Math modeling. Concept of a mathematical model

LECTURE 4

Definition and purpose of mathematical modeling

Under model(from the Latin modulus - measure, sample, norm) we will understand such a materially or mentally represented object, which in the process of cognition (study) replaces the original object, preserving some of its typical features that are important for this study. The process of building and using a model is called modeling.

The essence mathematical modeling (MM) consists of replacing the object (process) under study with an adequate mathematical model and subsequent study of the properties of this model using either analytical methods or computational experiments.

Sometimes it is more useful, instead of giving strict definitions, to describe a particular concept using a specific example. Therefore, we illustrate the above definitions of MM using the example of the problem of calculating specific impulse. In the early 60s, scientists were faced with the task of developing rocket fuel with the highest specific impulse. The principle of rocket propulsion is as follows: liquid fuel and oxidizer from the rocket tanks are supplied to the engine, where they are burned, and the combustion products are released into the atmosphere. From the law of conservation of momentum it follows that in this case the rocket will move with speed.

The specific impulse of a fuel is the received impulse divided by the mass of the fuel. Conducting experiments was very expensive and led to systematic damage to equipment. It turned out that it was easier and cheaper to calculate the thermodynamic functions of ideal gases, using them to calculate the composition of the escaping gases and the plasma temperature, and then the specific impulse. That is, to carry out the MM of the fuel combustion process.

The concept of mathematical modeling (MM) is one of the most common in scientific literature today. The vast majority of modern diploma and dissertation works are related to the development and use of appropriate mathematical models. Computer MM today is an integral part of many areas of human activity (science, technology, economics, sociology, etc.). This is one of the reasons for today's shortage of specialists in the field of information technology.

The rapid growth of mathematical modeling is due to the rapid improvement of computer technology. If 20 years ago only a small number of programmers were involved in numerical calculations, now the memory capacity and speed of modern computers make it possible to solve mathematical modeling problems accessible to all specialists, including university students.

In any discipline, a qualitative description of phenomena is first given. And then - quantitative, formulated in the form of laws establishing connections between various quantities (field strength, scattering intensity, electron charge, ...) in the form of mathematical equations. Therefore, we can say that in every discipline there is as much science as there is mathematics in it, and this fact allows many problems to be successfully solved using mathematical modeling methods.

This course is designed for students majoring in applied mathematics who are completing their graduate work under the supervision of leading scientists working in various fields. Therefore, this course is necessary not only as educational material, but also as preparation for the thesis. To study this course we will need the following sections of mathematics:

1. Equations of mathematical physics (cant mechanics, gas and hydrodynamics)

2. Linear algebra (elasticity theory)

3. Scalar and vector fields (field theory)

4. Probability theory (quantum mechanics, statistical physics, physical kinetics)

5. Special functions.

6. Tensor analysis (elasticity theory)

7. Mathematical analysis

MM in natural science, technology, and economics

Let us first consider various sections of natural science, technology, and economics in which mathematical models are used.

Natural science

Physics, which establishes the basic laws of natural science, has long been divided into theoretical and experimental. Theoretical physics deals with the derivation of equations that describe physical phenomena. Thus, theoretical physics can also be considered one of the areas of mathematical modeling. (Remember that the title of the first book on physics - “Mathematical Principles of Natural Philosophy” by I. Newton can be translated into modern language as “Mathematical Models of Natural Science”.) Based on the obtained laws, engineering calculations are carried out, which are carried out in various institutes, companies, design bureaus. These organizations develop technologies for the manufacture of modern products that are knowledge-intensive. Thus, the concept of science-intensive technologies includes calculations using appropriate mathematical models.

One of the most extensive branches of physics is classical mechanics(sometimes this section is called theoretical or analytical mechanics). This section of theoretical physics studies the movement and interaction of bodies. Calculations using formulas of theoretical mechanics are necessary when studying the rotation of bodies (calculation of moments of inertia, gyrostats - devices that keep the axis of rotation motionless), analyzing the movement of a body in airless space, etc. One of the sections of theoretical mechanics is called the theory of stability and underlies many mathematical models describing the movement of aircraft, ships, and missiles. Sections of practical mechanics - courses “Theory of Machines and Mechanisms”, “Machine Parts”, are studied by students of almost all technical universities (including Moscow State University).

Elasticity theory– part of a section continuum mechanics, which assumes that the material of an elastic body is homogeneous and continuously distributed throughout the entire volume of the body, so that the smallest element cut from the body has the same physical properties as the entire body. Application of the theory of elasticity - the course “strength of materials”, is studied by students of all technical universities (including Moscow State University). This section is required for all strength calculations. This includes the calculation of the strength of the hulls of ships, aircraft, rockets, the calculation of the strength of steel and reinforced concrete structures of buildings and much more.

Gas and hydrodynamics, like the theory of elasticity, is part of the section continuum mechanics, examines the laws of motion of liquids and gases. The equations of gas and hydrodynamics are necessary when analyzing the movement of bodies in a liquid and gaseous environment (satellites, submarines, missiles, projectiles, cars), when calculating the outflow of gas from the nozzles of rocket and airplane engines. Practical application of hydrodynamics - hydraulics (brake, steering wheel,...)

Previous sections of mechanics considered the movement of bodies in the macrocosm, and the physical laws of the macrocosm are not applicable in the microcosm, in which particles of matter move - protons, neutrons, electrons. Completely different principles apply here, and to describe the microworld it is necessary quantum mechanics. The basic equation describing the behavior of microparticles is the Schrödinger equation: . Here is the Hamiltonian operator (Hamiltonian). For a one-dimensional equation of particle motion https://pandia.ru/text/78/009/images/image005_136.gif" width="35" height="21 src=">-potential energy. The solution to this equation is a set of energy eigenvalues and eigenfunctions..gif" width="55" height="24 src=">– probability density. Quantum mechanical calculations are needed for the development of new materials (microcircuits), the creation of lasers, the development of spectral analysis methods, etc.

Solve a large number of problems kinetics, describing the movement and interaction of particles. Here we have diffusion, heat transfer, and the theory of plasma - the fourth state of matter.

Statistical physics considers ensembles of particles, allows us to say about the parameters of the ensemble based on the properties of individual particles. If the ensemble consists of gas molecules, then the properties of the ensemble derived by the methods of statistical physics are the equations of gas state, well known from high school: https://pandia.ru/text/78/009/images/image009_85.gif" width="16" height="17 src=">.gif" width="16" height="17"> is the molecular weight of the gas. K – Rydberg constant. The properties of solutions, crystals, and electrons in metals are also calculated using statistical methods. MM of statistical physics is the theoretical basis of thermodynamics, which underlies the calculation of engines, heating networks and stations.

