The question of the stability of systems has worried the minds of scientists for a long time. True, this was not due to economic problems, but to the functioning of the World system, the question of the stability of which before the discovery of the law gravity decided a priori. First, an assumption was made about the fundamental property of the world - stability and chaos, then a system was created. Let us trace some evolution of views on the problem of sustainability, especially since, in our opinion, this is related to the sustainability of socio-economic systems.

Newton was the first to build a dynamic model of the stability of the solar system and immediately ran into the question of its functioning. The scientist got out of the difficulty with the help of the "Great Clockmaker", who from time to time must return the planets to their orbits.

In the future, the concept of stability developed in parallel with the study of planetary motion. In particular, Lagrange considered motion to be stable if it occurs in a closed region of space. In 1773 Laplace formulated the stability theorem solar system: if the planets move in the same direction, their masses are of the same order, the eccentricities and inclinations are small, and the major semi-axes experience only small deviations relative to the mean position, then the eccentricities and inclinations of the orbits will remain small in the considered interval. However, it turned out that the theorem is not applicable for large time intervals and for different masses of planets, which is present in the real world. This is also observed in the development of socio-economic systems, where there are no structures of the same size, equal conditions for all organizations, but there is a desire to ensure the effective functioning of the organization for a long period.

Science owes the creation of a mathematically rigorous and consistent theory of motion stability to A. Poincare and the Russian mathematician A. Lyapunov, who owns the most successful definition of the concept of stability: it is not enough to deviate from it over the entire time interval of interest. If there is at least one (!) Movement, which at the initial moment differs little from the investigated, which gradually, albeit after a long period of time, noticeably deviates from it, then the studied movement is unstable. When talking about the stability of the solar system, as a rule, they mean the stability of movement major planets on an infinite or very large, comparable to her age, time interval, i.e. system stability occurs when no fundamental changes occur. In this case, the extreme manifestations of instability are leaving the solar system, falling into the Sun, or colliding with another planet. Such an event can significantly change the structure and dynamics of the solar system. It is not surprising that many scientists understand the stability of a system to be precisely the invariance of its elements and the environment. For example, the Soviet mathematicians L. N. Kolmogorov and V. I. Arnold and the American mathematician Yu. Moser developed a theory called the KAM theory. Its application to the solar system gives the following theorem: if the masses of the planets are small enough, the eccentricities and inclinations of the orbits are small, then for most initial conditions (excluding resonant and close to them)(highlighted by us. - Primel. auth.) the movements will be conditionally periodic, the eccentricities and inclinations will remain small, and the semi-major axes will forever oscillate near their original values. But such stability is possible only in the absence of resonances, which cannot be even in excessively large systems. What, then, is sustainability in the face of constant change? After all, if we take as an object of study the socio-economic system, where the functioning of the elements is currently taking place with infinite change at the micro and macro levels, then the above theorem rejects the stability of the economy by definition! In fact, there are quite a few types of stability: with respect to perturbations of the initial data (Lyapunov stability), with respect to permanent perturbations, structural, practical, orbital, Poincaré stability, Zhukovsky stability, Lagrange stability, etc. In these definitions lies the possibility to designate the stability of the system in the cases of the presence of perturbations both inside the system and outside it. It is no coincidence that two types of stability are most often used - with respect to perturbations of the initial data and with respect to permanent perturbations (sufficiently small external influences). Stability with respect to perturbation of the initial data is nothing else, as a continuous dependence of solutions on the initial data, fair on an infinite time interval. This dependence can be mathematically represented as follows:

where L- const Lipschitz; t- time included in time span , where solutions are considered.

The resulting estimate indicates the continuous dependence of the solutions on the initial data. This estimate shows that if the initial points * 10 and * 20 are close, i.e. small value 8 = || * 10 - * 2 o ||, then solutions *, ( t) and 2

(t) will also be close, but only on some finite time interval О t On this segment (1.1) will take the form:

If we want the distance between solutions X,(?) and x 2 (t) did not exceed set value p on the segment 0 t Г, then we get

The meaning of stability with respect to permanent perturbations is that any solution indignant systems close to elementary moment to the given solution of the unperturbed system remains close to it always, if permanent perturbations are sufficient small. Stability under constantly acting perturbations is precisely connected with finding out what happens on an infinite time interval with the solutions of the original and perturbed systems, leaving at the initial moment from the same point. Stability with respect to constantly acting perturbations is nothing else, as a continuous dependence of solutions on the right side of the equation on an infinite time interval ( . This is also indicated by the formula below:

The development of real systems is nonmonotonous and includes not only progressive directions, but also paths of degradation (which can be replaced by progress, or can lead to collapse), and directions of destruction. In the process of development, which consists of cyclically repeating stages of evolution and a jump, the system is constantly moving from a stable state to an unstable one and vice versa. For a long time it was believed that unstable movements are useless, unobservable, that they must disappear sooner or later, and that there is practically no particular benefit from them. In theoretical terms, this may be so. However, in practice, unstable movements can have every right to exist and be used in practice. Here we enter the area practical sustainability. Analysis of the concept of practical sustainability includes the following aspects:

• 1) analysis of practically admissible initial perturbations;
• 2) analysis of practically admissible subsequent deviations;
• 3) estimation of the time interval beyond which the evolution of the system is of no interest;
• 4) analysis of the maximum allowable external influences.

