UNIQUE NORMAL FORMS FOR NILPOTENT PLANAR VECTOR FIELDS

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.

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Unique normal forms for planar vector fields

Mathematische Zeitschrift ◽

10.1007/bf01159780 ◽

1988 ◽

Vol 199 (3) ◽

pp. 303-310 ◽

Cited By ~ 32

Author(s):

Alberto Baider ◽

Richard Churchill

Keyword(s):

Vector Fields ◽

Normal Forms ◽

Planar Vector Fields ◽

Planar Vector ◽

Unique Normal Forms

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Uniqueness and non-uniqueness of normal forms for vector fields

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500026482 ◽

1988 ◽

Vol 108 (1-2) ◽

pp. 27-33 ◽

Cited By ~ 13

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.

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Normal Forms and Bifurcation of Planar Vector Fields

10.1017/cbo9780511665639 ◽

1994 ◽

Cited By ~ 232

Author(s):

Shui-Nee Chow ◽

Chengzhi Li ◽

Duo Wang

Keyword(s):

Vector Fields ◽

Normal Forms ◽

Planar Vector Fields ◽

Planar Vector

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

Mediterranean Journal of Mathematics ◽

10.1007/s00009-021-01820-7 ◽

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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Orbital Reversibility of Planar Vector Fields

Mathematics ◽

10.3390/math9010014 ◽

2020 ◽

Vol 9 (1) ◽

pp. 14

Author(s):

Antonio Algaba ◽

Cristóbal García ◽

Jaume Giné

Keyword(s):

Vector Field ◽

Normal Form ◽

Vector Fields ◽

Center Problem ◽

Normal Form Theory ◽

Planar Vector Fields ◽

Planar Vector ◽

Form Theory ◽

Nilpotent Center

In this work we use the normal form theory to establish an algorithm to determine if a planar vector field is orbitally reversible. In previous works only algorithms to determine the reversibility and conjugate reversibility have been given. The procedure is useful in the center problem because any nondegenerate and nilpotent center is orbitally reversible. Moreover, using this algorithm is possible to find degenerate centers which are orbitally reversible.

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