Field theory describes using MM methods one of the main forms of matter – the field. In this case, the main interest is electromagnetic fields. The equations of the electromagnetic field (electrodynamics) were derived by Maxwell: , , . Here and https://pandia.ru/text/78/009/images/image018_44.gif" width="16" height="17"> - charge density, - current density. The equations of electrodynamics underlie calculations of the propagation of electromagnetic waves necessary to describe the propagation of radio waves (radio, television, cellular communications), and explain the operation of radar stations.

Chemistry can be presented in two aspects, highlighting descriptive chemistry - the discovery of chemical factors and their description - and theoretical chemistry - the development of theories that allow one to generalize established factors and present them in the form of a specific system (L. Pauling). Theoretical chemistry is also called physical chemistry and is, in essence, a branch of physics that studies substances and their interactions. Therefore, everything that has been said regarding physics fully applies to chemistry. The branches of physical chemistry will be thermochemistry, which studies the thermal effects of reactions, chemical kinetics (reaction rates), quantum chemistry (the structure of molecules). At the same time, chemistry problems can be extremely complex. For example, to solve problems of quantum chemistry – the science of the structure of atoms and molecules – programs are used that are comparable in scope to the country’s air defense programs. For example, in order to describe the UCl4 molecule, consisting of 5 atomic nuclei and +17 * 4) electrons, you need to write down the equation of motion - partial differential equations.

Biology

Mathematics really came to biology only in the second half of the 20th century. The first attempts to mathematically describe biological processes related to models of population dynamics. A population is a community of individuals of the same species occupying a certain area of ​​space on Earth. This area of ​​mathematical biology, which studies changes in population size under various conditions (presence of competing species, predators, diseases, etc.) subsequently served as a mathematical testing ground on which mathematical models in different areas of biology were “tested.” Including models of evolution, microbiology, immunology and other areas related to cell populations.
The very first known model formulated in a biological formulation is the famous Fibonacci series (each subsequent number is the sum of the previous two), which Leonardo of Pisa cited in his work in the 13th century. This is a series of numbers that describes the number of pairs of rabbits that are born each month if rabbits begin breeding from the second month and produce a pair of rabbits each month. The row represents a sequence of numbers: 1, 1, 2, 3, 5, 8, 13, 21, ...

1,

2 ,

3,

5,

8, 13, …

Another example is the study of ion transmembrane transport processes on an artificial bilayer membrane. Here, in order to study the laws of formation of the pore through which the ion passes through the membrane into the cell, it is necessary to create a model system that can be studied experimentally, and for which a physical description well developed by science can be used.

A classic example of MM is also the Drosophila population. An even more convenient model is viruses, which can be propagated in vitro. Modeling methods in biology are methods of dynamic systems theory, and means are differential and difference equations, methods of qualitative theory of differential equations, and simulation modeling.
Goals of modeling in biology:
3. Clarification of the mechanisms of interaction between system elements
4. Identification and verification of model parameters using experimental data.
5. Assessing the stability of the system (model).

6. Forecasting the behavior of the system under various external influences, various control methods, etc.
7. Optimal control of the system in accordance with the selected optimality criterion.

Technique

A large number of specialists are involved in improving technology, and in their work they rely on the results of scientific research. Therefore, the MM in technology is the same as the MM in natural science, which was discussed above.

Economics and social processes

It is generally accepted that mathematical modeling as a method of analyzing macroeconomic processes was first used by the physician of King Louis XV, Dr. Francois Quesnay, who in 1758 published the work “Economic Table”. This work made the first attempt to quantitatively describe the national economy. And in 1838 in the book O. Cournot"A Study of the Mathematical Principles of Wealth Theory" quantitative methods were first used to analyze competition in the product market under various market situations.

Malthus's theory of population is also widely known, in which he proposed the idea: population growth is not always desirable, and this growth is faster than the growth of the ability to provide the population with food. The mathematical model of such a process is quite simple: Let the population growth during the time https://pandia.ru/text/78/009/images/image027_26.gif" width="15" height="24"> be equal to . and - coefficients taking into account fertility and mortality (persons/year).

https://pandia.ru/text/78/009/images/image032_23.gif" width="151" height="41 src=">Instrumental and mathematical methods " href="/text/category/instrumentalmznie_i_matematicheskie_metodi/" rel ="bookmark">mathematical methods of analysis (for example, in recent decades, mathematical theories of cultural development have appeared in the humanities, mathematical models of mobilization, cyclical development of socio-cultural processes, a model of interaction between the people and the government, a model of the arms race, etc. have been constructed and studied).

In the most general terms, the process of MM of socio-economic processes can be divided into four stages:

formulation of a system of hypotheses and development of a conceptual model; development of a mathematical model; analysis of the results of model calculations, which includes comparing them with practice; formulation of new hypotheses and refinement of the model in case of discrepancy between the calculation results and practical data.

Note that, as a rule, the process of mathematical modeling is cyclical in nature, since even when studying relatively simple processes it is rarely possible to build an adequate mathematical model and select its exact parameters from the first step.

Currently, the economy is considered as a complex developing system, for the quantitative description of which dynamic mathematical models of varying degrees of complexity are used. One of the areas of research in macroeconomic dynamics is associated with the construction and analysis of relatively simple nonlinear simulation models that reflect the interaction of various subsystems - the labor market, the goods market, the financial system, the natural environment, etc.

The theory of catastrophes is developing successfully. This theory considers the question of the conditions under which a change in the parameters of a nonlinear system causes a point in phase space, characterizing the state of the system, to move from the region of attraction to the initial equilibrium position to the region of attraction to another equilibrium position. The latter is very important not only for the analysis of technical systems, but also for understanding the sustainability of socio-economic processes. In this regard, the findings are of interest about the importance of studying nonlinear models for management. In the book “The Theory of Catastrophes,” published in 1990, he writes, in particular: “... the current perestroika is largely explained by the fact that at least some feedback mechanisms have begun to operate (fear of personal destruction).”