If the perturbed solution under admissible initial perturbations (or admissible external perturbations) on given time interval deviates from the unperturbed solution within acceptable limits, then this

the unperturbed solution is called practically stable. We share the statement of A. Filatov: “The Universe, apparently, is theoretically unstable. Perhaps the development goes from bifurcation to bifurcation, and between them the system is practically stable. If this is so, then there is no theoretical stability in nature in principle, and in fact, only instability and its real embodiment - practical stability should be studied.

Let the set В 0 of admissible perturbations of the initial data and the set Bt admissible subsequent deviations of the perturbed solution from the unperturbed one. In many cases, as a set At 0 takes a set of the form:

and as a set Bt lots of

An unperturbed solution φ(?) is called practically stable if there exists such a set in the space of admissible parameter values ​​that the solutions y(t), starting in the set B 0 remain in the set AT, on the interval [? 0; ? 0 + L- This is clearly shown in Fig. 1.1.

Rice. 1.1.

We can draw an intermediate conclusion: by identifying a sufficiently large number of types of stability, we are trying to find an answer to the question: how can the system maintain its quality in the presence of perturbations? In our opinion, the answer is as follows: only as a result of constant changes (instabilities) through adaptation, it is possible to achieve the practical stability of the system.

If we continue to describe the functioning of the solar system as an example, we can say the following: in the presence of resonances, the evolution of a dynamical system can go in two ways:

• 1) the system will go through resonance, which will lead to a sharp abrupt change in the elements of the orbit, for example, its inclination;
• 2) the system will get stuck in resonance and will pass into a new state with a libration mode of motion, in which the positional elements (major axis, inclination) together or separately will experience fluctuations sometimes of a rather large amplitude.

Any of these scenarios could result in an object moving into a new orbit. And this is the main thing: the object will retain its quality and will develop, albeit in new conditions. Therefore, it can be stated with a certain degree of certainty that the stability of a system is its ability to constantly change, the ability of a system to keep its parameters in a certain range of values, allowing it to maintain qualitative certainty, including the composition, connections and behavior (but not equilibrium!). Such stability is formed in the process of adapting the system to the external and internal conditions that have changed as a result of the catastrophe and is maintained during most of the evolutionary stage.

For the first time, the sustainability of the enterprise began to be considered in the 20s. 20th century It has been argued that when operating with minimal deviations of the system from the standards at its input and output (or within given deviations), it is said to be stable.

An analysis of the economic literature shows that researchers have an almost unanimous opinion regarding the sustainability of an enterprise, although with different interpretations:

• - the state of stability is associated with a state of equilibrium, stability, balance, any deviations from equilibrium mean a decrease in stability (or its loss);
• - a prerequisite for the sustainability of the enterprise is development;
• - the stability of an organization is understood as the preservation of its relative integrity, structuredness and profitability, ensuring the reproduction of the workforce with all possible changes in the environment, as well as preventing the destruction of the structure during crisis phenomena.

As we can see, the definitions include the concepts of balance and development. But there is some contradiction here. In the economic encyclopedia, equilibrium is defined as the state of the economic system, characterized by the presence of balance, balancing multidirectional factors. Equilibrium can be unstable - short-term and stable - long-term. A. Borisov defines the economic equilibrium of an enterprise as optimal ratio in the production, exchange, distribution and consumption of resources necessary for the existence and development of the enterprise. However, it should be noted here that there are different points perspective on balance. In particular, S. Braginsky and J. Pevzner understand equilibrium as such a situation in which, given the invariance of external conditions and parameters, none of the participants in the economic process has an incentive to change their economic behavior.

The semantic meaning of the term "equal-" implies either the equality of any parts, indicators, characteristics of the enterprise or their optimal ratio. Equality, by definition, cannot reflect the dynamic nature of the existence of any system; it is contrary to development, which is associated with excess, addition, change, etc. The duration of equilibrium cannot be a measure of its stability, since maintaining equality does not mean maintaining stability, in some cases it even contradicts the growth or other directions of development of the enterprise, the achievement of which in unstable, difficult to predict conditions is possible only with constant variability in the system’s activity in order to “grope for » the right path of development. This circumstance can be illustrated by the following example: a tightrope walker in a circus is stable when he balances with all parts of the body, even invisible to the viewer.

