Nonlinear single-input single-output (SISO) systems operating under parametric uncertainty often exhibit bifurcations, multistability, and deterministic chaos, which significantly limit the effectiveness of classical linear, adaptive, and switching control methods. This paper proposes a novel synthesis framework for self-organizing control systems based on catastrophe theory, specifically within the class of elliptic catastrophes. Unlike conventional approaches that stabilize a predefined system structure, the proposed method embeds the control law directly into a structurally stable catastrophe model, enabling autonomous bifurcation-driven transitions between stable equilibria. The synthesis procedure is formulated using a Lyapunov vector-function gradient–velocity method, which guarantees aperiodic robust stability under parametric uncertainty. The definiteness of the Lyapunov functions is established using Morse’s lemma, providing a rigorous stability foundation. To support practical implementation, a data-driven parameter tuning mechanism based on self-organizing maps (SOM) is integrated, allowing adaptive adjustment of controller coefficients while preserving Lyapunov stability conditions. Simulation results demonstrate suppression of chaotic regimes, smooth bifurcation-induced transitions between stable operating modes, and improved transient performance compared to benchmark adaptive control schemes. The proposed framework provides a structurally robust alternative for controlling nonlinear systems in uncertain and dynamically changing environments.

The study object: the chemical engineering system for production of benzene and optimization of the system operation modes based on modeling. An approach to the effective solution of problems of optimization of operating modes of real chemical engineering systems was proposed. Since such systems are usually multicriterial and characterized by the fuzziness of initial information, an approach to the development of their models and optimization of their operating modes in a fuzzy environment was proposed. The essence of this approach lies in the construction of mathematical models and optimization of system operation modes based on the system analysis methodology using available information of deterministic, statistical, and fuzzy nature. Statements of the problems of optimization by means of chemical engineering systems in a fuzzy environment have been obtained by modifying various principles of optimality for working in a fuzzy environment. Based on a modification of the principles of maximin and Pareto optimality, a heuristic algorithm for solving the formulated optimization problem was proposed based on the use of knowledge and experience of decision-makers. The proposed method of model construction and an optimization algorithm were implemented in practice when constructing models of benzene and rectification columns of a chemical engineering system of production of benzene when formulating and solving the problem of optimizing their operation modes in a fuzzy environment. Analysis and comparison of optimization results allow us to conclude about the effectiveness of the proposed fuzzy approach to solving optimization problems in a fuzzy environment. As a result of optimization of the benzene production process, the benzene yield increased by 1.45 thousand t or by 1.1 %, the raffinate output volume increased by 0.4 thousand t in conditions of upholding constraints on benzene quality. The proposed approach makes it possible to assess the degree of upholding of fuzzy constraints

The article is devoted to probabilistic statistical methods for modeling economic systems, as well as the theoretical foundations of probabilistic methods. The use of Markov random processes, which play a key role in probabilistic modeling, in the context of economic and mathematical research is considered. The Markov process is a mathematical model that allows you to describe random events that occur in time or space, which makes it especially relevant for analyzing the dynamics of economic systems. In recent years, there has been a growing interest in the theory of Markov processes in various fields of natural sciences. The article focuses on concepts related to determining the probabilities of transition from one state to another within a homogeneous Markov chain. These concepts are based on the theoretical foundations of discrete Markov processes and their application in financial and economic systems. As an illustration of the application of these theoretical conclusions, an analysis of the banking system was carried out. Two methods were used to determine the probabilities of reaching a finite state: the first is based on the original probability distribution matrix, and the second is based on the path vector matrix and the transition probability matrix. Both approaches resulted in identical results, confirming their reliability. The obtained probabilistic estimates allow not only to predict possible scenarios for the development of the economic system, but also to make informed decisions to optimize it. Thus, Markov random processes become an important tool for forecasting and analysis in the field of economics.

The compilation of optimal production plans often requires consideration of the fuzziness in certain initial information. Therefore, a pertinent issue in the theory and practice is the solution of production plan optimization problems under conditions of fuzzy information using mathematical models and tools of fuzzy sets. The research aims to develop various formulations for optimization problems of production plans using the example of a feed production plan. The problem is addressed based on a modified model considering fuzzy constraints by adapting various optimality principles. The research focuses on the processes of feed mixture production at the Makinsk poultry farm, characterized by fuzzy constraints on feed composition. New formulations of optimization problems under fuzzy constraints are obtained in the form of fuzzy mathematical programming problems, and heuristic methods for their solution are proposed. Various scenarios are considered, where the original multi-criteria optimization problem can be reduced to a single-criterion problem. For each formulation of the optimization problem, modifications of Pareto optimality methods and the main criterion are applied, ensuring the effective utilization of expert information and the decision maker’s knowledge during the optimization process and the selection of the best solution.

Linear integrators are traditionally used in motion control systems to compensate for static effects and suppress low-frequency disturbances. However, their use is inevitably accompanied by phase delays that limit the performance and robustness of control systems, especially in conditions of parametric uncertainty. In this regard, nonlinear integrators have been considered for several decades as a promising alternative that can weaken phase constraints and improve the quality of transients. In this paper, the concept of nonlinear integrators is reinterpreted in the context of self-organizing motion control of precision stages. In contrast to traditional approaches focused primarily on frequency analysis and the method of describing the function, a method is proposed for the synthesis of a self-organizing control system for nonlinear SISO objects based on catastrophe theory, namely in the class of elliptical dynamics with the property of structural stability. The control action is formed in such a way that transitions between stable modes occur due to bifurcation-conditioned self-organization, without using external switching logic. To ensure strict analytical guarantees of stability, the Lyapunov gradient-velocity vector function method is used, which guarantees aperiodic robust stability, suppression of oscillatory and chaotic modes, as well as monotonic convergence of trajectories under conditions of parameter uncertainty. The parameters of the nonlinear integrator are adapted using Self-Organizing Maps (SOM), while any parameter changes are allowed only within the regions that meet the conditions of Lyapunov stability. This approach ensures the alignment of analytical and data-oriented methods without violating the structural stability of the system. The results of numerical experiments demonstrate the superiority of the proposed method in comparison with classical linear and adaptive regulators in problems of controlling the movement of stages, especially near bifurcation boundaries and with significant parametric uncertainty. The results obtained confirm that the integration of nonlinear integrators with catastrophe theory and self-organization mechanisms forms a promising basis for the creation of robust and high-precision motion control systems of a new generation.

In the field of mathematical modeling and numerical analysis, singular perturbation problems occupy an important place due to their wide range of applications in various scientific and engineering disciplines. These problems arise in systems with multiscale dynamics, where the complex interaction of fast and slow processes significantly affects the behavior of the solution. Sobolev-type equations are widely used in mathematical physics and fluid mechanics to describe phenomena of various nature, including the propagation of plasma and ion-acoustic waves, filtration processes in porous media, compaction of clay materials, second-order fluid models, thermodynamic processes, and the dynamics of long waves of small amplitude. This work focuses on the study of nonlinear singularly perturbed Sobolev initial-boundary value problems and the development of new difference schemes for their numerical solution. The paper explores the mathematical aspects of the singularly perturbed problem.