Silantev Danila Mikhailovich (MSTU «STANKIN»
Graduate student/PhD student
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The mathematical theory of catastrophes is a branch of singularity theory that studies the behavior of systems at critical points, when small changes in parameters can cause abrupt changes in the state of the system. The article discusses the application of catastrophe theory to the modeling of physical processes, where small disturbances lead to unexpected and abrupt changes in the behavior of the system. The research is based on a mathematical model describing the interaction of nonlinear functions and critical point analysis. The theory of catastrophes by Christopher Ziman and Rene Thom is used to determine the main types of elementary catastrophes (folding, fork, topological structural instability, etc.). Methods of differential equations and bifurcation theory were used for analysis. The model includes a description of the dynamic interaction of system parameters and tracking the stage of occurrence of catastrophic events based on a set of variables. The analysis showed that the mathematical theory of catastrophes effectively describes abrupt changes in various physical processes, such as the model of brittle destruction of materials, the behavior of a liquid at a critical temperature or the propagation of waves in an environment with abnormal properties. In particular, it was revealed that changing specific parameters can serve as a trigger for the birth of a new phase of the system. It is shown that transitions between states are nonlinear in nature and can be effectively modeled using a simplified classification of catastrophes. The work demonstrates that the theory of catastrophes is a powerful tool for modeling abrupt changes in physical systems. In the future, it is necessary to develop specialized models for specific processes, taking into account many factors that can lead to bifurcations and catastrophes in dynamic systems.
Keywords:mathematical theory of catastrophes, mathematical modeling, physics.
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Read the full article …
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Citation link:
Silantev D. M. MATHEMATICAL THEORY OF CATASTROPHES IN THE MODELING OF PHYSICAL PROCESSES // Современная наука: актуальные проблемы теории и практики. Серия: Естественные и Технические Науки. -2024. -№11/2. -С. 120-127 DOI 10.37882/2223-2966.2024.11-2.30 |
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