The optimal combination and ratio of any indicators of the object is a variable criterion, depending on the specific situation, and, therefore, cannot serve as a measure of the stability of the enterprise. More reasonable and consistent with the systematic approach is to consider the enterprise not as an equilibrium system, but as a system of economic relations, the structure of which is formed and changed in accordance with the specific conditions of the enterprise, which determines the balance and sustainability of its existence and development. For example, R. Garipov and M. Khannanov understand economic sustainability as a system of economic relations regarding the formation and use of financial resources, within which an enterprise overcomes objective external restrictions through internal restructuring and adaptation (and in some cases through active opposition), gaining the opportunity to carry out extended reproduction. A large group of authors argue that sustainability is “the ability of a system to carry out normal reproduction of the existing structure of economic relations. The possibility of achieving economic stability and the actual state of the economic system determines the balance of forces and the degree of realization of the economic interests of the subject, on which the preservation of the integrity of the system depends. This approach also takes into account development as the main condition for the existence of an enterprise, the influence of external and internal factors. But what is development? Development is associated with qualitative changes. In other words, change and development are varieties of the process of change, distinguished depending on the level of orderliness of this process. If we consider the object of development as a system, then qualitative changes should be understood as the emergence of new stable structural components - elements, connections, dependencies, i.e. the process of development is associated with the transformation of the structure of the system. Here I would like to draw attention to some methodological points.

Many systems have the property of development, and management systems are no exception. Development is the path that each specific system goes through from the moment of its inception. Development, as you know, is a natural, qualitative change and is characterized by irreversibility and direction. Like any system, the organization management system in its development goes through a number of successive stages, i.e. has its own life cycle: 1) occurrence; 2) becoming; 3) maturity; 4) transformation.

The emergence and formation represent a progressive change in the system, since this is the process of formation, organization of the management system. In turn, the transformation reflects the process of disorganization of the management system. The maturity period reflects the stationary state of the system, the realization of its potential. "The stationarity of the system is apparently equivalent to the stationarity of the structure" . During this period, the process of organization is balanced by an equal in strength, but opposite in direction, process of disorganization.

Emergence means the emergence of a new quality. But none new system management does not arise from scratch, even if the emergence of a system is associated with a revolutionary socio-economic transformation, it is still carried out on the basis of the previous system. Having arisen on the basis of old managerial relations, the management system has systemic qualities that are strengthened and expanded in the process of functioning and development. Gradually, the new control system is being “finished”, i.e. forms new subsystems that are necessary to implement their own functions and achieve their goals. “In the process of development of a phenomenon, the following regularity is usually observed: development proceeds at first not at the expense of all elements, but at the expense of a more or less narrow group of defining elements, followed by the further development of all other elements of the phenomenon.”

Any socio-economic system has historical continuity. As A. Averyanov notes, the process of emergence can be divided into two stages: “1) hidden, when new elements appear in the bowels of the old, their quantitative growth takes place; 2) explicit, when new elements form a new structure, i.e. quality" .

The emergence of any new evidence that the old in these conditions has exhausted itself, has ceased to meet the needs of the subject of management. This means that any organizational restructuring of the elements of the system does not lead to improvement, but to its transformation.

The emergence and development of a system is the emergence and development of its contradictions. Becoming is a contradictory unity of the processes of differentiation and integration: the differentiation of elements enhances their integration, and integration, in turn, creates the prerequisites for differentiation. As V. Svidersky writes, “... characteristic feature development as a complication is the unity of the processes of increasing the diversity of structural dependencies, on the one hand, and the integrity of elements within a given structure, on the other. This differentiation-integration process is an organizational process: "... the process of complication of the structure can be characterized as a process of differentiation and integration" .

A mature system is in a stable state, but this does not mean that the process of interaction of the contradictory aspects of this system has stopped, which causes further transformation. As the management system develops, its functions develop. The system specializes and begins to adapt to a certain way of interacting with the external environment. In the period of maturity, differentiation processes stop - a stable connection is formed between the elements of the system, structuring is completed. Like any other system, the control system can function successfully in the environment in which it was formed. The transition of the system to another environment will inevitably cause its transformation. This is the law of existence of any systems.

But even functioning in favorable external conditions does not exclude the aggravation of internal contradictions, which lead it out of a state of equilibrium. The control system is entering the final stage of its development - the stage of transformation.

The transformation of the management system means its transition to a new quality. The reason for the transformation is the contradiction between the form of connection of the elements of the system and their interaction with the external environment. The external environment affects the control system in such a way that it changes the way they interact with the environment. As V. Prokhorenko writes: “... a change in the internal structure of a thing is accompanied by a corresponding transformation of the totality of its external properties, and any change outside world corresponds to a certain (essential or insignificant) shift in the internal structure of a given body.

Since the functions of individual subsystems and elements change, their connections with the rest of the control system, which still function, also change. There is a decrease in the number of old elements and interactions, an increase in the number of new ones - one system, thus, is destroyed, and another arises. The process of transformation of one control system means the simultaneous process of the emergence of a new one.