(model parameters)

When building models of real objects and phenomena, one often has to deal with a lack of information. For the object under study, the distribution of properties, impact parameters and initial state are known with varying degrees of uncertainty. When building a model, the following options for describing uncertain parameters are possible:

Classification of mathematical models

(implementation methods)

Methods for implementing MM can be classified according to the table below.

Methods for implementing MM Very often the analytical solution for a model is presented in the form of functions. To obtain the values ​​of these functions for specific values ​​of the input parameters, their expansion into series (for example, Taylor) is used, and the value of the function for each value of the argument is determined approximately. Models that use this technique are called close. At numerical approach the set of mathematical relations of the model is replaced by a finite-dimensional analogue. This is most often achieved by discretizing the original relations, i.e., by moving from functions of a continuous argument to functions of a discrete argument (grid methods). The solution found after computer calculations is taken as an approximate solution to the original problem. Most existing systems are very complex, and for them it is impossible to create a real model described analytically. Such systems should be studied using simulation modeling. One of the main methods of simulation modeling is associated with the use of a random number sensor. Since a huge number of problems are solved using MM methods, methods for implementing MM are studied in more than one course. This includes partial differential equations, numerical methods for solving these equations, computational mathematics, computer modeling, etc. Pauling, Linus Carl (Pauling, Linus Carl), American chemist and physicist, awarded the 1954 Nobel Prize in Chemistry for his studies of the nature of chemical bonds and determination of the structure of proteins. Born February 28, 1901 in Portland (Oregon). He developed a quantum mechanical method for studying the structure of molecules (along with the American physicist J. Slayer) - the method of valence bonds, as well as the theory of resonance, which makes it possible to explain the structure of carbon-containing compounds, primarily aromatic compounds. During the period of the personality cult of the USSR, scientists involved in quantum chemistry were persecuted and accused of “Paulingism.” MALTHUS, THOMAS ROBERT (Malthus, Thomas Robert) (), English economist. Born in Rookery near Dorking in Surrey on February 15 or 17, 1766. In 1798 he published his work anonymously Experience on the law of population. In 1819 Malthus was elected a member of the Royal Society.

According to the textbook by Sovetov and Yakovlev: “a model (lat. modulus - measure) is a substitute object for the original object, which ensures the study of some properties of the original.” (p. 6) “Replacing one object with another in order to obtain information about the most important properties of the original object using a model object is called modeling.” (p. 6) “By mathematical modeling we understand the process of establishing a correspondence to a given real object with a certain mathematical object, called a mathematical model, and the study of this model, which allows us to obtain the characteristics of the real object under consideration. The type of mathematical model depends both on the nature of the real object and the tasks of studying the object and the required reliability and accuracy of solving this problem.”

Finally, the most concise definition of a mathematical model: "An equation expressing an idea."

Model classification

Formal classification of models

The formal classification of models is based on the classification of the mathematical tools used. Often constructed in the form of dichotomies. For example, one of the popular sets of dichotomies:

and so on. Each constructed model is linear or nonlinear, deterministic or stochastic, ... Naturally, mixed types are also possible: concentrated in one respect (in terms of parameters), distributed in another, etc.

Classification according to the way the object is represented

Along with the formal classification, models differ in the way they represent an object:

• Structural or functional models

Structural models represent an object as a system with its own structure and functioning mechanism. Functional models do not use such representations and reflect only the externally perceived behavior (functioning) of an object. In their extreme expression, they are also called “black box” models. Combined types of models are also possible, which are sometimes called “gray box” models.

Content and formal models

Almost all authors describing the process of mathematical modeling indicate that first a special ideal structure is built, content model. There is no established terminology here, and other authors call this ideal object conceptual model , speculative model or premodel. In this case, the final mathematical construction is called formal model or simply a mathematical model obtained as a result of the formalization of a given meaningful model (pre-model). The construction of a meaningful model can be done using a set of ready-made idealizations, as in mechanics, where ideal springs, rigid bodies, ideal pendulums, elastic media, etc. provide ready-made structural elements for meaningful modeling. However, in areas of knowledge where there are no fully completed formalized theories (the cutting edge of physics, biology, economics, sociology, psychology, and most other areas), the creation of meaningful models becomes dramatically more difficult.

Content classification of models

No hypothesis in science can be proven once and for all. Richard Feynman formulated this very clearly:

“We always have the opportunity to disprove a theory, but note that we can never prove that it is correct. Let's assume that you have put forward a successful hypothesis, calculated where it leads, and found that all its consequences are confirmed experimentally. Does this mean your theory is correct? No, it simply means that you failed to refute it.”

If a model of the first type is built, this means that it is temporarily recognized as truth and one can concentrate on other problems. However, this cannot be a point in research, but only a temporary pause: the status of a model of the first type can only be temporary.

Type 2: Phenomenological model (we behave as if…)

A phenomenological model contains a mechanism for describing a phenomenon. However, this mechanism is not convincing enough, cannot be sufficiently confirmed by the available data, or does not fit well with existing theories and accumulated knowledge about the object. Therefore, phenomenological models have the status of temporary solutions. It is believed that the answer is still unknown and the search for the “true mechanisms” must continue. Peierls includes, for example, the caloric model and the quark model of elementary particles as the second type.

The role of the model in research may change over time, and it may happen that new data and theories confirm phenomenological models and they are promoted to the status of a hypothesis. Likewise, new knowledge can gradually come into conflict with models-hypotheses of the first type, and they can be translated into the second. Thus, the quark model is gradually moving into the category of hypotheses; atomism in physics arose as a temporary solution, but with the course of history it became the first type. But the ether models have made their way from type 1 to type 2, and are now outside science.

The idea of ​​simplification is very popular when building models. But simplification comes in different forms. Peierls identifies three types of simplifications in modeling.

Type 3: Approximation (we consider something very big or very small)

If it is possible to construct equations that describe the system under study, this does not mean that they can be solved even with the help of a computer. A common technique in this case is the use of approximations (type 3 models). Among them linear response models. The equations are replaced by linear ones. A standard example is Ohm's law.

Here comes type 8, which is widespread in mathematical models of biological systems.

Type 8: Feature Demonstration (the main thing is to show the internal consistency of the possibility)

These are also thought experiments with imaginary entities, demonstrating that supposed phenomenon consistent with basic principles and internally consistent. This is the main difference from models of type 7, which reveal hidden contradictions.

