Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case

1981 ◽  
pp. 54-74 ◽  
Author(s):  
Henk Broer
Keyword(s):  
Normal Form ◽  
Vector Fields ◽  
Volume Preserving

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.

scholarly journals Normal Form Near Orbit Segments of Convex Hamiltonian Systems

2021 ◽  
Author(s):  
Shahriar Aslani ◽  
Patrick Bernard
Keyword(s):  
Normal Form ◽  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Vector Fields ◽  
Energy Surface ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
The Other ◽  
Finsler Metrics ◽  
The One

Abstract In the study of Hamiltonian systems on cotangent bundles, it is natural to perturb Hamiltonians by adding potentials (functions depending only on the base point). This led to the definition of Mañé genericity [ 8]: a property is generic if, given a Hamiltonian $H$, the set of potentials $g$ such that $H+g$ satisfies the property is generic. This notion is mostly used in the context of Hamiltonians that are convex in $p$, in the sense that $\partial ^2_{pp} H$ is positive definite at each point. We will also restrict our study to this situation. There is a close relation between perturbations of Hamiltonians by a small additive potential and perturbations by a positive factor close to one. Indeed, the Hamiltonians $H+g$ and $H/(1-g)$ have the same level one energy surface, hence their dynamics on this energy surface are reparametrisation of each other, this is the Maupertuis principle. This remark is particularly relevant when $H$ is homogeneous in the fibers (which corresponds to Finsler metrics) or even fiberwise quadratic (which corresponds to Riemannian metrics). In these cases, perturbations by potentials of the Hamiltonian correspond, up to parametrisation, to conformal perturbations of the metric. One of the widely studied aspects is to understand to what extent the return map associated to a periodic orbit can be modified by a small perturbation. This kind of question depends strongly on the context in which they are posed. Some of the most studied contexts are, in increasing order of difficulty, perturbations of general vector fields, perturbations of Hamiltonian systems inside the class of Hamiltonian systems, perturbations of Riemannian metrics inside the class of Riemannian metrics, and Mañé perturbations of convex Hamiltonians. It is for example well known that each vector field can be perturbed to a vector field with only hyperbolic periodic orbits, this is part of the Kupka–Smale Theorem, see [ 5, 13] (the other part of the Kupka–Smale Theorem states that the stable and unstable manifolds intersect transversally; it has also been studied in the various settings mentioned above but will not be discussed here). In the context of Hamiltonian vector fields, the statement has to be weakened, but it remains true that each Hamiltonian can be perturbed to a Hamiltonian with only non-degenerate periodic orbits (including the iterated ones), see [ 11, 12]. The same result is true in the context of Riemannian metrics: every Riemannian metric can be perturbed to a Riemannian metric with only non-degenerate closed geodesics, this is the bumpy metric theorem, see [ 1, 2, 4]. The question was investigated only much more recently in the context of Mañé perturbations of convex Hamiltonians, see [ 9, 10]. It is proved in [ 10] that the same result holds: if $H$ is a convex Hamiltonian and $a$ is a regular value of $H$, then there exist arbitrarily small potentials $g$ such that all periodic orbits (including iterated ones) of $H+g$ at energy $a$ are non-degenerate. The proof given in [ 10] is actually rather similar to the ones given in papers on the perturbations of Riemannian metrics. In all these proofs, it is very useful to work in appropriate coordinates around an orbit segment. In the Riemannian case, one can use the so-called Fermi coordinates. In the Hamiltonian case, appropriate coordinates are considered in [ 10,Lemma 3.1] itself taken from [ 3, Lemma C.1]. However, as we shall detail below, the proof of this Lemma in [ 3], Appendix C, is incomplete, and the statement itself is actually wrong. Our goal in the present paper is to state and prove a corrected version of this normal form Lemma. Our proof is different from the one outlined in [ 3], Appendix C. In particular, it is purely Hamiltonian and does not rest on the results of [ 7] on Finsler metrics, as [ 3] did. Although our normal form is weaker than the one claimed in [ 10], it is actually sufficient to prove the main results of [ 6, 10], as we shall explain after the statement of Theorem 1, and probably also of the other works using [ 3, Lemma C.1].

Representation of volume-preserving maps induced by solenoidal vector fields

1985 ◽  
Vol 28 (3) ◽  
pp. 1005 ◽  
Author(s):  
A. Thyagaraja ◽  
F. A. Haas
Keyword(s):  
Vector Fields ◽  
Solenoidal Vector ◽  
Volume Preserving

Unique Normal Form and the Associated Coefficients for a Class of Three-Dimensional Nilpotent Vector Fields

2017 ◽  
Vol 27 (14) ◽  
pp. 1750224
Author(s):  
Jing Li ◽  
Liying Kou ◽  
Duo Wang ◽  
Wei Zhang
Keyword(s):  
Normal Form ◽  
Vector Fields ◽  
Three Dimensional ◽  
Recursive Formula ◽  
Higher Order ◽  
Dimensional Vector ◽  
Symbolic Calculations

In this paper, we mainly focus on the unique normal form for a class of three-dimensional vector fields via the method of transformation with parameters. A general explicit recursive formula is derived to compute the higher order normal form and the associated coefficients, which can be achieved easily by symbolic calculations. To illustrate the efficiency of the approach, a comparison of our result with others is also presented.

Гиперболические поля Руссари с вырожденной квадратичной частью

2021 ◽  
Vol 76 (2(458)) ◽  
pp. 183-184
Author(s):  
Наталья Геннадьевна Павлова ◽  
Natalia Gennadievna Pavlova ◽  
Алексей Олегович Ремизов ◽  
Alexey Olegovich Remizov
Keyword(s):  
Normal Form ◽  
Vector Fields ◽  
Quadratic Part

A local normal form for Roussarie vector fields with degenerate quadratic part is presented.

Normal Forms for Codimension One Planar Piecewise Smooth Vector Fields

2014 ◽  
Vol 24 (07) ◽  
pp. 1450090 ◽  
Author(s):  
Tiago de Carvalho ◽  
Durval José Tonon
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Vector Fields ◽  
Normal Forms ◽  
Piecewise Smooth ◽  
Topological Equivalence ◽  
Smooth Vector ◽  
Codimension One ◽  
Piecewise Smooth Vector Fields ◽  
Piecewise Smooth Vector Field

In this paper, we are dealing with piecewise smooth vector fields in a 2D-manifold. In such a scenario, the main goal of this paper is to exhibit the homeomorphism that gives the topological equivalence between a codimension one piecewise smooth vector field and the respective C0-normal form.

scholarly journals Holomorphic normal form of nonlinear perturbations of nilpotent vector fields

2016 ◽  
Vol 21 (4) ◽  
pp. 410-436 ◽  
Author(s):  
Laurent Stolovitch ◽  
Freek Verstringe
Keyword(s):  
Normal Form ◽  
Vector Fields ◽  
Nonlinear Perturbations

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.

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