Genre: Economy

Publisher:"World"

Format: PDF

Quality: OCR

Number of pages: 335

Description: In the book offered to the attention of readers by V.-B. Zang "Synergetic Economy" an attempt is made to give a general idea of ​​the possibilities of a synergistic approach in the economy. At the same time, the main attention is paid to the consideration of relatively simple mathematical models of small dimensions, which, as a rule, can be investigated by analytical methods. The use of synergetic methods in the economy is not a tribute to fashion, but an urgent need to move forward beyond the limits outlined by the quasi-stationary approach, to look for new ways to use powerful modern computing tools to solve serious practical problems.
The mathematical toolkit of the book is a fairly compact set of methods that make it possible to conduct a very effective analysis of nonlinear models of real economic processes. The undoubted advantage of the approach used is that the analysis of the low-dimensional models discussed in the book is easy to comprehend, since the set of properties that are the most striking consequences of nonlinearity is rather limited. Therefore, the mathematical apparatus used in the book should become not only the alphabet for a new generation of economists, but at the same time a beacon to which the mathematical training programs of economic universities should be tuned. Apparently, it is in connection with this that V.-B. Zang recommends his book not only to specialists, but also to students of economic specialties. The scale of the task that the author set himself did not allow him to avoid some shortcomings. This concerns, first of all, the excessive conciseness of the presentation of fundamental hypotheses in the formulation of mathematical models, which, unfortunately, is inherent not only in this, but also in many other books on mathematical economics. It can be noted that; that the economic models in the book often serve as illustrations of well-known mathematical results. This places the models under consideration in a subordinate position in relation to the mathematical apparatus, which, of course, cannot but cause some feeling of dissatisfaction among economist readers. However, as a result of this approach of the author to the presentation of the material, the reader discovers, for example, that economic cycles are as natural as population fluctuations, and "leaps" in society, that is, changes of a revolutionary type, are like phase transitions for matter. So, this can be regarded as a deliberate methodological approach in the presentation of material, which forces readers to delve more carefully into those stingy lines that set out the main hypotheses and mathematical constructions of the models, and to show maximum independence in understanding not only the results presented, but also the mathematical task setting.
The reader should be quite critical of some subjective assessments (and self-assessments) of the author. For example, speaking about Haken's subordination principle, it is impossible not to mention another formulation of this principle - Tikhonov's theorem for systems of equations with singular perturbations. And in general, speaking of synergetics, it should be remembered that many of its results are directly related to the development of mathematical modeling, at the origins of which in our country were A. A. Dorodnitsyn, N. N. Moiseev, A. A. Samarsky and others (for For the convenience of readers, we provide at the end of this preface a small list of literature in Russian on this topic).
At the same time, we would like to draw the reader's attention to the main advantage of the book: on the whole, the author managed to give a broad panorama of the state of affairs in today's synergetics using the analysis of relatively simple models of dynamic economic processes as an example. Moreover, the book is aimed at developing a non-linear style of thinking among readers, which is important in any field of knowledge, including, of course, in the modern economy.
When working on the translation manuscript, we corrected the observed inaccuracies of the original without any special reservations, and where necessary, made footnotes. It should be especially noted that the publication of the book in Russian was carried out thanks to the initiative of the translator of the book N.V. Ostrovskaya, who supported her initiative to the Russian Foundation for Basic Research (head of the publishing department V.D. Novikov), employees of the Mir publishing house, as well as A.V. Fedotov, who took part in the translation of chapters 5 and 9.
We would also like to express our gratitude to the author of the book, Prof. V.-B. Zang for his attention to the Russian edition - he kindly sent, at our request, a list of typographical errors, which was taken into account in the Russian edition, and also answered a number of questions regarding the clarification of certain places in the text. In conclusion, we express the hope that the book will be useful to all readers interested in applications of nonlinear analysis methods in economics. Who knows, maybe among them will be those who, with its help, will find the very thread, unraveling which, it will be possible to get to a clear synergistic picture of the economic problems that we are all experiencing today and, having this picture in front of us, find real ways to decent economic development. Contents of a book
«Synergetic economy »

Time and change in economic theory

1. Economic evolution. Introduction
2. Equilibrium theories in economic analysis
3. Dynamic theories in economics
4. Samuelson's correspondence principle and its limitations
5. Volatility in economic analysis

1. Dynamics and balance
2. Classification of second-order differential systems
3. The principle of stability in linear approximation
4. Direct Lyapunov method
5. Structural stability
6. conservative systems
7. Bifurcation theory
8. Singularity theory
9. Catastrophe theory

1. Catastrophe theory and comparative static analysis
2. Modeling regional dynamics
3. Some examples of structural changes
   • Business cycles in the Kaldor model
   • Resource management
   • Dynamic mode selection and bifurcation
   • Sets of equilibria in the Wilson retail model
4. Bifurcation analysis of the economic growth model
5. Singularity theory in economic analysis

1. Theories of business cycles
2. Some mathematical results of the theory of limit cycles
   • The Poincaré-Bendixon theorem and its applications to economics
   • Hopf's bifurcation theorem
3. Keynes' simplified business cycle model
4. The nature of disequilibrium in a model without equilibria
5. Monetary cycles in the generalized Tobin model
6. Oscillations in Van der Plueg's Hybrid Growth Model
7. Optimal Periodic Employment Policy
8. Optimal economic growth associated with endogenous fluctuations
9. Remarks on possible subsequent bifurcations of limit cycles
10. Competitive Business Cycles in an Overlapping Generation Economy - Discrete Model

1. Chaos in deterministic systems
2. Economic chaos in a discrete system
3. Aperiodic optimal economic growth
4. Urban dynamics - Lorenz system
5. Chaos in the model of the international economy
6. Chaos and economic forecasting
   • Lyapunov exponents of differential equations
   • Lyapunov exponents for discrete mappings
   • Signal, Power Spectrum, Autocorrelation Function, and Poincaré Mapping

1. Random Processes and Economic Evolution
2. Stochastic processes. Introduction
   • Some concepts of probability theory
   • Stochastic processes
3. Birth-Death Processes and the Master Equation
4. Schumpeter's Non-Equilibrium Clock Model
5. Influence of noise on the trajectories of nonlinear stochastic systems near singular points
6. Impact of random external factors on a second-order system in a neighborhood of singular points

1. Spatially continuous economy and description of the process of city formation
2. The role of structural sustainability in a two-dimensional economy
3. Economic cycles in the spatial model "multiplier-accelerator" Puu
4. Spatial diffusion as a stabilizer
5. Separation and coexistence of heterogeneous groups of the population of the city
6. Urban formations such as traveling waves
7. Instabilities and city building
   • Model of morphogenesis
   • Brusselsator