One of the most famous of these experiments is Lobachevsky’s geometry (Lobachevsky called it “imaginary geometry”). Another example is the mass production of formally kinetic models of chemical and biological vibrations, autowaves, etc. The Einstein-Podolsky-Rosen paradox was conceived as a type 7 model to demonstrate the inconsistency of quantum mechanics. In a completely unplanned way, it eventually turned into a type 8 model - a demonstration of the possibility of quantum teleportation of information.

Example

Consider a mechanical system consisting of a spring attached at one end and a mass of mass m attached to the free end of the spring. We will assume that the load can only move in the direction of the spring axis (for example, movement occurs along the rod). Let's build a mathematical model of this system. We will describe the state of the system by the distance x from the center of the load to its equilibrium position. Let us describe the interaction of the spring and the load using Hooke's law (F = − kx ) and then use Newton's second law to express it in the form of a differential equation:

where means the second derivative of x by time: .

The resulting equation describes the mathematical model of the considered physical system. This model is called a "harmonic oscillator".

According to the formal classification, this model is linear, deterministic, dynamic, concentrated, continuous. In the process of its construction, we made many assumptions (about the absence of external forces, the absence of friction, the smallness of deviations, etc.), which in reality may not be fulfilled.

In relation to reality, this is most often a type 4 model simplification(“we will omit some details for clarity”), since some essential universal features (for example, dissipation) are omitted. To some approximation (say, while the deviation of the load from equilibrium is small, with low friction, for not too much time and subject to certain other conditions), such a model describes a real mechanical system quite well, since the discarded factors have a negligible effect on its behavior . However, the model can be refined by taking into account some of these factors. This will lead to a new model, with a wider (though again limited) scope of applicability.

However, when refining the model, the complexity of its mathematical research can increase significantly and make the model virtually useless. Often, a simpler model allows for better and deeper exploration of a real system than a more complex one (and, formally, “more correct”).

If we apply the harmonic oscillator model to objects far from physics, its substantive status may be different. For example, when applying this model to biological populations, it should most likely be classified as type 6 analogy(“let’s take into account only some features”).

Hard and soft models

The harmonic oscillator is an example of the so-called “hard” model. It is obtained as a result of a strong idealization of a real physical system. To resolve the issue of its applicability, it is necessary to understand how significant the factors that we have neglected are. In other words, it is necessary to study the “soft” model, which is obtained by a small perturbation of the “hard” one. It can be given, for example, by the following equation:

Here is some function that can take into account the friction force or the dependence of the spring stiffness coefficient on the degree of its stretching - some small parameter. Explicit function form f We are not interested at the moment. If we prove that the behavior of the soft model is not fundamentally different from the behavior of the hard one (regardless of the explicit type of perturbing factors, if they are small enough), the problem will be reduced to studying the hard model. Otherwise, the application of the results obtained from studying the rigid model will require additional research. For example, the solution to the equation of a harmonic oscillator is functions of the form , that is, oscillations with a constant amplitude. Does it follow from this that a real oscillator will oscillate indefinitely with a constant amplitude? No, because considering a system with arbitrarily small friction (always present in a real system), we get damped oscillations. The behavior of the system has changed qualitatively.

If a system maintains its qualitative behavior under small disturbances, it is said to be structurally stable. A harmonic oscillator is an example of a structurally unstable (non-rough) system. However, this model can be used to study processes over limited periods of time.

Versatility of models

The most important mathematical models usually have the important property versatility: Fundamentally different real phenomena can be described by the same mathematical model. For example, a harmonic oscillator describes not only the behavior of a load on a spring, but also other oscillatory processes, often of a completely different nature: small oscillations of a pendulum, fluctuations in the level of a liquid in U-shaped vessel or a change in current strength in an oscillatory circuit. Thus, by studying one mathematical model, we immediately study a whole class of phenomena described by it. It is this isomorphism of laws expressed by mathematical models in various segments of scientific knowledge that inspired Ludwig von Bertalanffy to create the “General Theory of Systems”.

Direct and inverse problems of mathematical modeling

There are many problems associated with mathematical modeling. First, you need to come up with a basic diagram of the modeled object, reproduce it within the framework of the idealizations of this science. Thus, a train car turns into a system of plates and more complex bodies from different materials, each material is specified as its standard mechanical idealization (density, elastic moduli, standard strength characteristics), after which equations are drawn up, and along the way some details are discarded as unimportant , calculations are made, compared with measurements, the model is refined, and so on. However, to develop mathematical modeling technologies, it is useful to disassemble this process into its main components.

Traditionally, there are two main classes of problems associated with mathematical models: direct and inverse.

Direct task: the structure of the model and all its parameters are considered known, the main task is to conduct a study of the model to extract useful knowledge about the object. What static load will the bridge withstand? How it will react to a dynamic load (for example, to the march of a company of soldiers, or to the passage of a train at different speeds), how the plane will overcome the sound barrier, whether it will fall apart from flutter - these are typical examples of a direct problem. Setting the right direct problem (asking the right question) requires special skill. If the right questions are not asked, a bridge may collapse, even if a good model for its behavior has been built. So, in 1879, a metal bridge across the River Tay collapsed in England, the designers of which built a model of the bridge, calculated it to have a 20-fold safety factor for the action of the payload, but forgot about the winds constantly blowing in those places. And after a year and a half it collapsed.

In the simplest case (one oscillator equation, for example), the direct problem is very simple and reduces to an explicit solution of this equation.

Inverse problem: many possible models are known, a specific model must be selected based on additional data about the object. Most often, the structure of the model is known, and some unknown parameters need to be determined. Additional information may consist of additional empirical data, or requirements for the object ( design problem). Additional data can arrive regardless of the process of solving the inverse problem ( passive observation) or be the result of an experiment specially planned during the solution ( active surveillance).

One of the first examples of a masterly solution to an inverse problem with the fullest use of available data was the method constructed by I. Newton for reconstructing friction forces from observed damped oscillations.

Additional examples

Where x s- the “equilibrium” population size, at which the birth rate is exactly compensated by the death rate. The population size in such a model tends to an equilibrium value x s, and this behavior is structurally stable.

This system has an equilibrium state when the number of rabbits and foxes is constant. Deviation from this state results in fluctuations in the numbers of rabbits and foxes, similar to the fluctuations of a harmonic oscillator. As with the harmonic oscillator, this behavior is not structurally stable: a small change in the model (for example, taking into account the limited resources required by rabbits) can lead to a qualitative change in behavior. For example, the equilibrium state may become stable, and fluctuations in numbers will die out. The opposite situation is also possible, when any small deviation from the equilibrium position will lead to catastrophic consequences, up to the complete extinction of one of the species. The Volterra-Lotka model does not answer the question of which of these scenarios is being realized: additional research is required here.

