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EXCEPTIONAL DIRECTIONS OF A HOMOGENEOUS POLYNOMIAL

Kamolidin Shodiev

Samarkand State University of Architecture and Construction, Department of Social and Natural

Sciences, Associate Professor, Doctor of Philosophy (PhD) in Economics,

Samarkand, Uzbekistan.

k.shodiev@samdaqu.edu.uz

Jumanazarov Ruzimurod

Samarkand branch of TSUE, Accounting and finance faculty, bachelor of accounting and

auditing, BH-223 group

shodiyevkamoliddin91@gmail.com

https://doi.org/10.5281/zenodo.14850526

Abstract.

Investigations are conducted into situations where the system's exclusive

directives coexist.

𝑑𝑋

𝑑𝑡

= 𝑋

2

(𝑥), 𝑋 = (𝑥

1

, 𝑥

2

, 𝑥

3

)

here

𝑋

2

(𝑥)

– uniform polynomial vector function of degree 2,

𝑎 𝑋

3

2

(𝑥)

multiplies

𝑋

3

Brief historical background
The character of the neighbourhood of an isolated singular point of a higher-order p-

dimensional system

𝑑𝑋

𝑑𝑡

= 𝑋

𝑚

(𝑥) = 𝜑(𝑡, 𝑥),

𝑋 = (𝑥

1

, 𝑥

2

, … , 𝑥

𝑛

) (1)

here

𝑋

𝑚

(𝑥)

– one-race class function

𝐶

1

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

𝑖

= 0

was also

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.

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1.

INTRODUCTION

Consider the system

𝑑𝑥

𝑑𝑡

= 𝑋

2

(𝑥), 𝑋 = (𝑥

1

, 𝑥

2

, 𝑥

3

) , (1)

here

𝑋

2

(𝑥)

- homogeneous polynomial vector-function of degree 2

We study the cases of coexistence of exceptional directions of system (1) when

𝑋

2

(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

𝑖

= 0

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

1

0

. Introducing in sestem (1), we introduce the substitution

𝑥 = 𝑟𝑢

𝑖

, (2)

here

𝑈 = (𝑢

1

, 𝑢

2

, 𝑢

3

)

– unit vector,

𝑟

- vector norm

𝑥,

we obtain

𝑎)

𝑑𝑢

𝑑𝑡

1

= 𝑋

2

(𝑢) − 𝑢𝑅(𝑢)

𝑏)

𝑑𝑟

𝑑𝑡

1

= 𝑟𝑅(𝑢)

}

, (3)

here

𝑅(𝑢) = 𝑋

2

(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

𝑆

2

or system (3a).

𝑢

1

2

+ 𝑢

2

2

= 1

of the system (3a) correspond to the exclusive directions of the integral

multimodel

𝑥

3

= 0

.

Substituting

𝑢

3

= 0

by the systim (3a) and further, introducing

𝑢

1

= 𝜇𝑢

2

( 𝑜𝑟 𝑢

2

= 𝜇̅𝑢

1

)

we obtain

𝑑𝜇

𝑑𝑢

2

=

𝑋

1

2

(𝜇, 1) − 𝜇𝑋

2

1

(𝜇, 1)

𝑢

2

[𝑋

2

2

(𝜇, 1) − 𝑢

2

2

𝑅(𝜇, 1)]

(4)

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3. NUMERICAL SOLUTION

Координаты особых точк экватора

𝑢

1

2

+ 𝑢

2

2

= 1

сферы

𝑆

2

будут иметь вид.

Coordinates of special points of the equator

𝑢

1

2

+ 𝑢

2

2

= 1

of the sphere

𝑆

2

will have the

form

(

𝜇

0

√1 + 𝜇

0

2

,

1

√1 + 𝜇

0

2

, 0) , (−

𝜇

0

√1 + 𝜇

0

2

, −

1

√1 + 𝜇

0

2

, 0),

here

𝜇

0

- real root of the level.

𝑋

1

2

(𝜇, 1) − 𝜇𝑋

2

2

(𝜇, 1) = 0, 𝑋

2

2

(𝜇, 1) ≠ 0

(5)

The extraordinary directions of the manifold

𝑥

3

= 0

are determined by the nature of these

special points.