1. Haken's principle of submission
2. Center manifold theorem
3. Singular perturbations
4. Relationship between fast and slow variables in economic analysis
5. Time scale in economic analysis
6. Human dynamics. Attempt to comprehend

1. Synergetic economy and its connection with synergetics
2. The connection of the synergetic economy with the traditional theory of economic dynamics
3. Competitive and planned economy from the perspective of a synergistic economy
4. Developed and developing models of the economy from the point of view of a synergistic economy
5. Chance and Necessity in Economic Life
6. The role of political decision in a chaotic world
7. Correlation between micro- and macroeconomics

Federal State Educational Institution of Higher Professional Education
"RUSSIAN ACADEMY OF CIVIL SERVICE UNDER THE PRESIDENT OF THE RUSSIAN FEDERATION"
VLADIMIR BRANCH

Department:
INFORMATION TECHNOLOGIES
TEST
on the course: "Concepts of modern natural science"
on the topic: "Synergetic approach in economic research»

Completed:
Morozova A.S.
extramural studies
1 course, group SPYu-211
Specialty "Jurisprudence"
Lecturer: Avdonina A.M.

Vladimir 2012
Work plan.
Introduction…………………………………………………………………………...3
Chapter 1. Synergetics as a new research method in economics…………...6
Chapter 2. Synergetic approach in economic research… ……...13
2.1. Fundamentals of synergetic research of economic systems..13
2.2. Types of sustainability of economic systems……………………….15
Conclusion………………………………………………………………………….21
List of used literature…………………………………………... 23

Introduction.

The functioning of any system, the economic system is no exception, is expressed through events, processes and phenomena that determine the outcome, a certain result. The result, in turn, has an estimated indicator - a negative or positive assessment.
Events, processes and phenomena can manifest themselves explicitly or implicitly. It should be emphasized that the probabilistic nature of events is their objective property, and not the result of observation over them. Each event and phenomenon has its own form of manifestation. Any events, regardless of someone's will, may or may not manifest themselves clearly. The cause of various forms of manifestation is an irritant, i.e. Circumstances are their own cumulative irritant.
In addition to a certain form of manifestation, there is an implicit (indefinite) form. Events will necessarily occur under a confluence of certain circumstances - a certain form of manifestation. Or the events will not happen (an indefinite form of manifestation), in this case, one can speak either about the absence of circumstances at all, or about the combination of not all circumstances. Therefore, both the forms and the causes of such manifestations must be evaluated. The very nature of the "coincidence" is no less interesting in the study of economic systems.
Synergetics is called upon to teach to recognize, predict, anticipate fluctuations. Synergetics puts forward a different concept of uncertainty, the concept of disequilibrium is introduced, which exists implicitly and potentially, but manifests itself in the appearance of ordered structures in the form of ordering and disordering. Disequilibrium is considered as a kind of "cauldron of circumstances" that are the source of the event (events).
Synergetics is a direction of scientific research through the knowledge of general patterns and principles underlying the processes of self-organization of systems of various nature. Under the object of research, synergetics considers systems (including economic systems - enterprises) that are potentially in states far from equilibrium, near special critical points (bifurcation points), where, under the influence of the most insignificant circumstances, the system can dramatically change its state. The subject of research in synergetics is the behavior of the system when approaching and passing through bifurcation points. Synergetics is intended to explain the reasons for the behavior of a system near bifurcation points. Critical points (bifurcation points) are such a state when the behavior of the system becomes unstable or unpredictable. Instability can manifest itself transcendentally under the influence of minor deviations, when the economic system changes its state abruptly. The state of the system after passing through the bifurcation point can be assessed as either anergic or rheotropic.
At the bifurcation point, there is an impact on the internal and external relations of the economic system. Connections are either broken or restored, the system degrades or adapts to new conditions of existence. The system can also acquire new qualities through a leap.
The combination of elements of the old and new quality at the bifurcation point creates a non-equilibrium state, i.e. at the bifurcation point, the goal, objectives and structure of the economic system can change. In the course of transformations in the system, adaptation to new conditions of existence can occur, the mechanism of the functioning of the economic system changes, or the system degrades, i.e. does not reach a new level and, as a result, the system stagnates. A symptom of the system's preparation for a jump is changes in information links, both external and internal.
In some cases, the goals of the system begin to dominate the goals of the external environment, i.e. the system subjugates the larger system. Thus, a monopolist is formed in the state economy, a monopoly enterprise forms the conditions for the functioning of its key markets (labor market, sales market, stock market).
The very fact of preparing the system for a jump does not yet determine the result; it is necessary to know the state of the system before the jump, the moment of transition from quantity to quality.
Weakening or restoration of what qualities of the economic system will occur after passing the bifurcation points? How will the consequences of relaxation affect the state of the economic system? Possible risks of passing bifurcation points. These are far from idle questions that need to be answered by the management company, investor and shareholder. The task of the analyst is to assist users of information on these issues.
Today, it is no longer enough for an analyst-economist to operate with mathematical methods, he needs to master philosophical categories. Economics, as a science, in addition to its own mathematical methods, already needs its own philosophy, economic philosophy, in order to understand specific causes and effects. A specific philosophy is needed to avoid epithets such as "the fatality of the system", "fatal circumstances", etc. in economics.

Chapter 1. Synergetics as a new research method in economics.

