Research of Four-Dimensional Dynamic Systems Describing Processes of Three’level Assimilation

Abstract

In this work a new nonlinear mathematical model of process of three level assimilation which is described by four-dimensional dynamic systems is offered. In case of constancy of coefficients special points of the dynamic system are found. The conditions on constant coefficients for which it is possible to find special points with all four coordinates non-negative are determined. Introducing some dependence among coefficients of the system, two first integrals are derived, and the four-dimensional system is reduced to a two-dimensional one.The sign-variable divergence theorem of a two-dimensional vector field in some one-coherent area of the first quadrant of the phase plane is proved. According to Bendixon’s criterion it is possible to have a closed integral curve completely lying in this area.Introduction. Mathematical modeling of physical processes has a long history. Mathematical modeling of physical processes involves the model adequacy, which is validated by Newton’s non-relative five laws of classical mechanics: mass conservation law; law of conservation of impulse; the law of conservation the momentum of impulse; the first law of thermodynamics, i.e. energy conservation law; the second law of thermodynamics, i.e. entropy conservation law [, , , , ].Creation of mathematical models is more original in social sphere, because, they are more difficult to substantiate [, , ].We created a new direction of mathematical modeling, i.e. “Mathematical Modeling of Information Warfare” [, , ]. In these models two antagonistic sides waging with each other information warfare and also the third peacekeeping side trying to extinguish information warfare are considered. Conditions on model parameters at which the third side will be able to force the conflicted sides to completion of information warfare are found.We also offered mathematical models of forecasting the results of political elections in case of two or three parties. Also models in case of change of selective subjects before the next elections have been considered [, , , , ].We proposed to create new nonlinear mathematical models of economic cooperation between two politically (not military opposition) mutually warring sides (two countries or a country and its legal region) which consider economic or other type of cooperation between different parts of population aimed to the peaceful resolution of conflicts [, ].Taking into consideration the important tendencies in the world, it is important to study demographic and assimilation of social processes through mathematical modeling.In [] we considered a new nonlinear continuous mathematical model of linguistic globalization. Two categories of the world’s population are considered: a category that hinders and a category leading to the dominant position of the English language. With a positive demographic factor of the population, which prevents globalization or a negative demographic factor of the population contributing to globalization, it is shown that the dynamic systems describing these processes allow the existence of two topologically not equivalent phase portraits (a stable node, a limit cycle).It is known that, in the world, a social process of assimilation of languages is hidden. This process, as a rule, considers expansion of an area of the dominating languages (state languages of economically powerful states) at the expenses of less widespread languages (state languages economically of rather weak states).According to this point of view, today, for less widespread languages (including classic languages) the conditions under which there will be no disappearance of the major languages are important, i.e. there will be no full assimilation of people talking in these languages.