We add the substoins

𝑦 =

𝑢

1

𝑢

3

, 𝑧 =

𝑢

2

𝑢

3

,

to the system (3a) in order to analyze the

exceptional directed systems (1), which do not reside on the manifold

𝑥

3

= 0

(i.e., for which

𝑢

3

≠ 0

). The system will then have the form of

{

𝑎)

{

𝑑𝑦

𝑑𝑧

= 𝑋

1

2

(𝑦, 𝑧, 1) − 𝑦𝑋

3

1

(𝑦, 𝑧, 1) = 𝜙(𝑦, 𝑧)

𝑑𝑧
𝑑𝜏

= 𝑋

2

2

(𝑦, 𝑧, 1) − 𝑧𝑋

3

1

(𝑦, 𝑧, 1) = 𝜑(𝑦, 𝑧)

𝑏)

𝑑𝑢

3

𝑑𝜏

= 𝑢

3

[𝑋

3

1

(𝑦, 𝑧, 1) − 𝑢

3

2

𝑅(𝑦, 𝑧, 1)]

(6)

Specific parts of the system (6) identified by system solutions

𝑢

3

= 0, 𝜙(𝑦, 𝑧) = 0, 𝜓(𝑦, 𝑧) = 0

, (7)

relate to the system's (1) unusual directions, which are not on the manifold

𝑥

3

= 0

unique points found in the system's solution

{

𝑋

3

1

(𝑦, 𝑧, 1) = 0,

𝑋

2

2

(𝑦, 𝑧, 1) = 0,

𝑋

1

2

(𝑦, 𝑧, 1) = 0,

, (8)

correspond to the system's special lines (1); as point O is not isolated in this instance, we

won't be considering it.

System (1) can have four, three, two, one, or no unusual directions that do not lie on the

polyhedral

𝑥

3

= 0

since system (7) can have four, three, two, one, or no solution.

The exceptional directions of type I, type II, or the first group, respectively, refer to the

exceptional directions of system (1) that correspond to the special points found in system (7), from
equation (5), or that correspond to the special points of the first group of the sphere

𝑆

2

.

Lemma 1: Exclusive directions of Type II can alone belong to the initial group.

Proof. All points of the zquator

𝑢

1

2

+ 𝑢

2

2

= 1

of the sphere

𝑆

2

are special points of the

first group, since the equator itself u_3=0. is a wound.

Lemma 2. The special points of the equator of the sphere

𝑆

2

, 𝑢

1

2

+ 𝑢

2

2

= 1

can only be

nodes, saddles or open saddle-nodes.

Proof. Let

𝜇 = 𝜇

0

be the

𝑘

-map action coryeus of the level (5), then, introducing the

substitution

𝜇 − 𝜇

0

= 𝜇̅

into the differential level (4), we have the Briault-Bouquet level.

[2]

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𝑑𝜇

𝑑𝑢

2

=

[𝑋

1

2

(𝜇, 1) − 𝜇 𝑋

2

2

(𝜇, 1)]

(𝑘)

𝜇=𝜇

0

∙ 𝜇

𝑘

+ ⋯

𝑢

2

[𝑋

2

2

(𝜇

0

, 1) + ⋯ ]

here

𝑘 = 1,3

̅̅̅̅ , 𝑋

2

2

(𝜇

0

, 1) ≠ 0

. For it at even to the origin of coordinates

𝑢

2

= 0 , 𝜇 = 0

, is an open saddle-node if

𝑘

is odd, and a knot (saddle) at

[𝑋

1

2

(𝜇, 1) − 𝜇 𝑋

2

2

(𝜇, 1)]

(𝑘)

𝜇=𝜇

0

𝑋

2

2

(𝜇

0

, 1) > 0 (< 0)

Следовательно, особые точки экватора

𝑢

1

2

+ 𝑢

2

2

= 1

сферы

𝑆

2

также будут

только лишь узлы, сдела или открытые седло-узелы.

Hence, the special points of the equator

𝑢

1

2

+ 𝑢

2

2

= 1

of the sphere

𝑆

2

will also be just

knots, made or open saddle-nodes.