Development is associated with a certain direction of the process. Progressive development is characterized by an increase in the level of organization of the system and its complication. The main thing in the direction of development is the emergence of new opportunities in the implementation of the main goals of the system: internal and external requirements.

Organization - open system, i.e. a system constantly striving to maintain a balance between internal capabilities and external forces environment (i.e. self-stabilizing) to maintain its steady state.

Stability - the ability of a system to approach an equilibrium state under the influence of internal and external disturbances through constant changes. Moreover, we believe that an enterprise always strives for sustainability, not only with minor deviations, as some authors believe. For example, A. Romantsov writes that "the stability of an industrial enterprise is the ability of a management system to ensure the functioning of an enterprise under the influence of external and internal factors in a state of equilibrium and return it to this state after minor deviations."

An analysis of the presented points of view allows us to conclude that the vast majority of authors focus on adaptation, on the adaptive nature of the enterprise's behavior under a certain state of the environment. Under the stability of any phenomenon or process is meant the non-susceptibility of its fluctuations and changes; hardness, durability, reliability; constancy, staying in one state; the ability to maintain a given state, despite the action of various forces. For example, M. Khannanov emphasizes that sustainability is achieved under such a “state of economic and social relations, in which there are no critical threats and the ability of the subject to adequately respond to these threats, as soon as they arise” is retained. But, as we have already written, the external environment today does not allow us to hope for the absence of catastrophic threats and for the possibility of adaptation: threats arise faster than enterprises have time to adapt to them and predict many of them. Modern conditions raise the question of the inefficiency of the adaptive behavior of the enterprise, they are extremely dynamic and difficult to predict, they require the organization to develop faster.

In his book The Theory of Catastrophes, A. Arnold gives a number of examples when a stable, steady state of a system's functioning is usually destroyed when it collides with an unstable regime (moreover, at the moment of collision, the convergence rate is infinitely high) or due to an increase (infinitely fast) of self-sustaining oscillations. This explains why it is so difficult to deal with a catastrophe when its signs have already become noticeable: the speed of its approach increases indefinitely as one approaches the catastrophe.

Let us give an example from the theory of rearrangements. The mathematical model of the theory of perestroika was created long before the perestroika of the economy in Russia at the end of the 20th century. The problems of perestroika lie in its non-linearity. The generally accepted methods of control, in which the results are proportional to the efforts, do not work here, and it is necessary to develop control actions based on the sometimes paradoxical conclusions of the nonlinear theory. From the point of view of the theory of restructuring, the change of the administrative system of managing the economy to a market one can be represented as follows (Fig. 1.2).

A. Arnold makes the following qualitative conclusions for a nonlinear system that is in a steady state, recognized as bad, on the assumption that there is a more preferable stable state of the system within the visibility range:

1) gradual movement towards a better state immediately leads to deterioration (point a in fig. 1.2). The rate of deterioration with a uniform movement towards a better state increases;

Rice. 1.2.

• 2) but as we move from a worse state to a better one, the resistance of the system to a change in its state grows;
• 3) the maximum resistance (point /;) is reached earlier than the worst state (point c), through which you need to go through to achieve a better state. As the resistance maximum passes, the state of the system continues to deteriorate (to point c);
• 4) when the system approaches the worst state on the path of restructuring, the resistance of the system, starting from a certain moment, decreases (point b). And as soon as the worst state of the system (point c) is passed, the resistance not only disappears completely, but the system begins to be attracted to the best state (the path to the point e)
• 5) the amount of deterioration required to transition to best condition, is comparable to the final improvement and increases as the system improves. An underdeveloped system can move into it almost without prior deterioration, while a developed system, due to its stability (read - ossification), is not capable of such a gradual, continuous improvement;
• 6) if the system can be immediately, abruptly, and not continuously, transferred from a bad stable state quickly enough to a good one, then it will continue to evolve by itself in the direction good condition. Only the intellectual potential of a person can initiate a leap.

The above laws are the objective laws of the functioning of nonlinear systems, which cannot be ignored. The considered simplest qualitative conclusions from the nonlinear theory of rearrangements are very important and at the same time very reliable: they depend little on the details of the functioning of the system.

In this context, a somewhat limited approach to the definition of sustainability is the emphasis on the financial sustainability of the enterprise, since here, but by definition, there can be no disequilibrium 1 . But it is precisely this approach that remains the main one in determining the sustainability of socio-economic systems.

