The formal normal form degenerate singular points in the case of case of focus

Сентябрь 13, 2022

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The formal normal form degenerate singular points in the case of case of focus

Содержание

2. OBJECTIVE In the work we study the problem for the case Have a normal form where
3. AIM OF WORK The direct study of a formal normal form with the feature type Bogdanov-Takens
4. Takens in 1974 for the system of equations of the form got a fairly simple formal
5. General( for all cases) the formal normal form was obtained in 2015. This Form looks very
6. THEOREM The field is formally equivalent to one of the following fields formal power series. Forms
7. DEFINITION Two analytical vector fields in are formally equivalent ,then and only then when there is
8. THE FUNCTIONAL EQUATION OF EQUIVALENCE AND THE SCHEME OF REDUCTION TO SYSTEMS OF LINEAR EQUATIONS Objective
9. We substitute the expansions in the main equation : 17aliya93@mail.ru
10. Get the system of equations. This system has a specific type which allows to solve it
11. INFERENCE After we solve this system of equations the formal normal form takes the form We
12. REFERENCES Formal normal form Zholandeka Strozhinoy — PWS Publishing, 1997 17aliya93@mail.ru
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Слайд 2

OBJECTIVE
In the work we study the problem for the case Have

a normal form where Formal normal form takes the form

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Слайд 3

AIM OF WORK
The direct study of a formal normal form with

the feature type Bogdanov-Takens for a particular case, for the purpose of comparison with the results of Zolondek- Strozhinoi.

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Слайд 4

Takens in 1974 for the system of equations of the form got

a fairly simple formal normal form but this formal normal form admits of further simplification.

PREFACE

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Слайд 5

General( for all cases) the formal normal form was obtained in

2015. This Form looks very complicated and has the form:

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Слайд 6

THEOREM
The field is formally equivalent to one of the following fields formal

power series. Forms are the only modulo substitutions

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Слайд 7

DEFINITION
Two analytical vector fields in are formally equivalent ,then and only

then when there is a formal diffeomorphisms

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Слайд 8

THE FUNCTIONAL EQUATION OF EQUIVALENCE AND THE SCHEME OF REDUCTION TO

SYSTEMS OF LINEAR EQUATIONS

Objective :to find a formal normal form as simple as possible species

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Слайд 9

We substitute the expansions in the main equation :
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Слайд 10

Get the system of equations. This system has a specific type

which allows to solve it consistently.

tab1.

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Слайд 11

INFERENCE
After we solve this system of equations the formal normal form

takes the form

We have 3 adjacent degree - 3 monoms from a formal normal form. Thus , our result is consistent with the result of Zolondek-Strozhinoi.

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Слайд 12

REFERENCES
Formal normal form Zholandeka Strozhinoy — PWS Publishing, 1997
17aliya93@mail.ru