For those who are still unfamiliar with the concept of synergetics and its ideas, it will be enough to present such a definition of synergetics (from the Greek synergetikos) - joint, coordinated, acting.
A more complete definition is synergetics (from the Greek syn - "together" and ergon - "action, effort"): a modern term introduced by representatives of the natural sciences to denote non-linear and other relationships that cannot be described exhaustively in terms of rational science, mainly the principle self-organization of natural, social and cognitive (ideological) systems. Constructive postulates of the theory of synergetics: the principle of productivity of chaos, recognition of the presence of branching points (bifurcations), the principle of resonant impact, etc.
In fact, the above definitions do not reveal the essence of synergetics in the context in which it is presented in the scientific literature.
Different sources under synergy are:
1. phenomena that arise from the combined action of several different factors, while each factor separately does not lead to this phenomenon;
2. the science of self-organization, which means spontaneous complication of the structure of the system. Spontaneously arising formations are also united under the general name - dissipative structures (the term was proposed by I.R. Prigogine);
3. the science of unexpected phenomena. We can say that any qualitative change in the state of the system (or mode of its operation) gives the impression of an unexpected;
4. a scientific direction that studies the connections between structural elements (subsystems) that are formed in open systems (biological, physico-chemical, and others) due to the intensive (flow) exchange of matter and energy with the environment in non-equilibrium conditions.
Most of all, synergetics is a scientific direction that is called upon to play the role of a metascience that allows private sciences to explain phenomena and their manifestations that go beyond their competence. Synergetics is necessary not on the external conjugation of sciences, but when combining the subject area of ​​particular sciences, through the development of the necessary research method. At the same time, the developed method does not form the basis of the methodology of synergetics, but is considered as a kind of non-universal tool for studying individual systems.
The term "synergetics" was introduced by G. Haken, who is engaged in research on the theory of lasers and nonequilibrium phase transitions. G. Haken admits that for the scientific explanation of some processes and phenomena, research is needed at the interscientific and interdisciplinary level, i.e. X-science is needed. The modern meaning of the term synergetics was formed historically. G. Haken's predecessors spoke only about particular examples, for example, I. Zabuski's conclusion about the need for a unified approach to nonlinear mathematical and physical problems.
G. Haken himself stated that the naming of the scientific direction he proposed was accidental and unprincipled, he writes: "" It is useful to have some suitable definition of self-organization ...". Nevertheless, G. Haken's synergetics was not ignored and found application in the study of various sciences.
G. Haken proposed the term "synergetics" to study the principles of self-organization of systems, regardless of their nature, for the most part it referred to open systems. According to Haken, synergetics is the study of systems consisting of a large number of elements, components or subsystems. Only a conceptual designation of the direction of scientific research, i.e. the lack of regulations, a philosophical definition of science, fixed such a name for X-science.
The viability of the term was predetermined initially for several reasons: firstly, the question of both interscientific and interdisciplinary research methods is ripe (it is obvious that the methodologies of scientific knowledge are different, so the implementation of such a project is possible only at a conceptual level); secondly, the pressure of the community, representing classical science on the old paradigms, caused adequate opposition, new paradigms in the sciences have always been formed despite opposition from the outside; thirdly, the name is lapidary; fourthly, the scientific community adopted the term due to the direct association of the term "synergetics" with self-organization.
The philosophical and applied concept of structure in systems assumes the existence of processes of self-organization through ordering and disordering. The state of the system at the time of its transformation is estimated by the stability of the system (both in minus and in plus). The transition points of the system from one state to another were called critical points or bifurcations.
open systems are characterized by the fact that in the process of their functioning there is an accumulation of energy within the system and a constant exchange of flows of matter or energy with the environment. The results of the system's activity, the internal forces of the system itself and the external forces acting on the system are directly in constant dynamics. Indirectly has alternatives and self-organization of the system, through the causes and consequences of the action of forces.
The state of the economic system is characterized by two qualities: firstly, the ability to relax, and secondly, the level of entropy condensation. To maintain a stationary state, the system needs constant energy, which is necessary for continuous ordering through the jump. If the system is linear and iterative, it can live indefinitely, i.e. life cycle is permanent. In a nonlinear system, after a certain number of iterations, irreversible processes begin. Because there are no ideal linear systems, discreteness will definitely manifest itself in the end. Permanence and discreteness are mandatory conditions that determine life cycle economic system.
The essence of an economic or socio-economic system is constant ordering, through disordering (a kind of iteration). The ordering process involves a range of possibilities that determines the purpose of the system at a particular moment.
The main sign of the onset of disorder is an ever-increasing increase in the number of bifurcations in the behavior of the system.
Synergetics considers any state vector of the system as a parameter characterizing the system. And there can be many such vectors, depending on the objectives of the study, the key parameters of the system are determined, which are monitored. The state vector is opposed by entropy, both concepts are causal.
The validity of the synergetic approach to the study of systems is confirmed by the fact that between the behavior various systems studied by various private sciences, there are analogies (biological, economic, and other systems).
A biological system, such as a person, obeys its own laws of functioning. It would be wrong to give the economic system a certain image of intellectuality, we are talking about the fact that the economic system as an artificial formation created by man for certain purposes is subject to certain economic laws. Therefore, when defining the synergy effect in order to consider economic systems, it is first of all necessary to approach this task based on the dynamic characteristics of the system, and not as a stationary formation.
Any system is constantly evolving (changing and adapting), and any dynamics does not imply equilibrium. Therefore, the assertion that the process of self-organization is aimed at achieving a state of equilibrium is inappropriate. If we consider the achievement of a state of equilibrium as an end in itself, then this is a dead end, then only degradation. It is advisable to consider the fact of achieving equilibrium as the moment of preparation for the jump, the transition to a new level of self-organization, which is the system's goal in itself. To describe the self-organization of systems, it is advisable to use the Darwinian theory of evolution: heredity, variability, selection.
Heredity means that there is some starting potential inherent in the system. The starting potential is the genetic memory of the system (absolute heredity, unconditioned reflex).
In the process of functioning, there is an accumulation of negative and positive experience, which in turn determine the acquired qualities, a conditioned reflex. The ability to gain experience characterizes the variability of the system, the development of reactions to various stimuli (randomness and uncertainty).
Heredity and variability indicate the presence of memory in the system: long (hereditary) and short (acquired) memory.
The concept of "selection" or natural selection when considering complex systems, including such as economic and socio-economic systems, indicates the viability of the system. Self-organization is subject to the action of laws specific to the system. To the number general laws the law of self-preservation applies, the specific laws of the development of economic systems are the law of competition, the law of increasing costs, the law of decreasing profitability, etc. The laws are correctly called the principles of selection of the external environment. A system that meets the requirements of the law is able to function in a given external environment, otherwise the discrepancy between the goal of the system and the external environment may turn out to be a factor in increasing disorganization within the system, turning into a crisis, an increase in irreversible changes in the structure of the system.
The concept of self-organization of systems is closely linked to relation and reflection. Attitude is the interconnection of phenomena and manifestations (external and internal relationships). Reflection is the presence of an independent force of response to stimuli.
The presence of relation and reflection indicate the presence of equilibrium in the system. The equilibrium of the system assumes that the system is able to adequately respond to stimuli: counteract external influences and extinguish internal contradictions.
In the absence of an adequate response to stimuli, the system first loses stability, then balance. After the loss of stability, the balance is lost instantly. In this case, the system occupies a new equilibrium position.
The evolution of the system is impossible in the absence of stimuli to the system, as well as the potential to adequately respond to stimuli. This statement allows us to conclude that the evolutionary process implies a staged development of the system. First, the properties of the system develop due to the genetic potential. The genetic potential, in turn, determines the inertia of the system in all its subsequent developments. One of the main functions of the genetic potential is to inhibit inversion. At this stage of development, the system organizes the structure, internal connections between subsystems, external connections with the external environment. Further, experience is accumulated, mechanisms of interaction with the external environment are developed. At this stage, changes in the structure of the system may occur, connections of the system may break or appear. The result of the stage is the accumulation of the potential of the system. At some point, the action of stimuli can be critical for the system. Depending on the results of the passage of the previous stages, the reaction of the system may be different; in any case, the system enters the bifurcation stage. If the system is capable of cumulating potential, the system is preparing for a jump, otherwise the system degrades, and its stagnation is a matter of time. The degradation of the system is the loss of energy, the transition to an inertial state and, as a result, disorder and destruction.
A random change in the structure of the system leads to its mutation, i.e. it continues to be reproduced in a new form. Mutation caused by random changes is characteristic of biological systems and is not characteristic of economic systems. The main difference between living nature and inanimate nature is the presence of the function of producing their own kind.