Lemma 3. The system (1) can have no exclusive directions on the polyobasis

𝑥

3

= 0

if

the identity is satisfied

𝑢

2

𝑋

1

2

(𝑢

1

, 𝑢

2

, 0) − 𝑢

1

𝑋

2

2

(𝑢

1

, 𝑢

2

) ≡ 0 (9)

Доказательство. В самом деле, при впалнении таждество (9) уровнение (5)

выполняется тождественно и на эквотор

𝑢

1

2

+ 𝑢

2

2

= 1

сферы

𝑆

2

нет особьч

точек,следоательно, на многообразие

𝑥

3

= 0

системы (1) нет исключитлльных

направлений.

Proof. In fact, by applying the identity (9), the level (5) is fulfilled identically and there

are no singular points on the equator

𝑢

1

2

+ 𝑢

2

2

= 1

of the sphere

𝑆

2

, hence there are no excluded

directions on the manifold

𝑥

3

= 0

of the system (1).

Lemma 4. The system (3a) on the equator

𝑢

1

2

+ 𝑢

2

2

= 1

of the sphere

𝑆

2

can have six,

four, two special points (two diametrically opposite points each) if

𝐷 < 0, 𝐷 = 0, 𝐷 > 0

respectively, where

𝐷 = [108 (

𝑏

110

− 𝑎

200

𝑏

200

)

2

−

2
3

(

𝑏

110

− 𝑎

200

𝑏

200

) (

𝑏

020

− 𝑎

110

𝑏

200

) − 2

𝑎

020

𝑏

200

]

2

+ [− (

𝑏

110

− 𝑎

200

𝑏

200

)

2

+

3(𝑏

020

− 𝑎

110

)

𝑏

200

]

3

𝑎

𝑖𝑗𝑘

, 𝑏

𝑖𝑗𝑘

-coefficients of uniform polynomials

𝑋

1

2

(𝑥), 𝑋

2

2

(𝑥)

respectively.

Proof. It follows from the fact that the level (5) can be represented in the form

𝑏

200

𝜇

3

+

(𝑏

110

− 𝑎

200

)𝜇

2

+ (𝑏

200

− 𝑎

110

)𝜇 − 𝑎

020

= 0

The latter contains three at

𝐷 < 0

, two at

𝐷 = 0

(and one doubly), and one unpleasant

solution at

𝐷 > 0

; so, the system (3a) has six at

𝐷 < 0

, four at

𝐷 = 0

, and two (diametrically

opposing) exceptional points at

𝐷 > 0

.

Lemma 5. If the system (3a) has six (two each diametrically opposite) special points on the

equator of the sphere

𝑆

2

, then all six cannot be of saddle type.

Proof. Comparably to Lemma 2's proof in [3]
Lemma 6: System (1) cannot contain four exceptional noprovolutions of type I Proof if

identity (9) is met. In the event where identity (9) is met, system 7 may be expressed as:

{

𝜙(𝑦, 𝑧) = 𝑎

0

+ 𝑎

1

𝑦 + 𝑎

2

𝑧 + 𝑦𝑓

1

(𝑦, 𝑧)

𝜓(𝑦, 𝑧) = 𝑏

0

+ 𝑏

1

𝑦 + 𝑏

2

𝑧 + 𝑧𝑓

1

(𝑦, 𝑧)

Since such a system is known to be incapable of having four solutions [4], system (1)

lacks four exceptional directions of type I while satisfying identity (9).

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2

0

In the event that system (1) possesses four distinct type I directions, it can be reduced

to the form by use of an unexpressed affine proobrozovanaya.