Notes

1. “A mathematical representation of reality” (Encyclopaedia Britanica)
2. Novik I. B., On philosophical issues of cybernetic modeling. M., Knowledge, 1964.
3. Sovetov B. Ya., Yakovlev S. A., Modeling of systems: Proc. for universities - 3rd ed., revised. and additional - M.: Higher. school, 2001. - 343 p. ISBN 5-06-003860-2
4. Samarsky A. A., Mikhailov A. P. Math modeling. Ideas. Methods. Examples. . - 2nd ed., revised. - M.: Fizmatlit, 2001. - ISBN 5-9221-0120-X
5. Myshkis A. D., Elements of the theory of mathematical models. - 3rd ed., rev. - M.: KomKniga, 2007. - 192 with ISBN 978-5-484-00953-4
6. Wiktionary: mathematical model
7. CliffsNotes
8. Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena, Springer, Complexity series, Berlin-Heidelberg-New York, 2006. XII+562 pp. ISBN 3-540-35885-4
9. “A theory is considered linear or nonlinear depending on what kind of mathematical apparatus - linear or nonlinear - and what kind of linear or nonlinear mathematical models it uses. ...without denying the latter. A modern physicist, if he had to re-create the definition of such an important entity as nonlinearity, would most likely act differently, and, giving preference to nonlinearity as the more important and widespread of the two opposites, would define linearity as “not nonlinearity.” Danilov Yu. A., Lectures on nonlinear dynamics. Elementary introduction. Series “Synergetics: from past to future.” Edition 2. - M.: URSS, 2006. - 208 p. ISBN 5-484-00183-8
10. “Dynamical systems modeled by a finite number of ordinary differential equations are called concentrated or point systems. They are described using a finite-dimensional phase space and are characterized by a finite number of degrees of freedom. The same system under different conditions can be considered either concentrated or distributed. Mathematical models of distributed systems are partial differential equations, integral equations, or ordinary delay equations. The number of degrees of freedom of a distributed system is infinite, and an infinite number of data are required to determine its state.” Anishchenko V. S., Dynamic systems, Soros educational journal, 1997, No. 11, p. 77-84.
11. “Depending on the nature of the processes being studied in the system S, all types of modeling can be divided into deterministic and stochastic, static and dynamic, discrete, continuous and discrete-continuous. Deterministic modeling reflects deterministic processes, that is, processes in which the absence of any random influences is assumed; stochastic modeling depicts probabilistic processes and events. ... Static modeling serves to describe the behavior of an object at any point in time, and dynamic modeling reflects the behavior of an object over time. Discrete modeling is used to describe processes that are assumed to be discrete, respectively, continuous modeling allows us to reflect continuous processes in systems, and discrete-continuous modeling is used for cases when they want to highlight the presence of both discrete and continuous processes.” Sovetov B. Ya., Yakovlev S. A., Modeling of systems: Proc. for universities - 3rd ed., revised. and additional - M.: Higher. school, 2001. - 343 p. ISBN 5-06-003860-2
12. Typically, a mathematical model reflects the structure (device) of the modeled object, the properties and relationships of the components of this object that are essential for the purposes of research; such a model is called structural. If the model reflects only how the object functions - for example, how it reacts to external influences - then it is called functional or, figuratively, a black box. Combined models are also possible. Myshkis A. D., Elements of the theory of mathematical models. - 3rd ed., rev. - M.: KomKniga, 2007. - 192 with ISBN 978-5-484-00953-4
13. “The obvious, but most important initial stage of constructing or selecting a mathematical model is obtaining as clear a picture as possible about the object being modeled and refining its meaningful model, based on informal discussions. You should not spare time and effort at this stage; the success of the entire study largely depends on it. It has happened more than once that significant work spent on solving a mathematical problem turned out to be ineffective or even wasted due to insufficient attention to this side of the matter.” Myshkis A. D., Elements of the theory of mathematical models. - 3rd ed., rev. - M.: KomKniga, 2007. - 192 with ISBN 978-5-484-00953-4, p. 35.
14. « Description of the conceptual model of the system. At this substage of building a system model: a) the conceptual model M is described in abstract terms and concepts; b) a description of the model is given using standard mathematical schemes; c) hypotheses and assumptions are finally accepted; d) the choice of procedure for approximating real processes when constructing a model is justified.” Sovetov B. Ya., Yakovlev S. A., Modeling of systems: Proc. for universities - 3rd ed., revised. and additional - M.: Higher. school, 2001. - 343 p. ISBN 5-06-003860-2, p. 93.

The computer has firmly entered our lives, and there is practically no area of ​​human activity where a computer is not used. Computers are now widely used in the process of creating and researching new machines, new technological processes and searching for their optimal options; when solving economic problems, when solving problems of planning and production management at various levels. The creation of large objects in rocket technology, aircraft manufacturing, shipbuilding, as well as the design of dams, bridges, etc. is generally impossible without the use of computers.

To use a computer in solving applied problems, first of all, the applied problem must be “translated” into a formal mathematical language, i.e. for a real object, process or system, its mathematical model must be built.

The word "Model" comes from the Latin modus (copy, image, outline). Modeling is the replacement of some object A with another object B. The replaced object A is called the original or modeling object, and the replacement B is called a model. In other words, a model is a substitute object for the original object, which provides the study of some properties of the original.

The purpose of modeling is to obtain, process, present and use information about objects that interact with each other and the external environment; and the model here acts as a means of understanding the properties and patterns of behavior of an object.

Mathematical modeling is a means of studying a real object, process or system by replacing them with a mathematical model that is more convenient for experimental research using a computer.

Mathematical modeling is the process of constructing and studying mathematical models of real processes and phenomena. All natural and social sciences that use mathematical apparatus are essentially engaged in mathematical modeling: they replace a real object with its model and then study the latter. As with any modeling, a mathematical model does not fully describe the phenomenon being studied, and questions about the applicability of the results obtained in this way are very meaningful. A mathematical model is a simplified description of reality using mathematical concepts.

A mathematical model expresses the essential features of an object or process in the language of equations and other mathematical tools. As a matter of fact, mathematics itself owes its existence to what it is trying to reflect, i.e. model, in your own specific language, the patterns of the surrounding world.