Bifurcation, despite the small time frame of the process, is accompanied by a large release of energy, under the influence of the law of self-preservation. Bifurcation can be called the engine of evolution, the engine is embedded in the system itself, in its long memory. All together define the process of self-organization as a continuous transformation of the system in order to survive, to acquire new properties. To survive, all systems go in the direction of complicating the organization through self-organization. Self-organization is understood as a purposeful process during which complex systems are created, reproduced or improved. For economic systems, their transformation into mega-systems, corporations is natural.

Chapter 2. Synergetic approach in economic research.

2.1. Fundamentals of synergetic research of economic systems.

Traditional research approaches practically do not consider the potential of the enterprise as a whole, there are only some elements that allow obtaining individual characteristics of financial and economic activity. The definition of financial stability is presented in the works of Russian scientists in the field of analysis - Gilyarovskaya L.T., Dontsova L.V., Netetsky V.V., Sheremet A.D., L.V., such tools for assessing the performance of an enterprise: the coefficient of autonomy (financial independence), financial dependence ratio, financing ratio, financial stability ratio, investment ratio, maneuverability ratio, financial leverage (leverage) and some others.
We should agree with Dontsova L.V. which states: “However, the bankruptcy of many Russian organizations is primarily associated with their involvement in the system of non-payments, due to the influence of external, practically uncontrollable factors. factors". To this it should be added that for successful forecasting, an important factor is the assessment of transformations both in the economic system and in the external environment that occurred after their interactions.
Traditional methods of analysis considered the economic system according to the laws of dialectics. Synergetics does not deny the laws of dialectics, but considers the regularities and principles of self-organization as the processes of the functioning of the system, in states far from equilibrium - when approaching bifurcation points or passing through them.
Such a scientific direction in research as synergetics allows you to take a fresh look at the processes taking place in the economic system. Synergetics presents self-organization of a system in the form of processes of the emergence of ordered spatio-temporal structures that are in states far from equilibrium, near special critical points (bifurcation points), where the behavior of the system becomes unstable. Instability can manifest itself under the influence of the most insignificant deviations, when the system or a key indicator characterizing the system can change its state dramatically. In the study of some economic indicator of the time series, the following components are distinguished, where:
- the trend component, indicating the influence of long-term factors, the trend function is determined based on the behavior of the indicator for previous points in time;
- seasonal component, reflects the seasonal pattern in the behavior of the studied economic indicator;
- the cyclical component, shows the patterns in economic development, constantly manifesting itself over long periods;
- a random component, indicates the presence of random and unaccounted for factors that do not appear outwardly, but influence the behavior of the indicator.
The random component cannot be revealed by dialectical methods of studying economic systems, since randomness is an infinite set of conditions under which an event is unpredictable. In contrast to chance, necessity is considered - this is a manifestation of an event that will definitely occur under certain circumstances.

2.2. Types of sustainability of economic systems.

In the theory of economic analysis, sustainability indicators are considered. The term "stability" in the literal interpretation - not subject to fluctuations. Borrowed from mechanics, stability in relation to economic systems implies the ability of an economic entity to follow a given direction in accordance with statutory requirements, tactical and strategic tasks. Stability, participating in the evaluation of any system, economic is no exception, is a dynamic performance.
Introduced to justify the synergetic approach to studying economic systems, the concept of stability is built on indicators that allow assessing the potential of an enterprise out of motion. Stability considers the ability of an enterprise to restore financial and economic activity, despite the negative impact of external and internal factors. The possibility of regeneration of the information links that form this system determines the relaxation stability or the stability of the state of the economic system, which is determined by the results of an assessment of dynamic and static stability. Consideration of all factors affecting the "stability" of the enterprise, the subject of additional research.
If stability is the ability to follow a given direction, then stability is a property of an object that provides resistance to various (external and internal) factors, in relation to an enterprise, such as the "tax field" (national tax policy), technological features of production, features of the raw materials market, features of the sales market , government business regulation policy, force majeure actions. Stability as an evaluation criterion involves the use of static characteristics.
Unlike dynamic stability, the assessment of the static characteristics of stability (stability) is primarily an analysis of external and internal factors that have an impact on the financial and economic activities of an enterprise. Assessment of the consequences of the influence of external and internal factors is an assessment of static characteristics, i.e. the economic system is considered "out of motion".
When considering stability, it is necessary to take into account the ability of the enterprise to maintain and restore the resulting financial and economic activity, despite the negative impact of external and internal factors, in some cases, as if breaking the internal and external connections of the system, and the possibility of regenerating the information links that form this system.
With many factors affecting the "stability" of an enterprise, the most important and positive factor that ensures a stable position for an economic entity should be considered the potential of the "niche" occupied by this enterprise in the market of identical offers. The potential of the "niche" of the sales market is, first of all, the demand for products and services rendered in the current and future terms. And therefore, the results of the analysis of the financial and economic activities of the enterprise should be adjusted with a significant adjustment for changes taking place in the business, i.e. taking into account the influence of the external environment.
We single out the main criteria for grouping dynamic and static indicators within the framework of a synergistic approach to research:
- dynamic include indicators that determine the state of the economic system at a particular moment;
- static indicators include indicators that consider the state of the system in isolation from its development (movement). In the proposed method, this is the potential of the system, which allows it to respond to various stimuli of the internal and external environment. Stability reveals characteristics that indicate the presence of potential (a set of means and capabilities) of the system.
Dynamic characteristics include:
1. indicators of economic activity:
- depreciable property;
- raw materials and materials;
- Money.
b). dynamics:
- cost;
- administrative and other expenses.
in). security:
- assets;
- own and borrowed funds.
2. indicators of financial activity:
a). condition and efficiency of use:
- own funds;
- borrowed money.
b). condition:
- mutual settlements with external debtors and creditors;
- mutual settlements with internal debtors and creditors.
in). dynamics:
- proceeds;
- other income.
The static features are:
1. system status indicators:
- dependence on investments and borrowings;
- security of production with means of labor;
- positioning in key markets (stock, sales, labor);
- dependence on counterparties.
2. system capacity:
- investment attractiveness of business;
- share of cash in current assets enterprises;
- profit and its dynamics;
- competitiveness of products;
- technological elasticity of production;
- the state of production assets.
Qualitative addition of methods economic analysis stability assessment methods will help avoid bankruptcy for enterprises with the potential to get out of a crisis, reduce the degree of risk for investors and more confidently assess the potential of an enterprise and predict results qualitatively financial and economic activities, as well as allow the development of an effective
etc.................