{

𝑑𝑥

1

𝑑𝑡

= (1+𝑎

1

)𝑥

2

+ (

1 − 𝛼

𝛽

+

1 − 𝛽

𝛼

𝑐 + 𝑏

1

) 𝑥

1

𝑥

2

+ 𝑐

1

𝑥

2

2

+ (𝑐

1

− 1)𝑥

1

𝑥

3

− 𝑐𝑥

3

𝑥

2

𝑑𝑥

2

𝑑𝑡

= 𝑘 [𝑥

1

2

− 𝑥

1

𝑥

3

+ (𝑎 − 𝑏

1

)𝑥

1

2

− (𝑎 − 𝑐

1

)𝑥

1

𝑥

3

+ (

1 − 𝛼

𝛽

+

1 − 𝛽

𝛼

𝑎 + 𝑎

1

) 𝑥

1

𝑥

2

]

𝑑𝑥

3

𝑑𝑡

= 𝑎

1

𝑥

1

𝑥

3

+ 𝑏

1

𝑥

2

𝑥

3

+ 𝑐

1

𝑥

3

2

here

𝑘 = 1,3

̅̅̅̅

. The system (6a) then takes the form of the equation

𝑑𝑧

𝑑𝑦

= 𝑘

𝑦(𝑦 − 1) + 𝑎𝑧(𝑧 − 1) + (

1 − 𝛼

𝛽

+

1 − 𝛽

𝛼

𝑎) 𝑦𝑧

𝑦(𝑦 − 1) + 𝑐𝑧(𝑧 − 1) + (

1 − 𝛼

𝛽

+

1 − 𝛽

𝛼

𝑐) 𝑦𝑧

each special point of which corresponds to the upper parts of the exceptional directions

of type I of system (1). The hemisphere

𝑆

2

has the following special points.

𝐴(0,0,1),

𝐵 (0,1,

1

√2

) ,

𝐶 (1,0,

1

√2

) ,

𝐸 (𝛼, 𝛽,

1

√𝛼

2

+ 𝛽

2

) , 𝐹

𝑖

(

𝜇

𝑖

√𝛼

2

+ 𝛽

2

,

1

√𝛼

2

+ 𝛽

2

, 0) , где 𝑘 = 1,3

̅̅̅̅

Lemma 7. a) If

𝐷 < 0, 1 − 𝛼 − 𝛽 > 0, 𝛼 > 0, 𝛽 > 0

(or

1 − 𝛼 − 𝛽 < 0, 𝛼𝛽 < 0

), then on the hemisphere

𝑆

2

two of the special points

𝐴, 𝐵, 𝐶, 𝐸

will be apti-saddles, the other two saddles, and of the special points

𝐹

𝑖

two will be nodes,

one saddle.

б) if

𝐷 = 0, 1 − 𝛼 − 𝛽 > 0, 𝛼 > 0, 𝛽 > 0

(or

1 − 𝛼 − 𝛽 < 0, 𝛼𝛽 < 0

), то на полусфере

𝑆

2

две из особых точек

𝐴, 𝐵, 𝐶, 𝐸

будут

антиседлами, две другие-седелами, а из особых точек

𝐹

𝑖

две с ливаются причем двукратная

особая точка

𝐹

1

= 𝐹

2

будет открытый седлом-узел,

𝐹

3

-узел

then on the hemisphere

𝑆

2

two of the special points

𝐴, 𝐵, 𝐶, 𝐸

will be antisaddles, the other

two will be saddles, and two of the special points

𝐹

𝑖

will be lysed, and the twofold special point

𝐹

1

= 𝐹

2

will be an open saddle-node,

𝐹

3

-node:

b) If

𝐷 > 0, 1 − 𝛼 − 𝛽 > 0, 𝛼 > 0, 𝛽 > 0

(or

1 − 𝛼 − 𝛽 < 0, 𝛼𝛽 < 0

) then of the special points of

𝐹

𝑖

two vanish, one will be a

node.

Proof. On the sphere

𝑆

2

the sum of indices of special points is 2 and on the hemisphere is

1 [5]. Considering also that if

1 − 𝛼 − 𝛽 > 0

,

𝛼 > 0 , 𝛽 > 0

or

1 − 𝛼 − 𝛽 < 0, 𝛼𝛽 < 0

then of the four special points

𝐴, 𝐵, 𝐶, 𝐸

two

are antisaddles and the other two are saddles, at

𝐷 < 0

we have the case a) at

𝐷 = 0

the case b)

at

𝐷 > 0

the case c) of the distribution of special points of the semi-equator of the sphere

𝑆

2

.

