Unique Normal Form for a Class of Three-Dimensional Nilpotent Vector Fields

2017 ◽  
Vol 27 (08) ◽  
pp. 1750131 ◽  
Author(s):  
Jing Li ◽  
Liying Kou ◽  
Duo Wang
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Vector Fields ◽  
Normal Forms ◽  
Linear Part ◽  
Three Dimensional ◽  
First Integral ◽  
Second Order ◽  
Lie Brackets ◽  
The Given

In this paper, the unique normal form for a class of three-dimensional nilpotent vector fields with symmetry is investigated. The key technique used is a combination of multiple Lie brackets, linear grading function, new notations of block matrices and first integral of the linear part of the given vector field which avoids complicated calculations. The first and the second order normal forms are computed successively, and the uniqueness of the second order normal form is proved under certain conditions.

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.

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.

Unitary vector fields are Fermi–Walker transported along Rytov–Legendre curves

2015 ◽  
Vol 12 (10) ◽  
pp. 1550111 ◽  
Author(s):  
Mircea Crasmareanu ◽  
Camelia Frigioiu
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Killing Vector ◽  
Three Dimensional ◽  
Ricci Soliton ◽  
Conformal Killing ◽  
Space Forms ◽  
Potential Vector ◽  
Legendre Curves ◽  
Complex Space Forms

Fix ξ a unitary vector field on a Riemannian manifold M and γ a non-geodesic Frenet curve on M satisfying the Rytov law of polarization optics. We prove in these conditions that γ is a Legendre curve for ξ if and only if the γ-Fermi–Walker covariant derivative of ξ vanishes. The cases when γ is circle or helix as well as ξ is (conformal) Killing vector filed or potential vector field of a Ricci soliton are analyzed and an example involving a three-dimensional warped metric is provided. We discuss also K-(para)contact, particularly (para)Sasakian, manifolds and hypersurfaces in complex space forms.

scholarly journals Flows of measures generated by vector fields

2018 ◽  
Vol 148 (4) ◽  
pp. 773-818
Author(s):  
Emanuele Paolini ◽  
Eugene Stepanov
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Borel Measure ◽  
Weak Sense ◽  
Positive Borel Measure ◽  
Smooth Structure ◽  
Smooth Vector ◽  
Integral Curves ◽  
The Given ◽  
Measurable Vector

The scope of the paper is twofold. We show that for a large class of measurable vector fields in the sense of Weaver (i.e. derivations over the algebra of Lipschitz functions), called in the paper laminated, the notion of integral curves may be naturally defined and characterized (when appropriate) by an ordinary differential equation. We further show that for such vector fields the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure ‘flows along’ the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.

scholarly journals ARBTools: A Tricubic Spline Interpolator for Three-Dimensional Scalar or Vector Fields

2019 ◽  
Author(s):  
Paul Walker ◽  
Ulrich Krohn ◽  
Carty David
Keyword(s):  
Magnetic Field ◽  
Vector Field ◽  
Vector Fields ◽  
Three Dimensional ◽  
Spline Method ◽  
Particle Trajectories ◽  
The Core ◽  
Numerical Integrators ◽  
Data Points ◽  
System Requirements

ARBTools is a Python library containing a Lekien-Marsden type tricubic spline method for interpolating three-dimensional scalar or vector fields presented as a set of discrete data points on a regular cuboid grid. ARBTools was developed for simulations of magnetic molecular traps, in which the magnitude, gradient and vector components of a magnetic field are required. Numerical integrators for solving particle trajectories are included, but the core interpolator can be used for any scalar or vector field. The only additional system requirements are NumPy.

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.

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.

A degenerate singularity generating geometric Lorenz attractors

1995 ◽  
Vol 15 (5) ◽  
pp. 833-856 ◽  
Author(s):  
Freddy Dumortier ◽  
Hiroshi Kokubu ◽  
Hiroe Oka
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Vector Fields ◽  
Lorenz Attractor ◽  
Degenerate Singularity ◽  
Field Singularity ◽  
Lorenz Attractors ◽  
Geometric Lorenz Attractor

AbstractA degenerate vector field singularity in R3 can generate a geometric Lorenz attractor in an arbitrarily small unfolding of it. This enables us to detect Lorenz-like chaos in some families of vector fields, merely by performing normal form calculations of order 3.

Equivalence of Stochastic Averaging and Stochastic Normal Forms

1990 ◽  
Vol 57 (4) ◽  
pp. 1011-1017 ◽  
Author(s):  
N. S. Namachchivaya ◽  
Gerard Leng
Keyword(s):  
Normal Form ◽  
Additive Noise ◽  
Normal Forms ◽  
Stochastic Averaging ◽  
Second Order ◽  
Lorenz Equations ◽  
Markovian Approximation ◽  
Flutter Instability

The equivalence of the methods of stochastic averaging and stochastic normal forms is demonstrated for systems under the effect of linear multiplicative and additive noise. It is shown that both methods lead to reduced systems with the same Markovian approximation. The key result is that the second-order stochastic terms have to be retained in the normal form computation. Examples showing applications to systems undergoing divergence and flutter instability are provided. Furthermore, it is shown that unlike stochastic averaging, stochastic normal forms can be used in the analysis of nilpotent systems to eliminate the stable modes. Finally, some results pertaining to stochastic Lorenz equations are presented.

scholarly journals Normal forms for elements of o(p, q) and Hamiltonians with integrals linear in momenta

1992 ◽  
Vol 33 (4) ◽  
pp. 486-507 ◽  
Author(s):  
Gerard Thompson
Keyword(s):  
Dynamical Systems ◽  
Normal Form ◽  
Lie Algebra ◽  
Normal Forms ◽  
First Integral ◽  
Inner Product ◽  
Indefinite Inner Product ◽  
Hamiltonian Dynamical Systems ◽  
Matrix Normal

AbstractWe solve the problem of finding a simultaneous matrix normal form for an element of the Lie algebra o(p, q) and the underlying indefinite inner product. The results are used to determine several classes of classical Hamiltonian dynamical systems which possess a first integral linear in the momentum variables.

Sign in / Sign up

Export Citation Format

Share Document