Orbital equivalence of singular points of vector fields on the plane

Богданов, Р. И.

doi:10.1007/bf01076031

Cited by 13 publications

(11 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Similarly to methods developed in [4] and [7], calculations of orbital equivalent triples (v, w2, ~) for a fixed pair (v, w2) are performed in two steps. First, we consider the transformation (.)…”

Section: Reduction To Bivector Fieldsmentioning

confidence: 99%

“…Such a calculation of j and s can be performed with the help of the integrating divisor of the vector field in a neighborhood of the singularity (see, e.g., [1] and [7]), in other words, with the help of the vector field's invariant measure. Such a calculation of j and s can be performed with the help of the integrating divisor of the vector field in a neighborhood of the singularity (see, e.g., [1] and [7]), in other words, with the help of the vector field's invariant measure.…”

mentioning

confidence: 99%

“…This article continues papers [6][7][8]. In these papers the local invariants of the plane vector field's phase portraits are calculated in terms of elementary functions of suitable smooth phase coordinates: the generator of the first integral's algebra, the module over the ring of the first integrals of invariant measures, and the stationary subalgebra of the vector field's phase portrait modulo the subalgebra generated by collinear to initial vector fields (see [6]).…”

mentioning

confidence: 99%

See 2 more Smart Citations

Local relative integral invariants determined by the phase portrait of a vector field on the plane

Богданов¹

1995

J Math Sci

Self Cite

View full text Add to dashboard Cite

by the phase portrait of a vector field on the UDC 517.92 We/Tx some invariant measure for a given vector hem on the plane. The pair (the phase portrait of the vector field, the invariant measure of the vector held) determines a Lie suba/gebra in the algebra of all smooth vector 5elds on the plane, namely the stationary suba/gebra of the pair. An element of the suba/gebra has a relative integral invariant, namely: the integral of the above measure along sets bounded by phase curves of the initial vector/Te/d and the element of suba/gebra. The main result of the paper is the following:Theorem. Relative integral invariants in the genera/situation (more exactly, for the finite-modal case) are expressed in terms of elementary functions of suitable phase coordinates.Degenerate relative integral invariants, for which the above theorem is not valid, appear in twoparametric families of the above objects in the general situation.* Unfortunately, this connection is not a straightforward one, because it depends not only on the parity of j, but also on the parity of the first integral's homogeneous part.

show abstract

Section: Reduction To Bivector Fieldsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Local relative integral invariants determined by the phase portrait of a vector field on the plane

Богданов¹

1995

J Math Sci

Self Cite

View full text Add to dashboard Cite

show abstract

“…This case was studied by Bogdanov [7] [8], and the versal deformation in the sense of orbital equivalence is given there. But, we shall describe our procedures in the computation of our normal forms of vector field singularities for this case for the following reason.…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

Normal forms for singularities of vector fields

Ushiki

1984

Japan J. Appl. Math.

100

View full text Add to dashboard Cite

A new method to compute normal forms of vector-field singularities is proposed. Normal forms for some degenerate singularities of vector fields are computed. These normal forms are simpler than those known as Arnold-Takens normal form. Parameters in the normal forms are uniquely determined from the original singularity in the category of (jets of) coordinate transformations.The first step in studying the singularities of vector fields is to reduce the singularities to simpler forms which are called normal forms. Normal form theory has been one of the most important tools for the study of singularities and the theory of bifurcations. There have been plenty of results and applications in this field: H. Poincar› [22] [23], G. Birkhoff [6], L. C. Siegel [25], S. Sternberg [26] [27], F. Takens [28] [29] [30], V. I. Arnold [2] [3] [4] [5], J. Guckenheimer [11], W. F. Langford [20] [21], G. Iooss and D. D. Joseph [19], etc ....As far as the author knows, however, nobody has executed the complete computation of normal forms of vector field singularities, not even for the second order, which should be implied by the general theory, i.e., the computation of orbits of jets of germs of vector fields under the action of the group of coordinate transformations.Our theory leads us, if we restrict ourselves to jets of finite order, to a complete classification of orbits, modulo higher order terms, in the space ofjets of vector field singularities under the action ofjets of local diffeomorphisms.The result is surprising. Most of the degenerate singularities are much simpler than what was known. These jets can be transformed, by a polynomial coordinate transformation, into a vector field whose jet is in the list stated below.It is known that most degenerate singularities are of infinite codimension in the category of C~%equivalence. For example, F. Takens [28] showed the existence of non-stabilizable jets of vector fields. F. Ichikawa [16] [17] gave algebraic criteria for a jet to be wild. According to his result, wildness of singularities is a very common property of degenerate vector fields.

show abstract

“…More specifically, The classification of vector fields near a singularity of type A $for generic perturbations can be found in Arnold [1], Bogdanov [3,4,5], and, for symmetric perturbations, in Carr [81 and Takens [14]. Such classification is not simple.…”

Section: Introduction Dmentioning

confidence: 99%