The use of natural science concepts, synergistic ideas and approaches allows in a new way look at such a complex area of ​​scientific and practical human activity as economics. Despite certain achievements of economic theories, economic forecasts very often do not correspond to the real development of the economy. To a large extent, this is due to the fact that the existing classical economic theory continues to remain in the block of the humanities. Built on disparate empirical facts, economic models of dynamic processes are based on linear concepts of reality. They satisfactorily reflect the dynamics of economic systems for narrow specific conditions and are not able to predict the ambiguous economic processes taking place in market economy. In particular, the model economic development, developed and positively tested for the economy of one specific country, leads to a completely different result after applying it to forecast the development of another country, which actually develops according to a different scenario, different from that predicted by the “foreign” model.

In his book "Synergetic Economy" V.B. Zang notes that "in pre-synergetic theories, the most important results in economic analysis have been obtained on the basis of the concept of an equilibrium mechanism." This approach is applicable to describe the dynamics of a system that is in a weak non-equilibrium state, when a system previously taken out of equilibrium slowly returns to equilibrium. The development of such a system can be viewed as a succession of rapidly changing stable non-equilibrium states. Analysis based on this concept is effective as long as the system remains linear. With an increase in the degree of non-equilibrium, all systems begin to show their important property - nonlinearity. The behavior of a nonlinear system is complex and ambiguous. Systems of any nature, physical, chemical, biological, economic, social, etc., including the most gigantic and complex of the Universe known to us, are non-linear.

Modern methods analysis of nonlinear dynamic systems took shape in a special scientific direction - synergetics. As already noted, synergetics studies the principles of evolution and self-organization of complex systems of various nature based on the construction of nonlinear models of the behavior of these systems. Dynamic models developed in natural sciences (physics, biology, etc.) for describing complex processes are now increasingly used in economics. Complex economic processes generated by the nonlinearity and instability of systems cannot be understood and predicted at the phenomenological level of classical economics. Such processes include, for example, economic cycles, economic crises, fluctuations in competition, pricing, urban development dynamics, international economy, and many other processes. The regularities of a number of such real phenomena of the modern economy can be established at the quantitative physical and mathematical level within the framework of synergetics. Therefore, the synergistic methodology of studying economic phenomena and processes allows you to gradually "transfer" economic science from the block of the humanities to the block of natural sciences. Such a task is solved by sections of synergetics - economic synergetics and synergetic economics.

At one time, the concepts of equilibrium and disequilibrium came to classical economics from natural science. AT last years From modern natural sciences, a stream of new terms and concepts, such as self-organization, nonlinearity, order parameters, chaos, entropy, bifurcation, catastrophe, limit cycle, phase space, dissipative structure, attractor, and many others, flows into the economy from modern natural science. The influence of natural science, and in particular physics, and the methods of physical and mathematical sciences on the development of the economy is also evidenced by the fact that out of 40 Nobel laureates in economics, almost all have a physical and mathematical education. Developed by the remarkable Soviet physicist Academician L.I. Mandelstam, "nonlinear physical thinking" begins to penetrate into economic science, and thus physics influences the formation of "nonlinear thinking" and the culture of mathematical thinking in modern economics, the development of "nonlinear intuition" among economists.

The main tool of a “nonlinearly thinking” specialist (physicist, chemist, economist, etc.) is the corresponding physical and mathematical models. Such models of systems describe whole classes of phenomena, united according to some attribute. Even the most successful model is not a copy of the real phenomenon, but only an expedient approximation. Mathematical models of economic processes are a system of nonlinear equations various types. Modern synergetic models are constructed by combining numerical and analytical methods. A synergistic approach to non-linear mathematical and physical problems can be defined as modern usage analysis and numerical machine mathematics to obtain solutions to reasonably posed questions regarding the mathematical and physical content of equations. The use of synergetic methods will allow the economy to go beyond the quasi-statistical approach and introduce the physical and mathematical language to solve real scientific and practical problems of economic development.

Springer Springer in Synergetics
Editor: Hermann Halcen
Wei Bin Zhang
Synergetic Economics
Time and Change in Nonlinear Economics
With 92 Figures
Springer-Verlag
Berlin Heidelberg New York London
Paris Tokyo Hong Kong Barcelona

AT .- B . Zang
Synergistic
ECONOMY
Time and Change in Nonlinear Economics
Translation from English
N. V. Ostrovskaya, edited by
V. V. Lebedeva and V. N. Razzhevaikin
MOSCOW "MIR" 1999

UDC 519.86
BBC 16.22.9
Z27
AT .- B . Zang
Z27 Synergetic economy. Time and change in non-linear economic theory: Per. from English. - M.: Mir 1999. -335 p., ill.
ISBN 5-03-003304-1
The Chinese economist's book was written during his tenure at the Swedish Institute for Advanced Study and was published in 1991 in the famous Springer Literature Series on Synergetics, edited by Hermann Haken. The book uses the modern mathematical apparatus of nonlinear analysis for problems of macroeconomic dynamics.
It will be useful for specialists in the field of macroeconomics, applied mathematicians, graduate students and students of economic universities.
BBC 16.22.9
The publication was supported by
Russian Foundation for Basic Research on the project
№97-06-87089
Editorial office of literature on mathematical sciences
Originally published in English under the title:
«Synergetic Economics»by
Wei Bin Zhang.
Copyright © Springer-Verlag Berlin Heidelberg
1991. All Rights Reserved.
© translation into Russian, "Mir",
1999
ISBN 5-03003304-1 ( Russian .)
ISBN 0-387-52904

