Type of model depends on the informational essence of the modeled system, on the connections and relations of its subsystems and elements, and not on its physical nature.

For example, mathematical descriptions ( model) the dynamics of the epidemic of an infectious disease, radioactive decay, the acquisition of a second foreign language, the release of products of a manufacturing enterprise, etc. can be considered the same in terms of their description, although the processes themselves are different.

The boundaries between models of various types are rather arbitrary. You can talk about different modes use models- imitation, stochastic, etc.

Typically the model includes: object O, subject (optional) A, task Z, resources B, environment modeling WITH.

The model can be formally represented as: M =< O, Z, A, B, C >.

The main propertiesany model:

• purposefulness - the model always reflects a certain system, i.e. has a purpose;
• finiteness - the model reflects the original only in a finite number of its relations and, in addition, the modeling resources are finite;
• Simplicity - the model displays only the essential aspects of the object and, in addition, should be easy to study or reproduce;
• approximation - the reality is shown by the model roughly or approximately;
• adequacy - the model must successfully describe the modeled system;
• visibility, visibility of its main properties and relationships;
• availability and manufacturability for research or reproduction;
• informativeness - the model should contain sufficient information about the system (within the framework of the hypotheses adopted when building the model) and should make it possible to obtain new information;
• preservation of the information contained in the original (with the accuracy of the hypotheses considered when constructing the model);
• completeness - the model must take into account all the basic connections and relationships necessary to ensure the goal of modeling;
• stability - the model should describe and ensure the stable behavior of the system, even if it is initially unstable;
• integrity - the model implements a certain system, i.e. whole;
• isolation - the model takes into account and displays a closed system of necessary basic hypotheses, connections and relationships;
• adaptability - the model can be adapted to various input parameters, environmental influences;
• controllability - the model must have at least one parameter, the changes of which can imitate the behavior of the modeled system under various conditions;
• the possibility of developing models (previous level).

Life cycle of the simulated system:

• collection of information about the object, hypothesis, preliminary model analysis;
• design of the structure and composition of models (submodels);
• construction of model specifications, development and debugging of individual sub-models, assembly of the model as a whole, identification (if necessary) of model parameters;
• model research - the choice of a research method and the development of a modeling algorithm (program);
• study of the adequacy, stability, sensitivity of the model;
• evaluation of modeling tools (spent resources);
• interpretation, analysis of modeling results and the establishment of some cause-and-effect relationships in the system under study;
• generation of reports and design (national economic) solutions;
• refinement, modification of the model, if necessary, and return to the system under study with new knowledge obtained using the model and modeling.

Modeling is a method of systems analysis.

Often in system analysis with a model approach to research, one methodical mistake can be made, namely, the construction of correct and adequate models (submodels) of the subsystems of the system and their logically correct linkage does not guarantee the correctness of the model of the entire system constructed in this way.

A model built without taking into account the relationships of the system with the environment and its behavior in relation to this environment can often only serve as another confirmation of Gödel's theorem, or rather, its corollary, which states that in a complex isolated system there can be truths and conclusions that are correct in this system and incorrect outside it.

The science of modeling consists in dividing the modeling process (systems, models) into stages (subsystems, submodels), a detailed study of each stage, relationships, connections, relationships between them and then effectively describing them with the maximum possible degree of formalization and adequacy.

In case of violation of these rules, we get not a model of the system, but a model of "own and incomplete knowledge".

Modeling is viewed as a special form of experiment, an experiment not on the original itself, i.e. a simple or ordinary experiment, but over a copy of the original. The isomorphism of the original and model systems is important here. Isomorphism - equality, sameness, similarity.

Modelsand modelingapplied in the main areas:

• in teaching (both models, modeling, and the models themselves);
• in the knowledge and development of the theory of the systems under study;
• in forecasting (output data, situations, system states);
• in management (the system as a whole, its individual subsystems), in the development of management decisions and strategies;
• in automation (system or its individual subsystems).

Let us consider some of the properties of models that allow, to one degree or another, either to distinguish or to identify a model with an original (object, process). Many researchers distinguish the following properties of models: adequacy, complexity, finiteness, clarity, truth, proximity.

Adequacy problem... The most important requirement for a model is the requirement of adequacy (correspondence) to its real object (process, system, etc.) with respect to the selected set of its characteristics and properties.

