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Nonlinear Dynamic Model of the Educational Process

Authors

О. P. Chornyi ORCID 0000-0001-8270-3284
Kremenchuk Mykhailo Ostrohradskyi National University ROR
L. V. Herasymenko ORCID 0000-0003-3725-8681
Kremenchuk Mykhailo Ostrohradskyi National University ROR
О. А. Chorna ORCID 0000-0002-7113-1572
Kremenchuk Mykhailo Ostrohradskyi National University ROR
DOI https://doi.org/10.31649/1997-9266-2026-187-4-218-227

Keywords

dynamics of the learning process, nonlinear modeling second-order cybernetic model parallel modeling distributed data processing educational environment

Abstract

The objective of the article is to substantiate an approach to nonlinear modeling of the dynamics of the educational process as a controlled system where learning outcomes are achieved through the combined action of a set of instructional stimuli and destabilizing factors of the educational environment, such as inertia, forgetting, and overload. The study employs methods of mathematical and simulation modeling, dynamic systems theory, local linearization, and numerical integration of differential equations. The controlled nonlinear “knowledge–disturbance” model has been proposed, in which the learning process is described through the interaction of two state variables: the normalized level of learning material acquisition and an integral indicator of the process intensity that reduce the efficiency of acquisition. This indicator generalizes the influence of external organization factors, cognitive overload, instability of attention, fatigue, stress, and other factors that are the cause of the uneven academic achievements over time.

The paper shows that the classical linear second-order cybernetic model is suitable for describing inertia, damping, and effective losses in the educational process; however, it requires the extension in order to reproduce nonlinear saturation effects, background disturbances, and mode changes under improved learning conditions. For this purpose, the nonlinear controlled model is coordinated with a second-order cybernetic model by means of local linearization in the neighborhood of the equilibrium point. It is established that the parameters of the equivalent second-order equation can be interpreted through the derivatives of the right-hand sides of the nonlinear system, while the nature of transient processes is determined by the eigenvalues of the Jacobian matrix.

Numerical modeling is carried out for a scenario involving a controlled change in learning conditions at a specified time instant. The results demonstrate the presence of three characteristic modes: initial adaptation, a quasi-stationary educational process with fluctuations, and transition to a new operating regime after the improvement of conditions. It is shown that reducing the intensity of destabilizing factors and increasing the efficiency of the instructional influence led to the increase in the average level of acquisition and decrease in process variability. The phase trajectory in the state space is proposed as an informative indicator of a change in the learning mode, since it reflects not only the average result but also the relationship between the achieved level of acquisition and the resource costs of the system.

The scientific novelty of the study lies in the coordination of a controlled nonlinear model of the learning process with a second-order cybernetic model, which makes it possible to combine the pedagogical interpretation of acquisition, forgetting, and the influence of disturbances with the formal apparatus for analyzing stability and transient modes. The proposed approach can be used for simulation analysis of educational processes, assessment of the impact of organizational and methodological interventions, comparison of learning-load scenarios, and as a theoretical basis for developing adaptive and personalized learning systems.

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How to Cite

[1]
“Nonlinear Dynamic Model of the Educational Process”, Вісник ВПІ, no. 4, pp. 218–227, Oct. 2026, doi: 10.31649/1997-9266-2026-187-4-218-227.

Author Biographies

О. P. Chornyi, Kremenchuk Mykhailo Ostrohradskyi National University

Dr. Sc. (Eng.), Professor, Professor of the Chair of Systems of Automatic Control and Electric Drive

L. V. Herasymenko, Kremenchuk Mykhailo Ostrohradskyi National University

Dr. Sc. (Pedag.), Professor, Professor of the Chair of Psychology, Pedagogy and Philosophy

О. А. Chorna, Kremenchuk Mykhailo Ostrohradskyi National University

Cand. Sc. (Eng.), Associate Professor, Associate Professor of the Chair of Computer Engineering and Electronics

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