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киберзащиты и кибератак зависит от эвристики исследователей. Возможности дальности лежат между двумя конечными состояниями. Если область действия описывает свойства проблемы, она может усложнить модель за счет множества деталей и стать неразрешимой. Другая крайность: если математическая модель проста, она не может предсказать и изучить свойства системы. Эти математические модели могут быть связаны с кибератаками, преследующими разные цели. Таким образом, пострадавшие системы можно анализировать, делать системные прогнозы, извлекать, проверять системную информацию и делать прогнозы на будущее. Математические методы доказали свою эффективность в различных приложениях упреждающих действий, технологиях кибербезопасности, управлении инцидентами безопасности и расследованиях. Однако эти методы нуждаются в систематизации, дальнейшем развитии и обогащении с точки зрения их использования для углубленных исследований в области кибербезопасности. Список использованной литературы:
1. Fuster G.G., Jasmontaite L. Cybersecurity Regulation in the European Union: The Digital, the Critical and Fundamental Rights. In The Ethics of Cybersecurity; Springer International Publishing, 2020; Volume 21, pp. 97115.
2. Kenneally E. Cyber Risk Economics Capability Gaps Research Strategy. 2018. Available online: https://www.dhs.gov/publication/cyrie-capability-gaps-research-strategy.
3. Goupil F., Laskov P., Pekaric I., Felderer M., Dürr A., Thiesse F. Towards Understanding the Skill Gap in Cybersecurity. In Proceedings of the 27th ACM Conference on Innovation and Technology in Computer Science Education, 2022; Volume 1, pp. 477-483.
4. Alzahrani N.M., Alfouzan F.A. Augmented Reality (AR) and Cyber-Security for Smart Cities - A Systematic Literature Review. Sensors 2022, 22, 2792.
5. Ma C. Smart City and Cyber-Security; Technologies Used, Leading Challenges and Future Recommendations. Energy Rep. 2021, 7, 7999-8012.
© Текяев М., Эсенмырадова С., Гылыджова Ч., Оразов М., 2024
УДК 53
Хусейнова М.,
Студентка.
Туркменский инженерно-технологический университет имени Огузхана.
Ашхабад, Туркменистан.
СИСТЕМЫ ОБЫКНОВЕННЫХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ И КЛАССИФИКАЦИЯ КРИТИЧЕСКИХ ТОЧЕК
Аннотация
Обыкновенные дифференциальные уравнения — это математические уравнения, в которых одна или несколько зависимых переменных выражаются через одну или несколько независимых переменных и их производных. Эти уравнения используются при моделировании природных явлений и находят применение в различных областях: от физики до биологии, от техники до экономики.
Ключевые слова:
обыкновенные дифференциальные уравнения, математические уравнения, физика, критические точки, анализ.
Annotation
Ordinary differential equations are mathematical equations where one or more dependent variables are expressed in terms of one or more independent variables and their derivatives. These equations are used in modeling natural phenomena and find applications in various fields ranging from physics to biology, engineering to economics.
Key words:
оrdinary differential equations, mathematical equations, physics, critical points, analysis.
Ordinary differential equations are mathematical equations where one or more dependent variables are expressed in terms of one or more independent variables and their derivatives. These equations are used in modeling natural phenomena and find applications in various fields ranging from physics to biology, engineering to economics.
An ordinary differential equation system consists of multiple ordinary differential equations interconnected together. These systems are commonly used to model complex dynamics and interactions. The analysis of such systems, particularly the classification of critical points, is important for understanding the stability and behavior of the systems.
Critical points are the fixed points of the equations in an ordinary differential equation system. These points carry important characteristics of the dynamics in the system and determine its behavior. The classification of critical points helps us understand how the behavior around these points changes.
The classification of critical points is often done using a linearization approach. According to this approach, the dynamics around the critical point are linearized, and then the linearized system is analyzed. This analysis helps determine the stability status of the critical point and the type of behavior around it.
In summary, the classification of critical points in ordinary differential equation systems is an important tool for understanding the behavior of systems. This classification plays a critical role in determining the stability of systems and analyzing their complex dynamics.
The classification of critical points is typically based on the eigenvalues of the linearized system around each point. These eigenvalues determine the local behavior of the system and can be used to categorize critical points into different types such as stable, unstable, or saddle points.
Stable critical points are those where small perturbations lead the system back towards the critical point, indicating a stable equilibrium. Unstable critical points, on the other hand, are points where small perturbations cause the system to move away from the critical point, indicating an unstable equilibrium. Saddle points, also known as semi-stable points, possess both stable and unstable directions, leading to complex behavior in their vicinity.
The stability analysis of critical points provides crucial insights into the long-term behavior of the system. By understanding the stability properties of critical points, we can predict the system's behavior over time and assess its resilience to external influences.
Moreover, the classification of critical points is not only limited to linear systems but can also be extended to nonlinear systems using advanced techniques such as Lyapunov stability theory and center manifold theory. These techniques allow for the analysis of more complex systems where linearization may not be applicable.
The classification of critical points plays a crucial role in understanding the dynamics of systems. This classification helps us predict the long-term behavior of a system and understand how it will respond under various conditions. For example, in an engineering application, identifying stable critical points of a system can enhance the reliability of design and help prevent undesirable outcomes.
However, the classification of critical points often involves complex mathematical calculations. While it may be straightforward for linear systems, analysis can be more challenging for nonlinear systems, and analytical solutions may not be available. In such cases, numerical methods and computer simulations can facilitate the
classification process and aid in understanding the behavior of more complex systems.
In conclusion, the classification of critical points is a significant subject in mathematical modeling and scientific research. It enables us to understand the behavior of natural phenomena, engineering systems, and many other fields. In the future, more advanced methods and techniques will continue to be developed for the analysis of more complex systems, contributing to the advancement of science and technology. Список использованной литературы:
1. Егоров А.И. Обыкновенные дифференциальные уравнения с приложениями. Москва, Физматлит, 2005.
2. Дмитриев В.И. Лекции по обыкновенным дифференциальным уравнениям. Москва, изд. КДУ, 2007.
3. Широв С. Дифференциальные уравнения высшего порядка. - Ашхабад., Наука, 2001.
4. Матвеев Н.М. Обыкновенные дифференциальные уравнения. Санкт-Петербург, 1996.
© Хусейнова М., 2024