The adequacy of the model is understood as the correct qualitative and quantitative description of the object (process) for the selected set of characteristics with some reasonable degree of accuracy. In this case, we mean the adequacy not in general, but the adequacy for those properties of the model that are essential for the researcher. Full adequacy means the identity between the model and the prototype.

A mathematical model can be adequate with respect to one class of situations (state of the system + state of the external environment) and not adequate with respect to another. A black box model is adequate if, within the chosen degree of accuracy, it functions in the same way as a real system, i.e. defines the same operator for converting input signals to outputs.

You can introduce the concept of the degree (measure) of adequacy, which will vary from 0 (lack of adequacy) to 1 (full adequacy). The degree of adequacy characterizes the proportion of the truth of the model relative to the selected characteristic (property) of the object under study. The introduction of a quantitative measure of adequacy makes it possible to quantitatively formulate and solve problems such as identification, stability, sensitivity, adaptation, and training of the model.

Note that in some simple situations, the numerical assessment of the degree of adequacy is not particularly difficult. For example, the problem of approximating a given set of experimental points by some function.

Any adequacy is relative and has its own limits of application. For example, the differential equation

reflects only the change in the frequency  of rotation of the turbocharger of the GTE with a change in fuel consumption G T and nothing more. It cannot reflect such processes as gas-dynamic instability (surge) of the compressor or oscillations of the turbine blades. If in simple cases everything is clear, in complex cases the inadequacy of the model is not so clear. The use of an inadequate model leads either to a significant distortion of the real process or properties (characteristics) of the object under study, or to the study of non-existent phenomena, processes, properties and characteristics. In the latter case, the verification of adequacy cannot be carried out at a purely deductive (logical, speculative) level. It is necessary to refine the model based on information from other sources.

The difficulty of assessing the degree of adequacy in the general case arises from the ambiguity and vagueness of the criteria of adequacy themselves, as well as because of the difficulty of choosing those signs, properties and characteristics by which the adequacy is assessed. The concept of adequacy is a rational concept, therefore, increasing its degree is also carried out at a rational level. Consequently, the adequacy of the model should be checked, controlled, refined in the process of research on particular examples, analogies, experiments, etc. As a result of the adequacy check, it is found out what the assumptions made lead to: either to an acceptable loss of accuracy, or to a loss of quality. When checking the adequacy, it is also possible to justify the legality of the application of the accepted working hypotheses when solving the problem or problem under consideration.

Sometimes the adequacy of the model M possesses collateral adequacy, i.e. it gives a correct quantitative and qualitative description not only of those characteristics for which it was built to imitate, but also of a number of side characteristics, the need for the study of which may arise in the future. The effect of the side adequacy of the model increases if it reflects well-proven physical laws, system principles, basic provisions of geometry, proven techniques and methods, etc. Perhaps that is why structural models, as a rule, have a higher collateral adequacy than functional ones.

Some researchers consider the goal as the object of modeling. Then the adequacy of the model, with the help of which the set goal is achieved, is considered either as a measure of proximity to the goal, or as a measure of the effectiveness of achieving the goal. For example, in an adaptive control system according to the model, the model reflects the form of movement of the system, which in the current situation is the best in the sense of the accepted criterion. As the situation changes, the model must change its parameters in order to be more adequate to the newly formed situation.

Thus, the property of adequacy is the most important requirement for the model, but the development of highly accurate and reliable methods for checking the adequacy remains an intractable task.

Simplicity and complexity... The simultaneous demands for simplicity and adequacy of the model are contradictory. From the point of view of adequacy, complex models are preferable to simple ones. In complex models, a greater number of factors can be taken into account that affect the studied characteristics of objects. Although complex models more accurately reflect the modeled properties of the original, they are more cumbersome, difficult to visualize and inconvenient to use. Therefore, the researcher seeks to simplify the model, since with simple models easier to operate. For example, approximation theory is the theory of correct construction of simplified mathematical models. When striving to build a simple model, the basic model simplification principle:

the model can be simplified as long as the basic properties, characteristics and patterns inherent in the original are preserved.

This principle points to the limit of simplification.

Moreover, the concept of simplicity (or complexity) of a model is a relative concept. A model is considered quite simple if modern research tools (mathematical, informational, physical) make it possible to carry out qualitative and quantitative analysis with the required accuracy. And since the capabilities of research tools are constantly growing, those tasks that were previously considered difficult can now be classified as simple. In the general case, the concept of simplicity of the model also includes the psychological perception of the model by the researcher.

