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

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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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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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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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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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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DEFINITION
Two analytical vector fields
in are formally equivalent ,then and only then when there is a formal diffeomorphisms
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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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We substitute the expansions in the main equation :
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Get the system of equations. This system has a specific type which allows to solve it consistently.
tab1.
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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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REFERENCES
Formal normal form Zholandeka Strozhinoy — PWS Publishing, 1997
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