CONTENT
1
In troduction .......................................................... ................................................. ............................................... 16 2
Time and changes in economic theory .............................................. ...................................... 24 2.1
Economic evolution. Introduction ................................................ ......................24 2.2
Equilibrium theories in economic analysis....................................................... ............25 2.3
Dynamic theories in economics .............................................................. ...............................27 2.4
The Samuelson correspondence principle and its limitations .........................................30 2.5
Instability in economic analysis .............................................................. ................32 3
ELEMENTS OF MATHEMATICAL THEORETICAL AND DYNAMIC SYSTEMS.... ...... 35 3.1
Dynamics and balance ............................................................... .............................................36 3.2
Classification of differential systems of the second order..............................................41 3.3
The principle of stability with respect to linear approximation.......................................................45 3.4
Lyapunov's direct method ............................................... ...............................................48 3.5
Structural stability .................................................................. ................................................52 3.6
Conservative systems .................................................................. .........................................56 3.7
Theory of bifurcations.................................................... ................................................. .60 3.8
Singularity theory .................................................................. ...................................................67 3.9
Catastrophe theory .................................................................. ................................................. .....72
Appendix: Some remarks on the theory of bifurcations.................................................... ...75 4
Equilibrium Sets and Structural Changes in Economic Systems .................................................. 78 4.1
Catastrophe theory and comparative static analysis.......................................................78 4.2
Modeling regional dynamics .................................................................. .................84 4.3
Some examples of structural changes .................................................................. ......................87 4.3.1
Business cycles in the Kaldor model .............................................. ..............87 4.3.2
Resource management................................................ ..............................89 4.3.3
Dynamic mode selection and bifurcation ..........................................91 4.3.4
Sets of equilibria in the Wilson model of retail trade.............92 4.4
Bifurcation Analysis of the Economic Growth Model.......................................................94 4.5
The Theory of Singularities in Economic Analysis....................................................... ......101 4.6
Remarks ................................................. ................................................. ..............103 5
Business cycles .................................................................. ................................................. ................. 104 5.1
Theories of economic cycles .............................................................. .................................104 5.2
Some Mathematical Results of the Theory of Limit Cycles....................................110 5.2.1
The Poincaré-Bendixson theorem and its applications to economics..........110 5.2.2
Hopf's bifurcation theorem.................................................... ...............114 5.3
Keynes' simplified business cycle model.................................................................... ............117 5.4
The nature of disequilibrium in the model without equilibria ..............................................122 5.5
Monetary cycles in the generalized Tobin model .............................................. ..126

5.6
Oscillations in Van der Plueg's Hybrid Growth Model..................................................................133 5.7
Optimal Periodic Employment Policy .................................................................. ..138 5.8
Optimal economic growth associated with endogenous fluctuations 142 5.9
Remarks on possible subsequent bifurcations of limit cycles ......... 145 5.10
Competitive business cycles in an economy with overlapping generations - a discrete model .............................................................. ...................................................149 6
Economic chaos in deterministic systems.................................................................. ................... 155 6.1
Chaos in deterministic systems............................................................... ......................155 6.2
Economic Chaos in a Discrete System....................................................... .............158 6.3
Aperiodic Optimal Economic Growth...............................................................166 6.4
Dynamics of cities - Lorentz system .............................................. ....................169 6.5
Chaos in the model of the international economy .............................................. ..............174 6.6
Chaos and economic forecasting .............................................................. .................176 6.7
Remarks ................................................. ................................................. ..............180
Appendix: Some Criteria for the Classification of Attractors....................................................180 6.7.1
Lyapunov Exponents of Differential Equations...............................181 6.7.2
Lyapunov Exponents for Discrete Mappings....................................182 6.7.3
Signal, power spectrum, autocorrelation function and display
Poincare 184 7
Stochastic processes and economic evolution .............................................................. .................. 187 7.1.
Random Processes and Economic Evolution...............................................187 7.2 .
Stochastic processes. Introduction ................................................ ...............190 7.2.1.
Some Concepts of Probability Theory...............................................191 7.2 .2.
Stochastic processes .................................................................. .........................193 7.3.
Birth-Death Processes and the Master Equation .......................................................... 197 7.4.
Schumpeter's Non-Equilibrium Clock Model............................................................... ............203 7.5.
Influence of noise on the trajectories of nonlinear stochastic systems near singular points.................................................................................. ................................................. ....................................213 7.6.
The impact of random external factors on a second-order system in the neighborhood of singular points.................................................................. ................................................. .................218 7.7.
Conclusions................................................. ................................................. .............222 8
Urbanization - sustainability, structural changes and chaos .................................................. 225 8.1
Spatially continuous economy and description of the process of city formation .............................................................. ................................................. .........................226 8.2
The Role of Structural Sustainability in a Two-Dimensional Economy .................................................231 8.3
Economic cycles in the spatial model " multiplier accelerator» Puu.................................................. ................................................. ....................239 8.4
Spatial diffusion as a stabilizer .............................................................. ......242 8.5
Separation and coexistence of heterogeneous groups of the population of the city .................... 245 8.6
Urbanistic formations of the type of traveling waves .............................................. ..252 8.7
Instability and city formation ............................................................... ......................256
Appendix: Structural Changes in the Two-Component Model.......................................257

8.7.1
Model of morphogenesis .............................................................. ..............................257 8.7.2
Brusselsator ............................................................ .........................................260 9
Haken's principle of subordination and the time scale in economic analysis .............................................. 268 9.1
Haken's principle of subordination ............................................................... .................................268 9.2
The center manifold theorem............................................................... .........................272 9.3
Singular perturbations .................................................................. ...............................................276 9.4
Relationship between fast and slow variables in economic analysis ..........280 9.5
Time scale in economic analysis .............................................................. ...........284 9.6
Human dynamics. Attempt to comprehend .............................................................. ............289
Application: Subordination principle for stochastic differential equations
.................................................................................................................................................... 291 10
Synergetic economy and its significance .............................................................. ......................................... 295 10.1
Synergetic economy and its connection with synergetics .................................................296 10.2
Connection of the synergetic economy with the traditional theory of economic dynamics | 297 10.3
Competitive and planned economy from the perspective of a synergistic economy 303 10.4
Developed and developing economic models in terms of synergistic economy 306 10.5
Chance and Necessity in Economic Life..................................................310 10.6
The role of political decision in a chaotic world..............................................................311 10.7
Correlation between micro- and macroeconomics...............................................313 eleven
Conclusions and prospects for further research ............................................................... ....................... 317

Foreword by the translation editors
All knowledge is only bringing the essence of life under the laws of reason.
Lev Tolstoy ,« War and Peace »
A characteristic feature of the current stage of development of economic science is its mathematization, which manifests itself in the replacement of the studied economic process with an adequate mathematical model and the subsequent study of the properties of this model either by analytical methods or on the basis of computational experiments. The use of mathematical models in economics has more than a century of history. For example, one of the first models of market competition (O. Cournot) was published in 1838, and half a century later, L. Walras already applied mathematical models when reading the course of political economy in
University of Lausanne. To date, various models of interaction between labor markets, goods and money markets, models of single-product and multi-product firms, a model of consumer behavior, a model of firm competition in the goods market, and others, which, in essence, are equilibrium models, have firmly entrenched in economic theory.
However, the vast majority of economic processes take place in time, as a result of which the corresponding mathematical models are, in principle, dynamic. One of the traditional approaches to predicting the development of economic processes is to study the shift in the equilibrium point of a dynamic system caused by a change in certain parameters of the model.
This (quasi-stationary) approach is based on the key concept of classical political economy - the "invisible hand" of Adam Smith. As is known, this concept is based on the hypothesis of the existence of an automatic equilibrium mechanism in competitive markets.
The use of a quasi-stationary approach to the analysis of the dynamic processes of the economy has led to the spread of the general