"Adequacy-Simplicity"

You can also highlight the degree of simplicity of the model, assessing it quantitatively, as well as the degree of adequacy, from 0 to 1. In this case, the value 0 will correspond to inaccessible, very complex models, and the value 1 - very simple. Let's break the degree of simplicity into three intervals: very simple, accessible, and inaccessible (very complex). We also divide the degree of adequacy into three intervals: very high, acceptable, and unsatisfactory. Let us construct Table 1.1, in which the parameters characterizing the degree of adequacy are plotted horizontally, and the degree of simplicity is plotted vertically. In this table, areas (13), (31), (23), (32) and (33) should be excluded from consideration either because of unsatisfactory adequacy or because of a very high degree of complexity of the model and the inaccessibility of studying it by modern means. research. Region (11) should also be excluded, since it gives trivial results: here any model is very simple and highly accurate. Such a situation can arise, for example, when studying simple phenomena obeying known physical laws (Archimedes, Newton, Ohm, etc.).

The formation of models in areas (12), (21), (22) must be carried out in accordance with some criteria. For example, in area (12), it is necessary to strive to ensure that there is maximum degree adequacy, in area (21) - the degree of simplicity was minimal. And only in area (22) it is necessary to optimize the formation of the model according to two contradictory criteria: minimum complexity (maximum simplicity) and maximum accuracy (degree of adequacy). In the general case, this optimization problem is reduced to the choice of the optimal structure and parameters of the model. A more difficult task is to optimize the model as a complex system consisting of separate subsystems connected to each other in some hierarchical and multi-connected structure. Moreover, each subsystem and each level have their own local criteria of complexity and adequacy, different from the global criteria of the system.

It should be noted that in order to reduce the loss of adequacy, it is more expedient to simplify the models:

a) on physical level while maintaining the basic physical relationships,

b) at the structural level while maintaining the basic systemic properties.

Simplification of the models at the mathematical (abstract) level can lead to a significant loss of the degree of adequacy. For example, truncation of a high-order characteristic equation to the 2nd - 3rd order can lead to completely incorrect conclusions about the dynamic properties of the system.

Note that simpler (rough) models are used to solve the synthesis problem, and more complex exact models are used to solve the analysis problem.

Finite models... It is known that the world is infinite, like any object, not only in space and time, but also in its structure (structure), properties, relations with other objects. Infinity manifests itself in the hierarchical structure of systems of various physical nature. However, when studying an object, a researcher is limited to a finite number of its properties, connections, resources used, etc. He seems to "cuts" from the infinite world some finite piece in the form of a specific object, system, process, etc. and tries to know the infinite world through the finite model of this piece. Is this approach to the study of the endless world legitimate? Practice answers positively to this question, based on the properties of the human mind and the laws of Nature, although the mind itself is finite, but the ways of knowing the world generated by it are endless. The process of cognition goes through the continuous expansion of our knowledge. This can be observed in the evolution of reason, in the evolution of science and technology, and in particular, in the development of both the concept of a system model and the types of models themselves.

Thus, the finiteness of systems models lies, first, in the fact that they reflect the original in a finite number of relations, i.e. with a finite number of connections with other objects, with a finite structure and a finite number of properties at a given level of study, research, description, available resources. Secondly, the fact that the resources (information, financial, energy, time, technical, etc.) of modeling and our knowledge as intellectual resources are finite, and therefore objectively limit the possibilities of modeling and the very process of knowing the world through models at this stage development of mankind. Therefore, the researcher (with rare exceptions) deals with finite-dimensional models. However, the choice of the dimension of the model (its degrees of freedom, state variables) is closely related to the class of problems to be solved. The increase in the dimension of the model is associated with problems of complexity and adequacy. In this case, it is necessary to know what is the functional relationship between the degree of complexity and the dimension of the model. If this dependence is power-law, then the problem can be solved by using high-performance computing systems. If this dependence is exponential, then the "curse of dimension" is inevitable and it is practically impossible to get rid of it. In particular, this refers to the creation of a universal method for finding the extremum of functions of many variables.

