Research of Non-linear Dynamic System Describing Interaction Between Colchian-Georgian and Svan Populaces

Abstract

This paper discusses the two stages of transformation of the Common-Kartvelian-speaking people: the first stage was 5000–2500 BC, when the entire populace spoke the Common Kartvelian language and lived in the South Caucasus; the second stage was 2500–1000 BC, when the entire populace was divided into three parts: Svan, Colchian-Georgian and the third part, Pelasgian tribe, was emigrated to various areas of the European continent. For the first stage, computer simulation is used in the case of variable coefficients of the equation and corresponding numerical values are obtained. The second stage is described by two different mathematical models: one part of the Common-Kartvelian-speaking people went to Europe and the process of their partial or complete assimilation on the European continent began. Two other parts of the populaces speaking on the Colchian-Georgian and Svan languages that arose as a result of the transformation of the Common-Kartvelian-speaking populace remained in the South Caucasus and Anatolia. To describe the process of interaction between the Colchian-Georgian and Svan peoples, a two-dimensional non-linear system of ordinary differential equations with variable coefficients is proposed. A special case of a two-dimensional system of ordinary differential equations with constant coefficients is considered. In two cases of certain interdependencies between the constant coefficients of the system, it is shown that the divergence of an unknown vector function in the physically significant first quarter of the phase plane changes sign when passing through any segment or half-line one. Taking into account Bendixson’s criterion, theorems were proved about the variability of the divergence of a vector field and the existence of closed trajectories in some simply connected domain that completely contains the starting point lying on one segment or half-line. Thus, it is shown that there is no assimilation of the Svan populace by the Colchian-Georgian populace and these two indigenous inhabitants peacefully coexist in the same territory after the transformation of the Common-Kartvelian-speaking populace.