As we have already noted, for the actual preservation of the organization, more significant activities are needed than those of which our organized whole consisted. An increase in activities can give us the environment, which, in turn, makes it necessary to change the internal relationships of the complex, its structure. A. Bogdanov, drawing a parallel between the social and the living, noted that in a living cell, growth processes change molecular bonds, and in society, the development of an organization leads to a change in structure. An enterprise needs to manage its activities, internal, and possibly external parameters in such a way as to ensure not just sustainable operation, but constantly create additional benefits, anticipating and ahead of future changes in the environment, realize emerging opportunities and reduce threats, while remaining internally and externally stable. Earlier we talked about the mathematical justification of practical stability. In the economic context, the practical stability of an organization depends not only on the number of activities-resistances concentrated in it, but also on the way they are combined, on the nature of their organizational ties, and the type of organizational structure. Even in economic theory in a certain discussion of the stability of the monetary unit, the main role is assigned to the person: "The stability of the monetary unit, in addition to its commodity content, is also influenced by factors related to the characteristics of the economic and monetary behavior of individuals" . We consider it necessary to dwell on this in more detail.

Any enterprise is a kind of structural formation with systemic properties. The most important feature of the system is that the elements that make up the system form, in interconnection, a single whole with qualitatively new properties. Considering this feature, it should be emphasized that the system is an ordered set of interconnected and interacting elements, naturally forming a single whole, possessing properties that are absent from the elements that form it. The system has integrity, activity, is capable of development and improvement of its organization. In this regard, the system-wide, integral properties of the system are singled out, which characterize its behavior: utility, efficiency, self-organization, safety, stability, manageability, reliability, survivability. V. A. Ostreikovskii presented the interrelationship of the integral properties of complex systems in an interesting way (Fig. 1.3).

Rice. 1.3.

dynamic systems 1

The integral properties of complex systems in the general case are not a simple sum of the properties of the elements included in the system. For realistic estimates of the state of the system, it is necessary to study all its properties. Any system must correspond to its environment, adapt to it, which makes it possible to talk about sustainable organized system, i.e. identify positive and negative changes.

In this context, sustainability can be viewed from two perspectives. On the one hand, sustainability can be understood as preservation, an unchanged state in relation to the perturbing influences of the external and internal environment of the organization, on the other hand, it can be considered as a process, a kind of “forward” movement, as a result of which development and improvement occur. organizational structures and systems.

In our opinion, the second is more obvious, since nothing is permanent, which means that in any organized system there are always elements of chaos that require coordination. Considering the stability of the monetary unit as an example, it is appropriate to recall the theory of agreements (conventions) - one of the areas of institutional theory within its French school, where an agreement is a certain form of coordination of the interaction of individuals, developed under the influence of the entire set of formal and informal norms and rules of social behavior . Individuals act in various forms of coordination or agreement regarding the observance of norms of social behavior, i.e. operate "in an environment consisting of many heterogeneous spheres or worlds" . In tektology, the organization is an “organizational complex”, the elements of which are various activities-resistances that are in a certain combination and interaction. Moreover, this relationship is quite flexible and mobile, it contributes to an easy rearrangement of elements; it is no coincidence that this nature of connections has been called "organizational plasticity". Organizational plasticity helps to increase the adaptability of the complex to new changing environmental conditions, which favors the sustainable development of the system. However, a plastic organization is fraught with one contradiction: the mobility of the elements of the system allows the destruction of the links between them, which causes an imbalance and leads to a kind of instability of the organization. Thus, organizational plasticity, on the one hand, leads to a complication of organizational forms, an increase in their adaptability, organization and flexibility, on the other hand, to a decrease in strength, stability, and the emergence of new “vulnerabilities”.

Being in constant interaction with the environment, the system gives up its activities, but at the same time takes the same amount from the environment. Essentially, the system is evolving. In our opinion, development is a way of existence of a complex in a changing environment.

Processes arise inside the system aimed at overcoming external influences and restoring balance. Thus, the preservation of the forms and stability of the entire system is possible only through progressive development, otherwise it simply will not survive under the influence of an increasingly complex environment.

Based on the existence of relationships and interactions between systems, i.e. on the existence of a coordinated development of systems, it can be argued that the stability of the organization depends on the level of organization of the system. The stability of the entire system is facilitated by the fact that one part of the system assimilates what is rejected by the other. In addition, the stability of the complex can be ensured through additional connections with other systems and an increase in the diversity of the dino system. The more diverse the system, the greater the chance that one of its destroyed elements can be replaced by another. “Nature, for all its infinity and eternity, has a beginning and an end ... Stability is the desire for balance, the interaction of beginning and end.” In other words, the normal state of the system is the non-equilibrium state. There are objective reasons for this, which we have already stated when speaking about a person, about the diversity of his states.

Continuing the conversation, we want to draw attention to the approach of K. Waltukh, who proceeds from the fact that in the process of production activity a person “systematically creates from objects found in nature such products that are either not generated by spontaneous natural shaping at all, or are generated only relatively rare." According to K. Waltukh, production is the production of information. Information, as a measure of diversity, creates uncertainty, relative disequilibrium, which contributes to stability. Information interaction by its nature is a resonant interaction. In mentioning the stability of the solar system, we emphasized the role of resonances in achieving stability. In this regard, this is how L. Amirkhanova defines the stability of the economic system: “The stability of economic systems is the ability to receive and process information in a timely manner, form resources and produce products with the required performance in accordance with consumer demand under the influence of disturbing factors of the internal and external environment” .