At mathematical modeling the study of an object is carried out through a model formulated in the language of mathematics using certain mathematical methods.

The path of mathematical modeling in our time is much more comprehensive than full-scale modeling. A huge impetus to the development of mathematical modeling was given by the advent of computers, although the method itself originated simultaneously with mathematics thousands of years ago.

Mathematical modeling as such does not always require computer support. Every specialist professionally involved in mathematical modeling does everything possible to analytically study the model. Analytical solutions (i.e., presented by formulas expressing the results of the study through the original data) are usually more convenient and more informative than numerical ones. The capabilities of analytical methods for solving complex mathematical problems, however, are very limited and, as a rule, these methods are much more complex than numerical ones.

A mathematical model is an approximate representation of real objects, processes or systems, expressed in mathematical terms and preserving the essential features of the original. Mathematical models in quantitative form, using logical and mathematical constructs, describe the basic properties of an object, process or system, its parameters, internal and external connections

All models can be divided into two classes:

1. real,
2. perfect.

In turn, real models can be divided into:

1. full-scale,
2. physical,
3. mathematical.

Ideal models can be divided into:

1. visual,
2. iconic,
3. mathematical.

Real full-scale models are real objects, processes and systems on which scientific, technical and industrial experiments are carried out.

Real physical models are models, dummies that reproduce the physical properties of the originals (kinematic, dynamic, hydraulic, thermal, electrical, light models).

Real mathematical ones are analog, structural, geometric, graphic, digital and cybernetic models.

Ideal visual models are diagrams, maps, drawings, graphs, graphs, analogues, structural and geometric models.

Ideal sign models are symbols, alphabet, programming languages, ordered notation, topological notation, network representation.

Ideal mathematical models are analytical, functional, simulation, and combined models.

In the above classification, some models have a double interpretation (for example, analog). All models, except full-scale ones, can be combined into one class of mental models, because they are a product of human abstract thinking.

Elements of Game Theory

In the general case, solving a game is a rather difficult problem, and the complexity of the problem and the amount of calculations required to solve it increases sharply with increasing . However, these difficulties are not of a fundamental nature and are associated only with a very large volume of calculations, which in some cases may turn out to be practically impossible. The principle aspect of the method of finding a solution remains for any the same one.

Let's illustrate this with the example of a game. Let's give it a geometric interpretation - already a spatial one. Our three strategies will be represented by three points on the plane ; the first lies at the origin of coordinates (Fig. 1). the second and third - on the axes Oh And OU at distances 1 from the beginning.

Axes I-I, II-II and III-III are drawn through the points, perpendicular to the plane . On the I-I axis are the payoffs for the strategy; on the II-II and III-III axes are the payoffs for the strategies. Every enemy strategy will be represented by a plane cutting off on axes I-I, II-II and III-III, segments equal to the winnings

with appropriate strategy and strategy . Having thus constructed all the enemy’s strategies, we obtain a family of planes over the triangle (Fig. 2).

For this family, you can also construct a lower bound for the payoff, as we did in the case, and find on this boundary the point N with the maximum height on the plane . This height will be the price of the game.

The frequencies of strategies in the optimal strategy will be determined by the coordinates (x, y) points N, namely:

However, such a geometric construction, even for a case, is not easy to implement and requires a lot of time and effort of imagination. In the general case of the game, it is transferred to -dimensional space and loses all clarity, although the use of geometric terminology in a number of cases may be useful. When solving games in practice, it is more convenient to use not geometric analogies, but calculated analytical methods, especially since these methods are the only ones suitable for solving the problem on computers.

All of these methods essentially come down to solving a problem through successive trials, but ordering the sequence of trials allows you to build an algorithm that leads to a solution in the most economical way.

Here we will briefly look at one calculation method for solving games - using the so-called linear programming method.

To do this, we first give a general formulation of the problem of finding a solution to the game. Let a game be given with T player strategies A And n player strategies IN and the payment matrix is ​​given

It is required to find a solution to the game, i.e. two optimal mixed strategies of players A and B

where (some of the numbers and may be equal to zero).

Our optimal strategy S*A should provide us with a gain no less than , for any behavior of the enemy, and a gain equal to , for his optimal behavior (strategy S*B).Similar strategy S*B should provide the enemy with a loss no greater than , for any of our behavior and equal for our optimal behavior (strategy S*A).

The value of the game in this case is unknown to us; we will assume that it is equal to some positive number. Believing this way, we do not violate the generality of reasoning; In order for it to be > 0, it is obviously enough that all elements of the matrix are non-negative. This can always be achieved by adding a sufficiently large positive value L to the elements; in this case, the price of the game will increase by L, but the solution will not change.

Let us choose our optimal strategy S*A. Then our average payoff under the opponent’s strategy will be equal to:

Our optimal strategy S*A has the property that, for any behavior of the enemy, it provides a gain no less than; therefore, any of the numbers cannot be less than . We get a number of conditions:

(1)

Let us divide inequalities (1) by a positive value and denote:

Then condition (1) will be written as

(2)

where are non-negative numbers. Because the quantities satisfy the condition

We want to make our guaranteed winnings as high as possible; Obviously, in this case, the right side of equality (3) takes on a minimum value.

Thus, the problem of finding a solution to the game comes down to the following mathematical problem: determine non-negative quantities , satisfying conditions (2), so that their sum

was minimal.

Usually, when solving problems related to finding extreme values ​​(maxima and minima), the function is differentiated and the derivatives are set equal to zero. But such a technique is useless in this case, since the function Ф, which need to minimize, is linear, and its derivatives with respect to all arguments are equal to one, i.e., they do not vanish anywhere. Consequently, the maximum of the function is achieved somewhere on the boundary of the range of changes in the arguments, which is determined by the requirement of non-negativity of the arguments and conditions (2). The technique of finding extreme values ​​using differentiation is also unsuitable in cases where the maximum of the lower (or minimum of the upper) limit of the winnings is determined to solve the game, as we did. for example, they did it when solving games. Indeed, the lower bound is made up of sections of straight lines, and the maximum is achieved not at the point where the derivative is equal to zero (there is no such point at all), but at the boundary of the interval or at the point of intersection of straight sections.

To solve such problems, which are quite often encountered in practice, a special apparatus has been developed in mathematics linear programming.

The linear programming problem is formulated as follows.

