This paper explores the role of invariants in the analysis of special points within various classes of differential equations. By leveraging invariants, the study provides a framework for simplifying the identification and characterization of equilibrium points, singularities, and other critical features. The results demonstrate that invariants offer powerful tools for analyzing the structure and solutions of differential equations, especially in complex systems.

Differential Equations, Invariants, Special Points

Arnold, V. I. (2012). Ordinary Differential Equations. Springer. A comprehensive text on the theory of ordinary differential equations, focusing on solutions and singularities.

Olver, P. J. (1993). Applications of Lie Groups to Differential Equations. Springer. Discusses the role of symmetries and invariants in the analysis of differential equations.

Guckenheimer, J., & Holmes, P. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer. A key work in non-linear differential equations, focusing on dynamical systems and bifurcation theory.

Teschl, G. (2012). Ordinary Differential Equations and Dynamical Systems. American Mathematical Society. Provides insight into linear and non-linear systems, including the role of invariants in stability analysis.

Noether, E. (1918). Invariante Variationsprobleme. Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen, Math-phys. Klasse, 1918, 235–257. A foundational paper establishing the link between symmetries and conserved quantities in differential equations.

Toʻxtasinov, B. T. (2019). Matematik analiz va differensial tenglamalar. Tashkent: Oʻzbekiston Milliy Universiteti. A foundational textbook on mathematical analysis and differential equations used in Uzbek higher education.

Nabiyev, A. X. (2016). Nozik chiziqli va no chiziqli differensial tenglamalar. Toshkent: Fan Nashriyoti. Focuses on linear and non-linear differential equations with practical applications in physics and engineering.

Sobirov, I. T., & Yusupov, A. S. (2020). Qiyosiy metodlar va differensial tenglamalar. Samarkand: SamDU Matbuoti. Discusses comparative methods in solving differential equations, with examples from Uzbek scientific research.

Ergashev, X. E. (2015). Matematik fizikada differensial tenglamalar va invariantlar. Tashkent: Fan va Texnologiya. Explores the application of invariants in mathematical physics, particularly in solving partial differential equations.

Saidov, S. I. (2021). Dinamik sistemalarda differensial tenglamalar va ular tatbiqi. Toshkent: Oʻzbekiston Respublikasi Fanlar Akademiyasi. Discusses the role of differential equations in dynamic systems, including the analysis of equilibrium points and bifurcations.