To preserve the system in a changing environment, a simple exchange equilibrium is not enough. Only an increase in the sum of activities can serve as a guarantee of stability, when new adverse effects meet not with the former, but with increased resistance. And the destruction of the system occurs precisely because of the decrease in the sum of these activities-resistances.

If the organization develops, then this leads to a further complication of the organization, the emergence of additional connections that lead to more stable structural relationships.

In cybernetics, stability characterizes the ability of a system to function in states at least close to equilibrium under constant external and internal disturbances. Equilibrium is defined as dynamic, i.e. it is not so much a state as a process characterized by some equilibrium trajectory of the system. In this case, the trajectory will be equilibrium if it steadily and in the shortest way in time or space leads the system to the goal. Achieving a precisely defined state of equilibrium and staying in this state for long periods of time is rather an exception, a limit that can only be approached. Although approaching such a limiting state requires many qualities from the system, which in combination are defined as stability.

In reality, there are not absolutely, but relatively stable states of the organization. Such states are not states of complete equilibrium, but are similar to equilibrium. In such a “quasi-equilibrium” state, there is a relatively weak exchange of energy between the system and the environment, but a relatively large information connection between them.

The greater the heterogeneity of internal links in the system, the less stable it is, and, conversely, with an increase in their homogeneity, the stability of the system increases. In the first case, the existing structural contradictions are preserved and more and more new ones are added to them, in the second, the ongoing destruction tears away from the complex, first of all, the least firmly connected with it elements, breaks the most contradictory ties. The complication of these connections, the growth of their heterogeneity reduce the harmony and stability of the entire system. And sooner or later, the development of the system leads to instability and crisis, as the parts of the whole become different, and the accumulated systemic contradictions outweigh the strength of additional connections between the parts and lead to their break, to a general disruption of organizational unity.

The stability of the structure depends on: 1) the presence of mechanisms designed to ensure that some of the most important characteristics of the system remain practically unchanged regardless of all kinds of external influences; 2) the presence of the so-called structural redundancy, i.e. the possibility of duplication of essential elements of the system. Such redundancy allows not to disrupt the functioning of the system under adverse external influences, and therefore to maintain the stability of the structure. However, there is a limit to such preservation. If the conditions of the external environment go beyond the boundaries in which the system with a given structure functions stably, then first there is a violation of the main functions, and then the structure as a whole. To prevent such a situation, systems can compensate for adverse disturbances due to more their varieties, the presence of wider boundaries of changes in each perturbation and efficiency in time. Essentially, the stability of the system is a consequence of the resolution of the crisis.

The crisis of any system is a transition from one stage of development to another, from one qualitative state to another with its own critical point. The cause of any crisis is the destruction of some internal connection, leading to the loss of stability of the equilibrium in which the system was located.

Generalization of the definitions of the sustainability of the socio-economic system

The concept of sustainable development first came into international use in 1987 after the publication and approval by the UN General Assembly of the report of the Commission on environment and development. Since the late 1980s, the theory and practice of sustainable development has been in the focus of attention of scientists and politicians both in Russia and abroad. However, there is no single interpretation of sustainable development. The most correct, in my opinion, is the following definition of sustainable development - it is a continuous process of meeting the needs of present and future generations. The beginning of Russia's transition to sustainable development was laid by the Decree of the President of the Russian Federation "On the concept of transition Russian Federation towards sustainable development” in 1996.

According to most researchers, it is the regions that should become the main “arena” for introducing the theory of sustainable development into practice, since they:

• act as the most manageable structure, occupying an equidistant position in the administrative space of the country (center - federal districts - regions - municipalities (districts) - citizen);
• are historically the most stable territorial entities;
• comparable in size to most countries in the world;
• acquired during the reform period the experience of combining the practice of stimulating market transformations in their territories with the policy of state regulation of these processes.

So, socio-economic system (region) is an integral set of interrelated and interacting social and economic institutions (subjects) and relations regarding the distribution and consumption of tangible and intangible resources, production, distribution, exchange and consumption of goods and services.

Currently, there are several dozen definitions regarding the sustainability of socio-economic systems, and their number continues to grow. This indicates both the complexity of the concept itself and the complexity of the object of study. In some cases, the object of study is the national economy (macroeconomics), in others - the regional economy (mesoeconomics), in the third - the economy of business entities (microeconomics), in the fourth - subsystems of the economy of one level or another. A critical analysis of the available definitions of the sustainability of socio-economic systems has shown that the generally accepted concept modern science not worked out. At least four distinct approaches can be identified.

