EXCEPTIONAL DIRECTIONS OF A HOMOGENEOUS POLYNOMIAL

Аннотация

Investigations are conducted into situations where the system's exclusive directives coexist. 𝑑𝑋 = 𝑋2(𝑥), 𝑋 = (𝑥 ,𝑥 ,𝑥 ) 𝑑𝑡 1 2 3 here 𝑋2(𝑥) – uniform polynomial vector function of degree 2, 𝑎 𝑋2(𝑥) multiplies 𝑋 3 3 Brief historical background The character of the neighbourhood of an isolated singular point of a higher-order p- dimensional system 𝑑𝑋 = 𝑋𝑚(𝑥) = 𝜑(𝑡,𝑥), 𝑋 = (𝑥 ,𝑥 ,…,𝑥 ) (1) 𝑑𝑡 1 2 𝑛 here 𝑋𝑚(𝑥) – one-race class function 𝐶1 degrees 𝑚 ≥ 1 а 𝑌(𝑡,𝑥) – vector function that does not contain members of degree t or higher, 𝑌(𝑡,𝑥) = 0. Under various assumptions concerning the function vector 𝜑(𝑡,𝑥) the system (2) is studied by many authors, the system (2) is called quasi-uniform. The case when 𝑌(𝑡,𝑥) = 0, 𝑛 = 2 system (2) was studied by Frister [2] V.V. Nemytsky [3] Sh.R. Sharipov [4] and others. It was found that the neighbourhood of the singular point consists of a combination of elliptic and hyperbolic prabolic regions (sectors), and the stability of the trivial solution 𝑋 = 0 was also 𝑖 considered. The research of homogeneous and pioneering systems of level (1), which has its roots in [5], [6], and the study of isolated special points in terms of raising the dimensionality of space, is of significant theoretical and practical significance. Many of the questions that were simplified in the historical note, like the cassification of integral manifolds next to isolated singular points, the general study of a trajectory's behavior in Euclidean or projective space, the investigation of isolated singular points' joint existence, the exceptional direction, the stability of zero solutions in the Lyapunov sense, and other questions, can be expanded upon. In the midst of a fresh transition, we have succeeded in examining the remarkable trajectory of system (1). Key words: Exceptional direction, homogeneous, vector-function, coexistence, manifold, knot, saddles, saddle, knot, knot, vector norm, special points, homogeneous polynomial, hemisphere. 164 RAQAMLI DUNYO: MATEMATIK VA INFORMATIK 2025 -Yil YONDASHUVLAR 13-Fevral Respublika ilmiy-uslubiy konferensiyasi 1. INTRODUCTION Consider the system 𝑑𝑥 = 𝑋2(𝑥), 𝑋 = (𝑥 ,𝑥 ,𝑥 ) , (1) 𝑑𝑡 1 2 3 here 𝑋2(𝑥)- homogeneous polynomial vector-function of degree 2 We study the cases of coexistence of exceptional directions of system (1) when 𝑋2(𝑥) has a multiplier 𝑥 . 3 Under various assumptions concerning the vector-functions, system (1) has been studied by many authors; the case when 𝑛 = 2 has been considered by V.V. Nemytsky [3], H. Forster [2], Sh.R. Sharipov [4] and others. It was established that the neighbourhood of a singular point consists of a combination of elliptic, hyperbolic, and parabolic regions (sectors), and the stability of the trivial solution 𝑥 = 0 𝑖 was also considered. The studies on the study of homogeneous and polynomial systems of differential levels are of tremendous theoretical and practical significance (1). This trend stems from studies of the joint existence of isolated points in terms of increasing the dimensionality of the space, as well as from [5], [6]. Many of the questions addressed in the historical note can also be applied to homogeneous and polynomial systems, including the classification of ittegral manifolds adjacent to isolated singular points, the general study of trajectory behavior in Euclidean or projective space, the study of the joint existence of isolated singular points, the stability of the zero solution in the sense of Lyapunov, and many more. The topic of this article has emerged as a result of achieving this goal. The method of limit sets [1] was applied to study the property of the trajectory in the vicinity of an isolated unique point of system (1). This note examines this approach by introducing a different transformation from the one presented in [1]. 2. PROBLEM STATEMENT Vivod 10. Introducing in sestem (1), we introduce the substitution 𝑥 = 𝑟𝑢 , (2) 𝑖 here 𝑈 = (𝑢 ,𝑢 ,𝑢 ) – unit vector, 𝑟- vector norm 𝑥, we obtain 1 2 3 𝑑𝑢 𝑎) = 𝑋2(𝑢)−𝑢𝑅(𝑢) 𝑑𝑡 1 , (3) 𝑑𝑟 𝑏) = 𝑟𝑅(𝑢) 𝑑𝑡 } 1 here 𝑅(𝑢) = 𝑋2(𝑢)𝑈 𝑑𝑡 = 𝑟𝑑𝑡 The extraordinary directions of system (1) correspond to the special points on the equator and to the special points on the sphere 𝑆2 or system (3a). 𝑢2 +𝑢2 = 1 of the system (3a) correspond to the exclusive directions of the integral 1 2 multimodel 𝑥 = 0. 3 Substituting 𝑢 = 0 by the systim (3a) and further, introducing 3 𝑢 = 𝜇𝑢 ( 𝑜𝑟 𝑢 = 𝜇̅𝑢 ) we obtain 1 2 2 1 𝑑𝜇 𝑋2(𝜇,1)−𝜇𝑋1(𝜇,1) 1 2 = (4) 𝑑𝑢 𝑢 [𝑋2(𝜇,1)−𝑢2𝑅(𝜇,1)] 2 2 2 2 165 RAQAMLI DUNYO: MATEMATIK VA INFORMATIK 2025 -Yil YONDASHUVLAR 13-Fevral Respublika ilmiy-uslubiy konferensiyasi 3. NUMERICAL SOLUTION Координаты особых точк экватора 𝑢2 +𝑢2 = 1 сферы 𝑆2 будут иметь вид. 1 2 Coordinates of special points of the equator 𝑢2 +𝑢2 = 1 of the sphere 𝑆2 will have the 1 2 form 𝜇 1 𝜇 1 0 0 ( , ,0),(− ,− ,0), √1+𝜇2 √1+𝜇2 √1+𝜇2 √1+𝜇2 0 0 0 0 here 𝜇 - real root of the level. 0 𝑋2(𝜇,1)−𝜇𝑋2(𝜇,1) = 0, 𝑋2(𝜇,1) ≠ 0 (5) 1 2 2 The extraordinary directions of the manifold 𝑥 = 0 are determin