 |
RU UA EN DonNTU Masters portal |
||||
 |
 |
||||
INTRODUCTION
The theory of dynamic systems are widely demanded by large spectrum of sciences - physics, biology, mechanics and, of course, economics. It allows you not only to determine the possible direction of development of the object, but also to develop a complex adaptive influences on the system to correct this trend.
The study of dynamic systems engaged by such scholars as Hayrer, Anishchenko, Verzhbitsky, Anosov. At the same time, the theory of dynamic systems are still unexplored, so that it remains relevant not only in the application, but also development.
The purpose of this work is to review the current state of the theory of dynamic systems and their stiffness, as well as the applicability of this theory to economic problems.
The study of dynamic systems engaged by such scholars as Hayrer, Anishchenko, Verzhbitsky, Anosov. At the same time, the theory of dynamic systems are still unexplored, so that it remains relevant not only in the application, but also development.
The purpose of this work is to review the current state of the theory of dynamic systems and their stiffness, as well as the applicability of this theory to economic problems.
THE NOTION OF A DYNAMIC SYSTEM
One of the important scientific problems of natural science is to solve the problem of predicting the behavior of the object in time and space on the basis of certain knowledge about its initial state. This problem reduces to finding some law that allows to determine the future of an object at any time t> t 0 using the available information about an object at the initial time t 0 in the space point x 0. Depending on the complexity of the object this law can be deterministic or probabilistic, can describe the evolution of the object only in time, in space, and can describe the spatial-temporal evolution.
The subject of our analysis are not objects at all, but dynamic systems in the mathematical sense of the term.
The dynamic system is any object or process for which there is a unique concept of state as a combination of some variables at a given time and set a law that describes the change (evolution) of the initial state over time. Dynamic systems are mechanical, physical, chemical and biological facilities, computing and information transformation processes occurring in accordance with specific algorithms.
A mathematical model of a dynamic system is given, if parameters (coordinates) of the system are introduced and the law of evolution is given. Depending on the degree of approximation of the same system may be associated with different mathematical models.
The study of real systems is reduced to the study of mathematical models, the improvement and development are determined by analysis of experimental and theoretical results when comparing them. In this regard, under the dynamic system we mean precisely its mathematical model. Exploring the same dynamic system (for example, price fluctuations), depending on the degree of various factors, we obtain different mathematical models.
The subject of our analysis are not objects at all, but dynamic systems in the mathematical sense of the term.
The dynamic system is any object or process for which there is a unique concept of state as a combination of some variables at a given time and set a law that describes the change (evolution) of the initial state over time. Dynamic systems are mechanical, physical, chemical and biological facilities, computing and information transformation processes occurring in accordance with specific algorithms.
A mathematical model of a dynamic system is given, if parameters (coordinates) of the system are introduced and the law of evolution is given. Depending on the degree of approximation of the same system may be associated with different mathematical models.
The study of real systems is reduced to the study of mathematical models, the improvement and development are determined by analysis of experimental and theoretical results when comparing them. In this regard, under the dynamic system we mean precisely its mathematical model. Exploring the same dynamic system (for example, price fluctuations), depending on the degree of various factors, we obtain different mathematical models.
Stiff dynamic systems
Cauchy problems for ordinary differential equations can be divided into the following types: soft, hard, ill-conditioned and rapidly oscillating. Each type makes specific requirements for numerical methods.
The examples of stiff systems are the problem of chemical kinetics, time-dependent processes in complex radiotsepyah systems arising when solving the equations of heat conduction and diffusion method of lines, and many others.
Stiff systems are quite difficult for the numerical solution. Classical explicit methods such as Adams and Runge-Kutta methods require unacceptable small step.
The examples of stiff systems are the problem of chemical kinetics, time-dependent processes in complex radiotsepyah systems arising when solving the equations of heat conduction and diffusion method of lines, and many others.
Stiff systems are quite difficult for the numerical solution. Classical explicit methods such as Adams and Runge-Kutta methods require unacceptable small step.
DYNAMIC SYSTEMS IN ECONOMICS
An example of a dynamic system in the economy can serve as a model for organizing an advertising campaign. The company began producing a new product or provide a new service. Of course, profits from future sales must cover the costs of expensive advertising. First, costs can exceed the profits, because consumers’ awareness about the product (service) is small. Further increasing the number of sales increases the profits, and, finally, saturation occurs and further continuation of the campaign would be meaningless. As soon as we are talking about saturation, it becomes clear that a dynamic system is stiff.
LITERATURE
1. Власов М.П., Шимко П.Д. Моделирование экономических процессов. – М.: ЮНИТИ, 2005 – 409с.
2. Хайрер Э., Ваннер Г. Решение ОДУ. Жесткие и дифференциально-алгебраические задачи. – М.: Мир, 1999г. – 686 с.
3. Анищенко В.С. Сложные колебания в простых системах. – М.: Мир, 1998 г. – 285 с.
4. Калиткин Н.Н. Численные методы решения жестких систем – М.: Мир, 1978. – 308 с.
5. Добрынин А.И., Тарасевич Л.С. Экономическая теория: учебник. – СПб: Питер, 2005. – 414 с.
6. Курс экономической теории: Учебное пособие//Под ред. А.В.Сидоровича – М.: Дело и Сервис, 2001. – 832 с.
7. Власов М.П., Шимко П.Д. Моделирование экономических процессов. – Ростов н/Д: Феникс, 2005. – 409 c.
8. Expert system for Ordinary Differential Equations. [Электронный ресурс]//Faculty of Mathematics, Western University of Timisoara
Режим доступа: http://web.info.uvt.ro/~petcu/epode/e314.htm
9. Ахмеров Р.Р. Очерки по теории обыкновенных дифференциальных уравнений. [Электронный ресурс]
Режим доступа: http://www.ict.nsc.ru/rus/textbooks/akhmerov/nm-ode/index.html
10. Фирсова А.А. Моделирование динамических систем в экономике. [Электронный ресурс]//Персональный сайт магистра ДонНТУ
Режим доступа: http://masters.donntu.ru/2009/fvti/firsova/diss/index.htm
2. Хайрер Э., Ваннер Г. Решение ОДУ. Жесткие и дифференциально-алгебраические задачи. – М.: Мир, 1999г. – 686 с.
3. Анищенко В.С. Сложные колебания в простых системах. – М.: Мир, 1998 г. – 285 с.
4. Калиткин Н.Н. Численные методы решения жестких систем – М.: Мир, 1978. – 308 с.
5. Добрынин А.И., Тарасевич Л.С. Экономическая теория: учебник. – СПб: Питер, 2005. – 414 с.
6. Курс экономической теории: Учебное пособие//Под ред. А.В.Сидоровича – М.: Дело и Сервис, 2001. – 832 с.
7. Власов М.П., Шимко П.Д. Моделирование экономических процессов. – Ростов н/Д: Феникс, 2005. – 409 c.
8. Expert system for Ordinary Differential Equations. [Электронный ресурс]//Faculty of Mathematics, Western University of Timisoara
Режим доступа: http://web.info.uvt.ro/~petcu/epode/e314.htm
9. Ахмеров Р.Р. Очерки по теории обыкновенных дифференциальных уравнений. [Электронный ресурс]
Режим доступа: http://www.ict.nsc.ru/rus/textbooks/akhmerov/nm-ode/index.html
10. Фирсова А.А. Моделирование динамических систем в экономике. [Электронный ресурс]//Персональный сайт магистра ДонНТУ
Режим доступа: http://masters.donntu.ru/2009/fvti/firsova/diss/index.htm