Unique normal forms for planar vector fields

1988 ◽  
Vol 199 (3) ◽  
pp. 303-310 ◽  
Author(s):  
Alberto Baider ◽  
Richard Churchill
Keyword(s):  
Vector Fields ◽  
Normal Forms ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
Unique Normal Forms

UNIQUE NORMAL FORMS FOR NILPOTENT PLANAR VECTOR FIELDS

2002 ◽  
Vol 12 (10) ◽  
pp. 2159-2174 ◽  
Author(s):  
GUOTING CHEN ◽  
DUO WANG ◽  
XIAOFENG WANG
Keyword(s):  
Normal Form ◽  
Vector Fields ◽  
Normal Forms ◽  
Unsolved Problem ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
Unique Normal Forms ◽  
Special Case

Further reduction of normal forms for nilpotent planar vector fields has been considered. Unique normal form for a special case of an unsolved problem for the Takens–Bogdanov singularity is given. Computations in Maple are used to conjecture the main results and some computations in the proof are also done with Maple.

Uniqueness and non-uniqueness of normal forms for vector fields

1988 ◽  
Vol 108 (1-2) ◽  
pp. 27-33 ◽  
Author(s):  
Alberto Baider ◽  
Richard Churchill
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Vector Fields ◽  
Normal Forms ◽  
Resonance Problem ◽  
Main Result Theorem ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
Result Theorem ◽  
The Relationship

SynopsisThe use of normal forms in the study of equilibria of vector fields and Hamiltonian systems is a well-established practice and is described in standard references (e.g. [1], [7] or [10]). Also well known is the fact that such normal forms are not unique, and the relationship between distinct normal forms of the same vector field has also been investigated, in particular by M. Kummer [8] and A. Brjuno [2,3] (also see [12]). In this paper we use this relationship to extract invariants of the vector field directly from an arbitrary normal form. The treatment is sufficiently general to handle the vector field and Hamiltonian cases simultaneously, and applications in these contexts are presented.The formulation of our main result (Theorem 1.1) is reminiscent of, and was heavily influenced by, work of Shi Songling on planar vector fields [11]. Additional inspiration was provided by M. Kummer's contributions to the 1:1 resonance problem in [9]. The authors are grateful to Richard Cushman for comments on an earlier version of this paper.

scholarly journals A New Normal Form for Monodromic Nilpotent Singularities of Planar Vector Fields

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Antonio Algaba ◽  
Cristóbal García ◽  
Jaume Giné
Keyword(s):  
Normal Form ◽  
Vector Fields ◽  
New Technique ◽  
Center Problem ◽  
New Normal ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
A New Technique

AbstractIn this work, we present a new technique for solving the center problem for nilpotent singularities which consists of determining a new normal form conveniently adapted to study the center problem for this singularity. In fact, it is a pre-normal form with respect to classical Bogdanov–Takens normal formal and it allows to approach the center problem more efficiently. The new normal form is applied to several examples.

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