Given a system of linear equations:

(4)

It is required to find non-negative values ​​of quantities that satisfy conditions (4) and at the same time minimize the given homogeneous linear function of quantities (linear form):

It is easy to see that the game theory problem posed above is a special case of a linear programming problem with

At first glance, it may seem that conditions (2) are not equivalent to conditions (4), since instead of equal signs they contain inequality signs. However, it is easy to get rid of the inequality signs by introducing new dummy non-negative variables and writing conditions (2) in the form:

(5)

The form Ф that needs to be minimized is equal to

The linear programming apparatus makes it possible to select values ​​using a relatively small number of successive samples , satisfying the stated requirements. For greater clarity, we will demonstrate here the use of this apparatus directly on the material of solving specific games.

PREFACE

The purpose of the course on modeling hoisting and transport systems is to teach the basics of modeling hoisting and transport machines (HTM), which includes the compilation of mathematical models of HTM, software implementation of models on a computer, as well as obtaining, processing and analysis of modeling results.

For independent familiarization with the listed issues, the following literature is recommended: Braude V.I., Ter-Mkhitarov M.S. “System methods for calculating lifting machines”, Ignatiev N.B., Ilyevsky B.Z., Klaus L.P. “Modeling machine systems”, Rachkov E.V., Silikov Yu.V. “Lifting and transport machines and mechanisms”, as well as reference books and tutorials on numerical methods of computational mathematics and the use of the MathCad mathematical editor.

§1. MAIN GOALS, DEFINITIONS AND PRINCIPLES OF MATHEMATICAL MODELING, TYPES OF MODELS

1.1 Basic definitions

Modeling is a theoretical and experimental method of cognitive activity; it is a method of studying and explaining phenomena, processes and systems (original objects) based on the creation of new objects - models.

Modeling is the replacement of the object under study (original) with its conventional image or another object (model) and the study of the properties of the original by studying the properties of the model.

Depending on the method of implementation, all models can be divided into 4 groups: physical, mathematical, subject-mathematical and combined [, ].

A physical model is a real embodiment of those properties of the original that interest the researcher. Physical models are also called layouts, so physical modeling is called prototyping.

A mathematical model is a formalized description of a system (or process) using some abstract language (mathematically), for example, in the form of graphs, equations, algorithms, mathematical correspondences, etc.

Subject-mathematical models are analog, i.e. in this case, for modeling, the principle of the same mathematical description of processes, real and occurring in the model, is used.

Combined models are a combination of a mathematical or subject-mathematical and physical model. They are used when the mathematical description of one of the elements of the system under study is unknown or difficult, and also, according to the modeling conditions, it is necessary to introduce a physical model (for example, a simulator) as an element.

Mathematical modeling is the replacement of the original with a mathematical model and the study of the properties of the original using this model.

A system is a combination of several objects (elements) interconnected, forming a certain integrity.

An element is a relatively independent part of the system, considered at this level of analysis as a single whole, intended to implement a certain function.

The system has the following, so-called "system" properties:

structure, i.e. a strictly defined order of combining elements into groups;

purposefulness or functionality, i.e. the presence of a purpose for which the system was created;

efficiency, the ability to achieve goals with the least expenditure of resources;

stability, the ability to maintain the characteristics of its properties unchanged within certain limits when external conditions change.

Currently, in technology, the concept of “human-machine system” (HMS) is used to study the operation of machine complexes and machines, i.e. mixed system, an integral part of which, along with technical objects, is a human operator [, ]. In addition, HMS interacts with the environment. Thus, to model the PTS, it is necessary to consider the Man-Machine-Environment system, which can be displayed by the following graph (Fig. 1).

R
is. 1 Graph of the Man-Machine-Environment system.

The arrows on the graph depict the flows of energy, matter and information that are exchanged between the elements of the system.

The processes occurring in technical systems are formed by a set of simple operations. Operations are transformations of input physical quantities into output ones in a low-level element of the system (Fig. 2).

In each element of the system (E i), the transformation of input influences (X i) into output influences (Y i) occurs, and the output influences of one element can be the input of the next. The connection of elements into a structural diagram according to the nature of the transfer of influences occurs sequentially or in parallel.

Rice. 2 Block diagram of the system.

Lifting and transport systems (HTS), studied in this course, will be called systems that include a person, the environment and hoisting and transport machines (HTM).

PTMs are machines designed to move cargo over relatively short distances without processing it. PTMs are used to facilitate, speed up, and increase the efficiency of reloading operations.

1.2 Principles and types of mathematical modeling

Mathematical models must have the following properties:

adequacy, property of correspondence between the model and the object of research;

reliability, ensuring the specified probability of modeling results falling into the confidence interval,

accuracy, insignificant (within the permissible error) discrepancy between the simulation results and the indicators of real objects (processes);

stability, the property of correspondence of small changes in output parameters to small changes in input parameters;

efficiency, the ability to achieve a goal with low expenditure of resources;

adaptability, the ability to easily adapt to solve various problems.

To achieve these properties, there are some principles (rules) of mathematical modeling, a number of which are given below.

The principle of purposefulness is that the model must ensure the achievement of strictly defined goals and, first of all, reflect those properties of the original that are necessary to achieve the goal.

The principle of information sufficiency consists in limiting the amount of information about an object when creating its model and searching for the optimum between the input information and the modeling results.

It can be illustrated by the following diagram.

All possible simulation cases are located in column 2. Feasibility principle

is that the model must ensure achievement of the set goal with a probability close to 1 and in a finite time. This principle can be expressed in two terms
, (1)

And
Where
- probability of achieving the goal, - time to achieve the goal,

and - acceptable values ​​of the probability and time of achieving the goal. Aggregation principle

is that the model should consist of 1st level subsystems, which, in turn, consist of 2nd level subsystems, etc. Subsystems must be designed as separate independent blocks. Such a model construction allows the use of standard calculation procedures, and also makes it easier to adapt the model to solve various problems.

Parameterization principle

consists in replacing, when modeling, certain parameters of subsystems described by functions corresponding to numerical characteristics.

The modeling process using these rules consists of the following 5 steps (stages).

Define modeling goals.

Development of a conceptual model (calculation scheme).

Formalization.

Implementation of the model.

Analysis and interpretation of simulation results. is the use of a mathematical model in the form of equations supplemented by a system of constraints that connect input variables with output parameters. Analytical modeling is used if there is a complete formulation of the research problem and it is necessary to obtain one final result corresponding to it.