Definition	Ator
Resilience as the security, stability, reliability, integrity and strength of a system
The stability of the national economy is determined based on the criterion its security, stability, ability for constant renewal and self-improvement	L.I. Abalkin
Sustainability acts as a guarantor of the integrity of the country and is inextricably linked with the reliability of the state's monetary system	A. Livshits
Sustainability is understood as such a state of the elements of any economic, ecological or other system, when their initial states with a high degree of reliability determine their future states.	A.L. Bobrov
The stability of the economic system in the general sense is the property of this system to maintain its integrity and stability. relative to a given vector of development in the long term in a changing environment	T.M. hemp plant
The stability of the country's national economy as a single system means the strength and reliability of its elements, economic and organizational ties between them, the ability to withstand internal and external loads.	D.V. Gordienko
Stability as relative immutability of the system
Economic sustainability is the ability of the system to maintain and reproduce (restore) the original (or close to it) state in the process internal and external influences on it	A.G. Shelomentsev V.D. Kalashnikov
Sustainability as one of the main dynamic characteristics economic system, revealing the property of the system to return to the equilibrium, initial or close to it steady state after any internal or external influence	CM. Ilyasov
Economic sustainability is considered as a permanent, strong position of the system, provided with effective mechanisms of self-regulation and self-development.	T.G. Krasnova
The stability of territorial systems is defined as the relative invariance of the main parameters of the territorial social economic system, its ability to keep them within the given limits under deviating influences from outside and inside	A.L. Gaponenko
The stability of a system is the ability to remain relatively unchanged over time. certain period time in spite of internal and external indignation	N.F. Reimers
Resilience is “a necessary condition under which the system must return to a state of equilibrium after any small shock”	M. Blaug
Stability - the ability of a system to return to its original state after an impact on it from the outside	O.V. Kolomiichenko V.E. Rokhchin
Sustainability as the ability of a socio-economic system to maintain a dynamic balance
Stability is “an integrated property of the system to maintain dynamic balance when the parameters of the external and internal environment"	N.V. Chaikovskaya
The economic stability of the economic system of the region is an integrated property of the system to maintain dynamic balance when the parameters of the external and internal environment change within acceptable limits	V.A. Cretinin E.S. Bodryashov
Sustainability as the ability of a system to evolve
The sustainability of the socio-economic system is the ability to effectively use, autonomously modify the resources of its development, continuously increase the indicators of its positive change without increasing or minimizing the costs of basic, non-renewable resources	B.K. Esekin Sh. Sapargali
The stability of economic systems (including regional ones) is understood as the ability to relatively quickly return to the original state or reach a new, higher point on the development trajectory	M.Yu. Kalinchikov
Stability is the ability of a system to function in states close to equilibrium, under conditions of constant external and internal disturbances.	L.L. Terekhov
Sustainability is considered as the ability of a system to function stably in a certain mode of activity.	A.I. Druzhinin HE. dunaev
The stability of the regional economy is its ability to consistently perform developmental functions under the deviating influence of internal and external factors and at the same time ensure an acceptable quality and efficiency of results.	A.M. Ozina
System stability is the ability of a dynamic system to keep moving along the intended trajectory (to maintain the intended mode functioning), despite the perturbations affecting it	L.I. Lopatnikov
The stability of the country's economy is its ability to simultaneously solve the problems of stabilization and development	J. Kornai
The sustainability of the economic system is “a system of economic relations that ensures the long-term development of the economic system with the presence of self-regulation mechanisms (stabilization and balance) that can achieve a comprehensive solution to economic, social and environmental issues in the context of the globalization of the world economy	E.V. Makarova

At the same time, under sustainable development of the regional socio-economic system, we will understand its ability to function stably and develop in the long term in a rapidly changing internal and external environment, achieving the goal of the socio-economic development of the region, namely, ensuring positive dynamics of the level and quality of life of the population on the basis of sustainable and balanced reproduction social, economic, resource and economic potentials.

Nikonorov V.M.
PhD in Economics, Associate Professor of the Higher School of Economics
Saint Petersburg politechnical University Peter the Great

Nikonorov V.M.
Ph.D., Associate Professor VSHUB
St. Petersburg Polytechnic University of Peter the Great

Annotation: The author considered in detail the types of mathematical stability. He proposed to apply a system of linear homogeneous differential equations With constant coefficients. The author pointed out that in this case it is possible to apply the Routh, Hurwitz, Mikhailov criteria to assess the stability of the system solution.

abstract: The author has considered in details types of mathematical stability. I have suggested to apply to the description of difficult social and economic system the system of the linear uniform differential equations with constant coefficients. The author has specified that in this case use of criteria of Raus, Gurvits, Mikhaylov for assessment of stability of the solution of system is possible.

Keywords: Solution, stability, initial data, external disturbances, practical stability.

keywords: Decision, stability, initial data, external indignations, practical stability.

Relevance. There is a sufficient number of studies on the sustainability of a complex socio-economic system in economic level, for example, . But, as it seems to the author, the mathematical aspect of the stability of a complex socio-economic system has not been sufficiently considered. Application mathematical apparatus to assess the sustainability of a complex socio-economic system will ensure optimal management of the corresponding complex socio-economic system.

