Nonlinear mathematical model of pedagogical system functioning

The purpose of this article is to study the crisis in pedagogical systems from the point of view of an internal observer. The aim of the work is to build and investigate a mathematical model describing the course of crises in pedagogical systems. When building the model, a synergetic methodology, system and process approaches are used. For the mathematical analysis of various social phenomena, systems of differential equations are used to investigate the dynamics of the process. The paper considers a system of nonlinear differential equations in three-dimensional space that describes the functioning of the pedagogical system during the crisis. Numerical and topological methods of nonlinear dynamics, the method of Lyapunov characteristic exponents and the theory of strange attractors by Lorentz were used to study it. Numerical modeling of system solutions for various sets of control parameters (system coefficients) makes it possible to determine the region of stability (asymptotic stability), limit cycles, bifurcation points, and describe possible trajectories of development of the pedagogical system. Mathematical modeling deepens the knowledge about the essence of crises, the peculiarities of their course, makes it possible to study qualitative and numerical modeling, and also allows predicting possible effective measures to combat crisis phenomena and develop new approaches in the management of pedagogical systems.