the idea that the development of any complex system can be viewed as a change from one stable state to another with a short period of transition between them. However, an analysis of real economic dynamics based on this approach may turn out to be erroneous, since the period of non-equilibrium development of many economic processes may turn out to be too long to be neglected. Perfectly understanding the importance of studying economic processes in dynamics, a classic of modern economic science
A. Marshall justified the use of a quasi-stationary approach to assess changes in the market by the fact that "our analysis is still in its infancy."
Note that this approach is effective only for the time being, until, for some reason, the nature of the stationary state does not change radically. Such changes, called bifurcations, already belong to the field of application of methods of nonlinear dynamic analysis, the development of which leads to the growing spread of this point of view: "The world is a constant development, eternal instability, and periods of stabilization are only brief stops along the way."
Dynamic mathematical models, which have proven themselves in physics and then in biology, have much in common, although they retain the specific features of each of these sciences. Now models of this class are increasingly used in sociology and economics. To date, the modern methodology for the analysis of nonlinear dynamic systems has taken shape in a new scientific direction called synergetics. This interdisciplinary science is aimed at identifying the general principles of evolution and self-organization of complex systems in various fields of knowledge based on the construction and study of nonlinear dynamic mathematical models. Important concepts of synergetics are "catastrophe", "bifurcation", "limit cycle", "strange attractor", "dissipative structure", "traveling wave", etc. Arising from the use of relatively simple nonlinear models, these concepts allow us to penetrate deeper in the essence of many processes and phenomena. Physics, chemistry, and biology abound in examples of the successful application of this methodology. These include phase transitions between aggregate states of matter, turbulent fluid flows, structures in media in the presence of autocatalytic reactions, life waves and combustion waves, fluctuations in the number of natural populations, etc.
It is not surprising that this universal methodology, which has arisen relatively recently and has proven itself in the natural sciences, has begun to penetrate into the traditional humanities, and

primarily to the economy. Without fear of making a mistake, it can be argued that any branch of economic science can be attributed to the field of applications of synergetics, since when considering any dynamic economic process, some active, i.e., feedback element is always present as an acting factor. Therefore, if we want to look beyond the horizon of a narrow world in which everything seems to be stable and in which there is no place for catastrophes and restructuring, we cannot do without using a synergistic approach.
In the book offered to the attention of readers by V.-B. Zang "Synergetic Economy" an attempt is made to give a general idea of ​​the possibilities of a synergistic approach in the economy. At the same time, the main attention is paid to the consideration of relatively simple mathematical models of small dimensions, which, as a rule, can be investigated by analytical methods. The use of synergetic methods in the economy is not a tribute to fashion, but an urgent need to move forward beyond the limits outlined by the quasi-stationary approach, to look for new ways to use powerful modern computing tools to solve serious practical problems.
The mathematical toolkit of the book is a fairly compact set of methods that make it possible to conduct a very effective analysis of nonlinear models of real economic processes. The undoubted advantage of the approach used is that the analysis of the low-dimensional models discussed in the book is easy to comprehend, since the set of properties that are the most striking consequences of nonlinearity is rather limited. Therefore, the mathematical apparatus used in the book should become not only the alphabet for a new generation of economists, but at the same time a beacon to which the mathematical training programs of economic universities should be tuned. Apparently, it is in connection with this that V.-B. Zang recommends his book not only to specialists, but also to students of economic specialties.
The scale of the task that the author set himself did not allow him to avoid some shortcomings. This concerns, first of all, the excessive conciseness of the presentation of the fundamental hypotheses in the formulation of mathematical models, which, unfortunately, is inherent not only in this, but also in many other books on mathematical economics. It can be noted that; that the economic models in the book often serve as illustrations of well-known mathematical results. This places the models under consideration in a subordinate position in relation to the mathematical apparatus, which, of course, cannot but cause some feeling of dissatisfaction among economist readers. However

as a result of this approach of the author to the presentation of the material, the reader discovers, for example, that economic cycles are as natural as population fluctuations, and “leaps” in society, that is, changes of a revolutionary type, are like phase transitions for matter. So this can be regarded as a deliberate methodological approach in presenting material, which forces readers to delve more carefully into those mean lines in which the main hypotheses and mathematical constructions of models are stated, and to show maximum independence in understanding not only the results presented, but also mathematics. -1eskoy statement of the problem.
The reader should be quite critical of some subjective assessments (and self-assessments) of the author. For example, speaking of the principle of subordination
Haken, it is impossible not to mention another formulation of this principle - Tikhonov's theorem for systems of equations with singular perturbations. And in general, speaking of synergetics, it should be remembered that many of its results are directly related to the development of mathematical modeling, at the origins of which in our country were A. A. Dorodnitsyn, N. N. Moiseev, A. A. Samarsky and others (for For the convenience of readers, we provide at the end of this preface a small list of literature in Russian on this topic).
At the same time, we would like to draw the reader's attention to the main advantage of the book: on the whole, the author managed to give a broad panorama of the state of affairs in today's synergetics using the analysis of relatively simple models of dynamic economic processes as an example. Moreover, the book is aimed at developing a non-linear style of thinking among readers, which is important in any field of knowledge, including, of course, in modern economics.
When working on the translation manuscript, we corrected the observed inaccuracies of the original without any special reservations, and where necessary, made footnotes. It should be especially noted that the publication of the book in Russian was carried out thanks to the initiative of the translator of the book N.V.
Ostrovskaya, who supported her initiative to the Russian Foundation for Basic Research (Head of the Publishing Department V.D. Novikov), employees of the Mir publishing house, and A.V. Fedotov, who took part in the translation of 5 and
9 chapters.
We would also like to express our gratitude to the author of the book, Prof. V.-B. Zang for his attention to the Russian edition - he kindly sent, at our request, a list of typographical errors, which was taken into account in the Russian edition, and also answered a number of questions regarding the clarification of certain places in the text. In conclusion, we express the hope that the book will be useful to all readers interested in applications of nonlinear analysis methods in economics. Who knows, maybe among them will be those who, with its help, will find the very thread, unraveling which, it will be possible to get to a clear synergistic picture of the economic problems that we are all experiencing today and, having this picture in front of us, find real ways to decent economic development.