As noted above, an increase in the dimension of the model leads to an increase in the degree of adequacy and, at the same time, to a complication of the model. Moreover, the degree of complexity is limited by the ability to operate with the model, i.e. by the means of modeling available to the researcher. The need to move from a rough simple model to a more accurate one is realized by increasing the dimension of the model by attracting new variables that are qualitatively different from the main ones and which were neglected when building a rough model. These variables can be classified into one of the following three classes:

fast-flowing variables, the extent of which in time or space is so small that, when roughly considered, they were taken into account by their integral or averaged characteristics;

slow-flowing variables, the extent of change of which is so great that in rough models they were considered constant;

small variables (small parameters), the values ​​and influence of which on the main characteristics of the system are so small that they were ignored in rough models.

Note that the division of the complex motion of the system in terms of speed into fast and slow motion makes it possible to study them in a rough approximation independently of each other, which simplifies the solution of the original problem. As for small variables, they are usually neglected when solving the synthesis problem, but they try to take into account their influence on the properties of the system when solving the analysis problem.

When modeling, one strives, if possible, to single out a small number of main factors, the influence of which is of the same order and is not too difficult to describe mathematically, and the influence of other factors can be taken into account using averaged, integral or "frozen" characteristics. In this case, the same factors can have significantly different effects on various characteristics and properties of the system. Usually, taking into account the influence of the above three classes of variables on the properties of the system turns out to be quite sufficient.

Approximation of models... It follows from the above that the finiteness and simplicity (simplification) of the model characterize the qualitative difference (at the structural level) between the original and the model. Then the approximation of the model will characterize the quantitative side of this difference. You can introduce a quantitative measure of approximation by comparing, for example, a rough model with a more accurate reference (complete, ideal) model or with a real model. The proximity of the model to the original is inevitable, it exists objectively, since the model as another object reflects only certain properties of the original. Therefore, the degree of approximation (proximity, accuracy) of the model to the original is determined by the statement of the problem, the purpose of modeling. The pursuit of increasing the accuracy of the model leads to its excessive complexity, and, consequently, to a decrease in its practical value, i.e. opportunities for her practical use... Therefore, when modeling complex (human-machine, organizational) systems, accuracy and practical meaning are incompatible and mutually exclusive (L.A. Zade's principle). The reason for the inconsistency and incompatibility of the requirements for the accuracy and practicality of the model lies in the uncertainty and vagueness of knowledge about the original itself: its behavior, its properties and characteristics, about the behavior of the environment, about the thinking and behavior of a person, about the mechanisms of goal formation, ways and means of achieving it, etc. .d.

The truth of the models... There is some truth in every model, i.e. any model in some way correctly reflects the original. The degree of truth of the model is revealed only by practical comparison of it with the original, because only practice is the criterion of truth.

On the one hand, any model contains the unconditionally true, i.e. definitely known and correct. On the other hand, the model also contains the conditionally true, i.e. true only under certain conditions. A typical mistake in modeling is that researchers apply certain models without checking the conditions for their truth, the limits of their applicability. This approach will lead to incorrect results.

Note that any model also contains the supposedly true (plausible), i.e. something that can be either true or false under conditions of uncertainty. Only in practice is the actual relationship between true and false in specific conditions established. For example, in hypotheses as abstract cognitive models, it is difficult to identify the relationship between true and false. Only practical testing of hypotheses makes it possible to reveal this relationship.

When analyzing the level of truth of the model, it is necessary to find out the knowledge contained in them: 1) accurate, reliable knowledge; 2) knowledge that is reliable under certain conditions; 3) knowledge assessed with a certain degree of uncertainty (with a known probability for stochastic models or with a known membership function for fuzzy models); 4) knowledge that cannot be assessed even with a certain degree of uncertainty; 5) ignorance, i.e. what is unknown.

Thus, the assessment of the truth of the model as a form of knowledge is reduced to identifying the content in it as objective reliable knowledge that correctly reflects the original, and knowledge that approximate the original, as well as what constitutes ignorance.

Model control... When building mathematical models of objects, systems, processes, it is advisable to adhere to the following recommendations:

Modeling should begin with the construction of the roughest models based on the selection of the most significant factors. At the same time, it is necessary to clearly understand both the goal of modeling and the goal of cognition using these models.

It is advisable not to involve artificial and difficult-to-verify hypotheses in the work.