The object of research is a complex socio-economic system.

The subject of the study is the stability of a complex socio-economic system in the mathematical aspect.

The purpose of the study is to consider the existing definitions of the mathematical stability of a complex system and propose a conjugation for an existing complex socio-economic system.

Research methods: analysis, comparison, isomorphism.

A systematic approach to the study of a complex socio-economic system is considered in.

First, it is necessary to determine the stability of which indicator is considered in this study. In other words, what signal of this complex socio-economic system will be considered as the output and, accordingly, what approaches are possible to ensure the stability of the selected signal. If we choose to consider such complex socio-economic systems as retail trade, education, health care, law enforcement, then we can assume that the output signal of the corresponding complex socio-economic system is the benefit supplied by this system to the population of the country:

1) retail trade - food and non-food products;

2) education - educational service;

3) healthcare - medical service;

4) law enforcement - ensuring the safety of citizens from dashing people.

Then, using the example of retail trade, it can be assumed that this complex socio-economic system will be stable if it ensures the final consumption of the country's population in terms of food and non-food products in a predetermined fixed range. Accordingly, as input signals in this system the following can be suggested:

• products Agriculture countries;
• products of own industry;
• import;
• fixed assets of retail trade;
• employed in the retail trade.

This complex socio-economic system depends on time, that is, it is a dynamic system. The system can be described by a system of differential equations. For simplicity of calculations, it is possible to use linear homogeneous differential equations in total derivatives.

Accordingly, it is appropriate to speak about the stability of the solution of this system of linear homogeneous differential equations (hereinafter - SLDE) on an unlimited time interval. However, the existence of the human population cannot be eternal, so the time interval can be limited. For example, the remaining life of the Sun. Since the existing types of mathematical stability refer specifically to the stability of the solution, we consider different kinds mathematical stability (Table 1).

Table 1

Types of system solution stability

№	Type of sustainability	Brief definition	Add-ons
1	According to Lyapunov	There is a particular solution to the system of differential equations (hereinafter - SDE) at time t0 and x0 (unperturbed solution). If the solution of the SDE with a slight change in x0 by δ (the perturbed solution with the perturbation of the initial data by δ) is sufficiently close to the unperturbed solution, then this particular solution of the SDE is stable.	Analyzing the behavior of a real complex economic system (hereinafter - SES), we are faced with the fact that the initial conditions cannot be changed, they have already been passed.
2	Regarding external disturbances (Demidovich B.P.)	Permanent external perturbations appear in the SDE (the right side in the SDE system). If solution (2) at time t0 is close to solution (1) at time t0 and remains similar throughout the entire time interval, then solution (1) is stable with respect to constant external perturbations μF(t, x).	The initial conditions of a real SES have already been passed and we cannot know how the system would develop if these initial conditions were aggravated by a constant external disturbance. There is no possibility of EMM verification,
3	According to Zhukovsky	This is a kind of stability according to Lyapunov, if the speed of passage of the time interval is changed.	Accordingly, at the economic level, the initial conditions are already fixed here, there is no way to change them.
4	Practical	If the allowable deviation of solution (2) from solution (1) and the time interval of the study are predetermined, and when this time interval passes, the allowable deviation is within the specified limits, then this is practical stability.	There is no way to compare calculated and actual data. It is impossible to carry out a repeated full-scale experiment with SES.
5	Attractors	The solution of the SDE on the phase plane (in the phase space) can tend to some point (a stable node is an attractor, an unstable node is a repeller). That is, the solution of a complex economic system in the time interval tends to an attractor.	The objectives of the study are rather to determine the stability of the system over a given time interval, which is closer to practical stability.
6	For the SLODE system with a constant matrix	If we simplify the economic model, describe it as a system of linear homogeneous differential equations with a constant matrix A, then the system is stable when the roots of the matrix A have non-positive real parts Re λ j (A) ≤ 0, (j=1,…,n) (3)	You can use this simplification when constructing mathematical model complex economic system, since it leads to the application of the criteria of Hurwitz, Routh, Mikhailov.

All considered types of stability of a dynamical system are somehow connected with stability according to Lyapunov. If it is possible to describe a complex socio-economic system by a system of SLODE with a constant matrix (respectively, with constant coefficients), then the question of its stability can be solved using the criteria of Hurwitz, Routh, Mikhailov.

Research results.

1. The types of stability of complex socio-economic systems are studied at the mathematical level.
2. A variant of assessing the stability of the solution of complex socio-economic systems is proposed.
3. A preliminary description of a complex socio-economic system by a system of SLOD with constant coefficients will make it possible to apply the criteria of Routh, Hurwitz, Mikhailov to study the stability of a solution.

The further direction of research is to describe a complex socio-economic system by the system of SLDE with constant coefficients and to investigate the stability of the solution of this complex socio-economic system.

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