Research of nonlinear dynamic systems describing the process of territorial stability of the state.

The article proposes a new nonlinear mathematical model that describes the process of possibility (instability of the state) or impossibility (stability of the state) of separating the region from the state. The model is described by the Cauchy problem for a nonlinear two-dimensional dynamic system. The model assumes that only two categories of citizens live in a particular region of a state: the first category, which is a supporter of the center (unionists) and opposes the secession of the region; the second - supporters of the separation of the region (separatists), i.e. its separation from the center, in order to form a new independent state. The general mathematical model implies the presence of both federal and external sides, which influence the separatists and unionists, respectively, by various factors, in order to change their opinion (will). Natural conditions are proposed under which secession of the region is considered possible (for example, the presence of a majority or qualified majority of citizens of the region who support separatism). In the case of variable coefficients of the model, under some restrictions, in quadratures, an exact solution to the Cauchy problem was found for a two-dimensional dynamic system, and in the case of constant coefficients, conditions were found under which secession of the region is impossible. In the particular case, the absence of external stakeholders for the region, in the case of constant coefficients and opposite values of demographic factors of the sides, the problem is actually reduced to a predator-victim model and is described by a nonlinear two-dimensional dynamic system of the Lotka-Volterra type. At the same time, conditions were found for the coefficients of attracting opponents to allies, demographic factors and baseline conditions under which it is impossible to separate the region. The article also proposes a new nonlinear mathematical model that describes in a certain politically conflicting region of the state the presence of three groups of the population with different political priorities. One part of the population (unionists) is politically oriented towards preserving the region as part of the state, the second part of the region supports the idea of separatism, separation of the region from the state in order to form a new independent state (separatists), and the third part of the region supports the idea of irredentism of the region, that is, separation in order to join another, possibly bordering state (irredentists). Weak (a simple majority of the region's population) and strong (a qualified majority of the region's population) conditions are offered, which in a legal sense may not have direct consequences, but may determine the aspirations of the majority of the region's population. The model is described by a nonlinear three-dimensional dynamic system with variable coefficients. With some constraints on model parameters, accurate analytical solutions are found. Additional conditions were also found under which: the region remains within the state; possible secession of the region; the irredentism of the region is possible, that is, its accession to another state. [ABSTRACT FROM AUTHOR]