It is necessary to control the dimension of the variables, adhering to the rule: only quantities of the same dimension can be added and equated. This rule must be used at all stages of the derivation of certain ratios.

It is necessary to control the order of the quantities added to each other in order to highlight the main terms (variables, factors) and discard insignificant ones. At the same time, the property of the “roughness” of the model should be preserved: the rejection of small values ​​leads to a small change in quantitative conclusions and to the preservation of qualitative results. The same applies to the control of the order of the correction terms in the approximation of nonlinear characteristics.

It is necessary to control the nature of functional dependencies, adhering to the rule: to check the safety of the dependence of the change in direction and speed of some variables on changes in others. This rule allows a deeper understanding of the physical meaning and correctness of the derived relationships.

It is necessary to control the behavior of variables or some ratios when approaching the parameters of the model or their combinations to extremely permissible (singular) points. Usually, at an extreme point, the model is simplified or degenerated, and the relationships acquire a more visual meaning and can be more easily verified, and the final conclusions can be duplicated by some other method. Investigations of extreme cases can serve for asymptotic representations of the behavior of the system (solutions) under conditions close to extreme.

It is necessary to control the behavior of the model under certain conditions: the satisfaction of the function as a model with the set boundary conditions; the behavior of the system as a model under the action of typical input signals on it.

It is necessary to control the receipt of side effects and results, the analysis of which may give new directions in research or require a restructuring of the model itself.

Thus, constant control over the correct functioning of the models in the process of research allows avoiding gross errors in the final result. In this case, the identified shortcomings of the model are corrected during the simulation, and not calculated in advance.

Each modern man daily encounters the concepts of "object" and "model". Examples of objects are both objects that are accessible to touch (book, earth, table, pen, pencil) and inaccessible (stars, sky, meteorites), objects of artistic creation and mental activity (composition, poem, problem solving, painting, music, etc.) other). Moreover, each object is perceived by a person only as a single whole.

An object. Views. Specifications

Based on the above, we can conclude that the object is part of the external world, which can be perceived as a whole. Each object of perception has its own individual characteristics that distinguish it from others (shape, scope, color, smell, size, and so on). The most important characteristic an object is a name, but for a complete qualitative description of it, one name is not enough. The more fully and in detail an object is described, the easier it is to recognize it.

Models. Definition. Classification

In his activities (educational, scientific, artistic, technological), a person daily uses existing and creates new models of the external world. They allow you to form an impression about processes and objects that are inaccessible for direct perception (very small or, conversely, very large, very slow or very fast, very distant, and so on).

So, a model is some object that reflects the most important features of the studied phenomenon, object or process. There can be several variations of the models of the same object, just as several objects can be described by one single model. For example, a similar situation occurs in mechanics, when different bodies with a material shell can be expressed, that is, by the same model (man, car, train, plane).

It is important to remember that no model can fully replace the depicted object, since it displays only some of its properties. But sometimes, when solving certain problems of various scientific and industrial trends, the description appearance models can be not only useful, but the only opportunity to present and study the characteristics of the object.

Scope of application of modeling items

Models play an important role in various spheres of human life: in science, education, trade, design and others. For example, without their use, design and assembly are impossible. technical devices, mechanisms, electrical circuits, cars, buildings and so on, since without preliminary calculations and creating a drawing, the release of even the simplest part is impossible.

Models are often used for educational purposes. They are called descriptive. For example, from geography, a person gets an idea of ​​the Earth as a planet by studying the globe. Visual models are also relevant in other sciences (chemistry, physics, mathematics, biology, and others).

In turn, theoretical models are in demand in the study of natural and (biology, chemistry, physics, geometry). They reflect the properties, behavior and structure of the objects being studied.

Modeling as a process

Modeling is a method of cognition, which includes the study of existing and the creation of new models. The subject of knowledge of this science is a model. are ranked according to various properties. As you know, any object has many characteristics. When creating a specific model, only the most important ones for solving the problem are highlighted.

The process of creating models is artistic creation in all its diversity. In this regard, virtually every artistic or literary work can be considered as a model of a real object. For example, paintings are models of real landscapes, still lifes, people, literary works are models of human lives, and so on. For example, when creating a model of an airplane in order to study it, it is important to reflect the geometric properties of the original in it, but its color is absolutely unimportant.