Lemma 8. а) If

1 − 𝛼 − 𝛽 > 0, 𝛼𝛽 < 0 𝑘(𝑎 − 𝑐) > 0

or

1 − 𝛼 − 𝛽 > 0,

𝛼 > 0,

𝛽 > 0, 𝑘(𝑎 − 𝑐) > 0, (1 − 𝛼 − 𝛽) < 0,

𝛼 < 0, 𝛽 < 0, 𝑘(𝑎 − 𝑐) < 0),

then

𝐷 = 0

, if

𝐷 ≠ 0

, then

𝐷 > 0 𝑜𝑟 𝐷 < 0

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b) On the hemisphere of the four special points

𝐴, 𝐵, 𝐶, 𝐸

three of them are antisaddles and

a saddle, on the semi-equator

𝑢

1

2

+ 𝑢

2

2

= 1

we have saddles and an open saddle-node at

𝐷 = 0

and at

𝐷 > 0

a saddle.

Proof. By direct calculations it can be obtained analogously to the proof of limm7.
Lemma 9. а)

1 − 𝛼 − 𝛽 < 0, 𝛼 > 0, 𝛽 > 0, 𝑘(𝑎 − 𝑐) < 0 𝑜𝑟 1 − 𝛼 − 𝛽 > 0, 𝛼 < 0,

𝛽 < 0, 𝑘(𝑎 − 𝑐) > 0 𝑜𝑟 1 − 𝛼 − 𝛽 > 0, 𝛼 𝛽 < 0,

𝑘(𝑎 − 𝑐) < 0, 𝑡ℎ𝑒𝑛 𝐷 > 0

c) If condition a) is fulfilled, there are three made and antiseeded special points

𝐴, 𝐵, 𝐶, 𝐸

on the hemisphere

𝑆

2

from the chtyre special points A,B,C,E, and there are three nodes on the

semi-equator

𝑢

1

2

+ 𝑢

2

2

= 1

-three nodes.

We denote by

𝑁(𝑎) 𝑁(𝑐) 𝑎𝑛𝑑 𝑁(𝑐𝑦)

the number of exceptional directions of the system

(1) corresponding to the special points of the antisaddle type, make, and open the saddle-node of

the sphere

𝑆

2

respectively.

4. RESULT AND DISCUSION.

Then the following theorems follow from the proved lemmas
Theorem

1

а)

If

1 − 𝛼 − 𝛽 > 0, 𝛼 > 0, 𝛽 > 0, (1 − 𝛼 − 𝛽 < 0,

(𝛼𝛽 < 0) 𝐷 < 0, 𝑡ℎ𝑒𝑛 𝑁(𝑎) = 4, 𝑁(𝑐) = 3;

b)

𝐷 = 0 то 𝑁(𝑎) = 3, 𝑁(𝑐) = 2, 𝑁(𝑐𝑦) = 1;

c)

𝐷 > 0, то 𝑁(𝑎) = 3, 𝑁(𝑐) = 2.

Theorem 2 а) if

1 − 𝛼 − 𝛽 > 0, 𝛼𝛽 < 0, (1 − 𝛼 − 𝛽 < 0, 𝛼 < 0 𝛽 < 0)

𝐷 = 0 𝑡ℎ𝑒𝑛 𝑤𝑒 𝑔𝑒𝑡 𝑁(𝑎) = 3, 𝑁(𝑐) = 2 𝑁(𝑐𝑦) = 1

b)

𝐷 > 0, 𝑡ℎ𝑒𝑛 𝑁(𝑎) = 3, 𝑁(𝑐) = 2

Theorem 3. If

1 − 𝛼 − 𝛽 < 0, 𝛼 > 0, 𝛽 > 0 𝑜𝑟 1 − 𝛼 − 𝛽 < 0,

𝛼 < 0,

𝛽 < 0 𝑜𝑟 𝑒𝑙𝑠𝑒 1 − 𝛼 − 𝛽 > 0, 𝛼𝛽 < 0, 𝑡ℎ𝑒𝑛 𝑤𝑒 𝑔𝑒𝑡

𝑁(𝑎) = 4, 𝑁(𝑐) = 3

5. CONCLUSION

In the future, the results obtained for one-rd polynomial systems (1) can be extended to

generalised homogeneous and generalised polynomial systems for

𝑛 ≥ 3.

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