List additional literature
1. Arnold V. I. Theory of catastrophes . Moscow: Nauka, ed. 3rd, add., 1990.-128s.
2. Akhromeeva G. S., Kurdyumov S. P., Malinetsky G. G., Samarsky A. A.
Non-stationary structures and diffusion chaos . M.: Nauka, 1992. -
542 p.
3. Ivanilov Yu. P., Lotov A. V. Mathematical models in economics . M.:
Science, 1979.- 304 p.
4. Lebedev V. V. Mathematical modeling of social - economic processes . M.: Izograph, 1997. - 224 p.
5. Loskutov A. Yu., Mikhailov A. S. Introduction to synergetics . Moscow: Nauka, 1990.
- 270 s.
6. Petrov A. A., Pospelov I. G., Shananin A. A. Experience of mathematical modeling of the economy . M.: Energoatomizdat, 1996. - 544 p.
7. Romanovsky Yu. M., Stepanova N. V., Chernavsky D. S., Mathematical biophysics . M.: Nauka, 1984. - 304 p.
8. Samarsky A. A., Mikhailov A. P. Mathematical modeling . M.:
Science, 1997. - 320 p.
9. Moiseev N. N. Mathematical problems of system analysis . M.: Nauka,
1981.
10. Tikhomirov N. P., Raitsin V. Ya., Gavrilets Yu. N., Spiridonov Yu. D. Mo -
division of social processes . Tutorial. M.: REA, 1993.
D . uh . n ., to . f .- m . n ., prof . AT . AT . Lebedev
D . f .- m . n. AT . H . Razzeeaykin

Foreword
This book is about the dynamics of economic and other social systems. It was written at the Swedish Institute for Advanced Study and focuses on the semantic side of the problems of economic evolution and rapid structural change.
The analysis carried out here is closely related to synergetics. This means that the doctor
Zang focuses on the fact that economic and other social variables can be subdivided into subsets of fast and slow variables. It has been established that some of the slow variables have the meaning of collective ones, i.e. can play the role of order parameters in economic and social systems.
With greater or lesser involvement of the mathematical apparatus, such a subdivision is also present in earlier attempts at dynamic analysis of the economy. Something similar was done by Alfred Marshall in his textbook as early as the nineteenth century, and by Paul Samuelson in The Foundations of Economic Analysis in 1940. However, they did not assume the possibility of an exact solution of the problems raised, which is implied by the approach to economics developed here. Doctor
Zang is not only committed to this direction, but specifically shows how synergetic methods work in the dynamic analysis of the most important large-scale problems of economic development. One of his most important findings is that with the proposed subdivision of interacting subsystems into fast and slow ones, it is possible to achieve predictability of their behavior, which otherwise should be recognized as unpredictable, i.e. chaotic. In addition, the analysis carried out shows that variables influencing order variables can become a tool of strategic policy. Most of these variables are of the slow type and, therefore, can themselves be considered as order parameters at the level of the economic system. The latter automatically means that these variables influence the adoption of strategically important decisions, i.e. become an instrument of future-oriented policy.
Predicting the future is certainly important, but it can easily turn into baseless fantasies if it is not based on a sound methodological foundation. One of the foundation stones in this foundation was laid by Dr. Zang with his book.
AkeE . Andersson , Professor of Economics at Umea University, Director of the Swedish Institute for Advanced Study

To my parents ,
who were so saddened by the long absence of their son
... if orthodox economics reaches an impasse, the cause is not to be found in the general structure, which has been brought with great care earlier to logical harmony and consistency, but in the lack of clarity and generality in the premises.
J . M . Keynes (1936)
From the author
Time changes not only the economic structure of society, but makes its own amendments to key economic ideas. It's too early to judge today historical significance contribution to economic science of the newest economists, since classical economists such as Ricardo, Malthus, Marx, Walras and Marshall lived in a different time and belong to other cultures. Time is the best arbiter.
Only time gives us enough wisdom to recognize that other ideas, which at first seemed so significant and promising, are rather superficial.
Not only general public, but many economists are increasingly losing faith in the possibility of applying economics to reality, although the level economic knowledge has recently grown significantly: apparently, there is no simple relationship between scientific knowledge and trust in science.
One can think of many reasons why economics fails in its attempt to explain reality. On the one hand, myself real world has become more complex in recent decades. Technology, the institutions of society, the quality of life, the aspirations of people, their mores, which in the past changed relatively slowly, now, as a rule, change much faster. This feature of modern society makes it difficult, if not impossible, to attempt to explain economic life in terms of pure science. On the other hand, traditional theoretical economics has its own internal limitations: it is limited mainly to static and externally stabilized economic systems.
Non-linear unstable processes, such as regular and irregular oscillations, which are the main objects of our study, are considered random or insignificant phenomena in traditional analysis.

This book addresses issues related to evolution and change in non-linear, unstable economic systems. We will focus on such aspects of dynamic economic systems as non-linearity, instability, bifurcations and chaos. To analyze the characteristics of nonlinear dynamic economic systems, we propose a new theory - "synergistic economy", based on Haken's synergetics. Synergetic economics emphasizes the interaction of linearity and non-linearity, stability and instability, continuity and discontinuity, permanence and structural change, as opposed to the properties of pure linearity, stability, continuity and permanence. Non-linearity and instability in a synergistic economy are considered more as sources of diversity and complexity of economic dynamics, rather than as sources of noise and random phenomena, as is done in traditional economy.
In a sense, this book aims to complete the task that Paul A. Samuelson set himself when he wrote his seminal Foundations of Economic Analysis. He divided the development of analytical economics into about five major stages. The first is associated with the name of Walras, in which we find the culmination of the idea of ​​a deterministic equilibrium and a static level. Pareto and others took the next step, which formed the basis of the theory of comparative statics. The third step related to maximizing the action economic object, was made by Johnson, Slutsky, Hicks and Allen.
The fourth achievement is related to the discovery of the principle of correspondence. "The fifth step, which is natural to take after we have examined the response of the system to a change in given parameters, is to investigate its behavior as a function of time." Moreover, Samuelson emphasized that “the benefit of any theoretical construction is to understand the behavior of economic variables depending on certain data or parameters. This is true for both dynamics and statics. Therefore, the next logical step should be precisely the creation of a theory of comparative dynamics. It should not only include the theory of comparative statics as a special case, as well as all the sections of economic theory listed above, but cover a much wider area.
(Samuelson, 1946). The fifth step will be developed in this book.
This book is intended for students of economics and research economists.
It may also be useful to scientists interested in applications of nonlinear dynamical theory to economic problems.
Stockholm, July 1990
AT .- B . Zang

Thanks
I am very much indebted to my teacher, Prof. Ake E. Andersson. It is easy to see his influence in the pages of this book, as well as in my entire professional biography. I am grateful to him for the Introduction to this book.
I would also like to express my deep gratitude to Prof. Hermann
Haken, prof. Björge Joansson and Prof. Thien Puu for their valuable comments. I am grateful to Dr. A. M. Laee and Mrs. I. Kaiser, employees of the publishing house
Springer, for cooperation.
I also want to express my gratitude to CERUM University of Umea and
To the Institute for Advanced Study in Stockholm for providing the conditions and creating a favorable intellectual environment that stimulated this research. I am grateful to CERUM and the Institute for Advanced Study for financial support.