The same objects are studied by different sciences from different points of view, and accordingly, their types of models for study will also differ. For example, physics studies the processes and results of the interaction of objects, chemistry - the chemical composition, biology - the behavior and structure of organisms.

Time factor model

With respect to time, models are divided into two types: static and dynamic. An example of the first type is a one-time examination of a person in a clinic. It displays a picture of his state of health on this moment, while his medical record will be a dynamic model, reflecting the changes taking place in the body over the course of a certain period time.

Model. Model views relative to shape

As it is already clear, the models can differ in different characteristics. So, all currently known types of data models can be conditionally divided into two main classes: material (subject) and informational.

The first type conveys the physical, geometric and other properties of objects in material form (anatomical dummy, globe, building model, and so on).

The types differ in the form of implementation: symbolic and figurative. Figurative models (photographs, drawings, etc.) are visual realizations of objects fixed on a specific medium (photographic, film, paper or digital).

They are widely used in the educational process (posters), in the study of various sciences (botany, biology, paleontology, and others). Sign models are implementations of objects in the form of symbols of one of the well-known language systems. They can be presented in the form of formulas, text, tables, diagrams, and so on. There are cases when, creating a sign model (types of models convey specifically the content that is required to study certain characteristics of an object), several well-known languages ​​are used at once. An example in this case there are various graphs, diagrams, maps and the like, where both graphic symbols and symbols of one of the language systems are used.

In order to reflect information from various spheres of life, three main types are used information models: network, hierarchical and tabular. Of these, the most popular is the latter, which is used to record various states of objects and their characteristic data.

Tabular model implementation

This type of information model, as mentioned above, is the most famous. It looks like this: it is an ordinary rectangular table consisting of rows and columns, the graphs of which are filled with symbols of one of the well-known sign languages ​​of the world. Are applied tabular models in order to characterize objects with the same properties.

With their help, both dynamic and static models can be created in various scientific fields. For example, tables containing mathematical functions, various statistics, train schedules, and so on.

Mathematical model. Types of models

Mathematical models are a separate type of information models. All kinds usually consist of equations written in the language of algebra. The solution of these problems, as a rule, is based on the process of finding equivalent transformations that contribute to the expression of a variable in the form of a formula. There are also exact solutions for some equations (square, linear, trigonometric, and so on). As a consequence, to solve them, it is necessary to apply solution methods with an approximate specified accuracy, in other words, such types of mathematical data as numerical (half-division method), graphical (plotting), and others. It is advisable to use the method of half division only under the condition that the segment is known where the function takes polar values ​​at certain values.

And the plotting method is unified. It can be used both in the above-described case and in a situation where the solution can only be approximate, and not exact, in the case of the so-called "rough" solution of equations.

Adequacy- the degree of conformity of the model to the investigated real object. It can never be complete. In practice, a model is considered adequate if it achieves the objectives of the study with satisfactory accuracy.

Complexity- quantitative characteristics of the properties of the object that describe the model. The higher it is, the more complex the model is. However, in practice, one should strive for the simplest model that allows one to achieve the required study results.

Potentiality- the ability of the model to give new knowledge about the object under study, to predict its behavior.

Mathematical models.

The main stages of building a mathematical model:

1. a description of the functioning of the system as a whole is drawn up;

2. a list of subsystems and elements is drawn up with a description of their functioning, characteristics and initial conditions, as well as interaction with each other;

3. a list of external factors affecting the system and their characteristics is determined;

4. the indicators of the efficiency of the system are selected, i.e. such numerical characteristics of the system that determine the degree of compliance of the system with its purpose;

5. a formal mathematical model of the system is drawn up;

6. a machine mathematical model is compiled, suitable for studying the system on a computer.

Requirements for the mathematical model:

Requirements are determined primarily by its purpose, i.e. the nature of the task:

A "good" model should be:

1. purposeful;

2. simple and understandable for the user;

3. sufficient from the point of view of the possibilities of solving the task;

4. easy to handle and manage;

5. reliable in the sense of protection against absurd answers;

6. Allows gradual changes in the sense that, being simple at first, it can become more complex when interacting with users.

Mathematical models. Mathematical models represent a formalized representation of a system using an abstract language, using mathematical relationships that reflect the process of the system's functioning. To compile mathematical models, you can use any mathematical means - algebraic, differential, integral calculus, set theory, theory of algorithms, etc. In essence, all mathematics is created for the compilation and study of models of objects and processes.