Simulation modeling is the use of a mathematical model to describe the functioning of a system over time under various combinations of system parameters and various external influences. Simulation modeling is used if there is no final formulation of the problem and it is necessary to study the processes occurring in the system. Simulation modeling assumes compliance with a time scale. Those. events on the model occur at time intervals proportional to events on the original with a constant proportionality coefficient.

Based on the use of tools to implement the model, one more type of modeling can be distinguished, computer modeling. Computer modeling is mathematical modeling using computer technology.

1.3 Classification of mathematical models

All mathematical models can be divided into several groups according to the following classification criteria.

Depending on the type of system being modeled, models can be static or dynamic. Static models are used to study static systems, dynamic models are used to study dynamic ones. Dynamic systems are characterized by having multiple states that change over time.

According to the purposes of modeling, models are divided into load, managerial and functional. Load models are used to determine the loads acting on the elements of the system, management models are used to determine the kinematic parameters of the system under study, which include speeds and movements of system elements, functional models are used to determine the coordinates of the model in the space of possible functional states of the system.

Based on the degree of discretization, models are divided into discrete, mixed and continuous. Discrete models contain interconnected elements whose characteristics are concentrated at points. These can be masses, volumes, forces and other influences concentrated at points.

Model (from the Latin modulus - measure) and modeling are general scientific concepts. Modeling from a general scientific point of view acts as a way of cognition through the construction of special objects, systems - models of the studied objects, phenomena or processes. In this case, one or another object is called a model when it is used to obtain information regarding another object - a prototype of the model.

The modeling method is used in virtually all sciences without exception and at all stages of scientific research. The heuristic power of this method is determined by the fact that with the help of the modeling method it is possible to reduce the study of the complex to the simple, the invisible and intangible, the visible and tangible, etc.

When studying an object (process or phenomenon) using the modeling method, we can select as a model those properties that currently interest us. The scientific study of any object is always relative. In a specific study, it is impossible to consider an object in all its diversity. Consequently, the same object can have many different models, and none of them can be said to be the only true model of this object.

It is customary to distinguish four main properties models:

· simplification in comparison with the object being studied;

· ability to reflect or reproduce the object of research;

· the ability to replace the object of research at certain stages of its cognition;

· the ability to obtain new information about the object being studied.

The study of various phenomena or processes by mathematical methods is carried out using a mathematical model. Mathematical model is a formalized description in the language of mathematics of the object under study. Such a formalized description can be a system of linear, nonlinear or differential equations, a system of inequalities, a definite integral, a polynomial with unknown coefficients, etc. The mathematical model must cover the most important characteristics of the object under study and reflect the connections between them.

Before creating a mathematical model of an object (process or phenomenon), it is studied for a long time using various methods: observation, specially organized experiments, theoretical analysis, etc., that is, the qualitative side of the phenomenon is studied quite well, the relationships in which the elements of the object are located are revealed. Then the object is simplified, and the most essential ones are singled out from the variety of its inherent properties. If necessary, assumptions are made about existing connections with the outside world.

As stated earlier, any model is not identical to the phenomenon itself; it only provides some approximation to reality. But the model lists all the assumptions that underlie it. These assumptions may be crude and yet provide a completely satisfactory approximation to reality. Several models, including mathematical ones, can be built for the same phenomenon. For example, you can describe the movement of the planets of the solar system using:

8 Kepler's model, which consists of three laws, including mathematical formulas (ellipse equation);

8 of Newton's model, which consists of one formula, but nevertheless it is more general and accurate.

In optics, several models of light were considered: corpuscular, wave and electromagnetic. Numerous quantitative patterns were derived for them. Each of these models required its own mathematical approach and appropriate mathematical tools. Corpuscular optics used the means of Euclidean geometry and came to the conclusion of the laws of reflection and refraction of light. The wave model of the theory of light required new mathematical ideas and, purely computationally, new facts related to the phenomena of diffraction and interference of light that had not previously been observed were discovered. Geometric optics, associated with the corpuscular model, turned out to be powerless here.

The constructed model must be such that it can replace an object (process or phenomenon) in research and must have similar features with it. Similarity is achieved either through similarity in structure (isomorphism) or analogy in behavior or functioning (isofunctionality). Based on the similarity of the structure or function of the model and the original, modern technology tests, calculates and designs the most complex systems, machines and structures.

As mentioned above, many different models can be built for the same object, process or phenomenon. Some of them (not necessarily all) may be isomorphic. For example, in analytical geometry, a curve in a plane is used as a model for the corresponding equation in two variables. In this case, the model (curve) and the prototype (equation) are isomorphic to systems (points lying on the curve and corresponding pairs of numbers satisfying the equation),

In the book “Mathematics Conducts an Experiment,” academician N.N. Moiseev writes that any mathematical model can arise in three ways:

· As a result of direct study and understanding of an object (process or phenomenon) (phenomenological) (example - equations describing the dynamics of the atmosphere, ocean),

· As a result of some process of deduction, when a new model is obtained as a special case of a more general model (asymptomatic) (example - equations of hydro-thermodynamics of the atmosphere),

· As a result of some process of induction, when the new model is a natural generalization of “elementary” models (ensemble model or generalized model).

The process of developing mathematical models consists of the following stages:

· formulation of the problem;

· determination of the purpose of modeling;

· organizing and conducting research of the subject area (research of the properties of a modeling object);

· model development;

· checking its accuracy and compliance with reality;

· practical use, i.e. transfer of knowledge obtained using the model to the object or process under study.

Modeling as a way of understanding the laws and phenomena of nature acquires particular importance in the study of objects that are not fully accessible to direct observation or experimentation. These also include social systems, the only possible way to study which is often modeling.

There are no general methods for constructing mathematical models. In each specific case, it is necessary to proceed from the available data, target orientation, take into account the objectives of the study, and also balance the accuracy and detail of the model. It should reflect the most important features of the phenomenon, the essential factors on which the success of the modeling mainly depends.

When developing models, it is necessary to adhere to the following basic methodological principles for modeling social phenomena:

· the principle of problematicity, which implies a movement not from ready-made “universal” mathematical models to problems, but from real, actual problems - to the search and development of special models;

· the principle of systematicity, which considers all the relationships of the modeled phenomenon in terms of the elements of the system and its environment;

· the principle of variability in the formalization of management processes associated with specific differences in the laws of development of nature and society. To explain it, it is necessary to reveal the fundamental difference between models of social processes and models describing natural phenomena.