The means of abstract description of systems also include the languages ​​of chemical formulas, diagrams, drawings, maps, diagrams, etc. The choice of the type of model is determined by the characteristics of the system under study and the goals of modeling, since the study of the model allows you to get answers to a certain group questions. Other information may require a different type of model. Mathematical models can be classified as deterministic and probabilistic, analytical, numerical and simulation.

Deterministic modeling displays processes in which the absence of any random influences is assumed; stochastic modeling displays probabilistic processes and events. In this case, a number of realizations of the random process are analyzed and the average characteristics, i.e., a set of homogeneous realizations, are estimated.

Analytical a model is a formalized description of a system that allows one to obtain an explicit solution to an equation using a well-known mathematical apparatus.

Numerical model characterized by a dependence of this kind, which allows only particular solutions for specific initial conditions and quantitative parameters of the models.

Simulation model is a set of system descriptions and external influences, algorithms for the functioning of the system or the rules for changing the state of the system under the influence of external and internal disturbances. These algorithms and rules do not make it possible to use the available mathematical methods of analytical and numerical solution, but they allow simulating the process of the system's functioning and making calculations of the characteristics of interest. Simulation models can be created for a much wider class of objects and processes than analytical and numerical ones. Since VS are used for the implementation of simulation models, universal and special algorithmic languages ​​serve as the means of formalized description of IM. MI are most suitable for the study of VS at the systemic level.

8. The structure of the model. Modeling is the reproduction of the characteristics of one object on some other object, specially created for their study. The latter is called a model.

The structure of the model (and the physical one as well) is understood as a scoop of el-in included in the model and the connections between them. Moreover, the model (its elements) may have the same or a different physical nature. The proximity of structures is one of the main features in modeling. In each concrete case, the model can fulfill its role when the degree of its corresponding to the object is determined strictly enough. Simplifying the structure of the model reduces accuracy.

Type of model depends on the informational essence of the modeled system, on the connections and relations of its subsystems and elements, and not on its physical nature.

For example, mathematical descriptions ( model) the dynamics of the epidemic of an infectious disease, radioactive decay, the acquisition of a second foreign language, the release of products of a manufacturing enterprise, etc. can be considered the same in terms of their description, although the processes themselves are different.

The boundaries between models of various types are rather arbitrary. We can talk about different modes of use models- imitation, stochastic, etc.

Typically the model includes: object O, subject (optional) A, task Z, resources B, environment modeling WITH.

The model can be formally represented as: M =< O, Z, A, B, C > .

The main propertiesany model:

purposefulness - model always displays some system, i.e. has a purpose;

limb - model displays the original only in a finite number of its relations and, in addition, resources modeling are finite;

simplicity - model displays only the essential aspects of the object and, in addition, should be easy to study or reproduce;

approximation - reality is displayed model rough or rough;

adequacy - model must successfully describe the system being modeled;

visibility, visibility of its main properties and relationships;

availability and manufacturability for research or reproduction;

informativeness - model should contain sufficient information about the system (within the framework of the hypotheses adopted when constructing model) and should provide an opportunity to receive new information;

preservation of the information contained in the original (with the accuracy considered when constructing model hypotheses);

completeness - in model all the basic connections and relationships necessary to ensure the goal must be taken into account modeling;

stability - model should describe and ensure the stable behavior of the system, even if it is initially unstable;

integrity - model implements some system, i.e. whole;

isolation - model takes into account and displays a closed system of necessary basic hypotheses, connections and relationships;

adaptability - model can be adapted to various input parameters, environmental influences;

manageability - model must have at least one parameter, the changes of which can imitate the behavior of the modeled system under various conditions;

development opportunity models(previous level).

Life cycle of the simulated system:

collection of information about the object, hypothesis, preliminary model analysis;

design of structure and composition models(submodels);

building specifications model, development and debugging of individual sub-models, assembly model in general, identification (if needed) of parameters models;

study model- choice of research method and development of an algorithm (program) modeling;

study of adequacy, stability, sensitivity model;

assessment of funds modeling(spent resources);

interpretation, analysis of results modeling and the establishment of some causal relationships in the studied system;

generation of reports and design (national economic) solutions;

clarification, modification model, if necessary, and return to the system under study with new knowledge obtained using model and modeling.