×

zbMATH — the first resource for mathematics

Singularities of vector fields on \({\mathbb{R}}^ 3\) determined by their first non-vanishing jet. (English) Zbl 0661.58005
The authors present a result which gives a sufficient condition for a vector field X on \({\mathbb{R}}^ 3\) to be equivalent at a singularity to the first non-vanishing jet \(j^ kX(p)\) of X at p. This condition - which only depends on the homogeneous vector field defined by \(j^ kX(p)\)- is stated in terms of the blown-up vector field \(\bar X\) (which is defined on \(S^ 2\times {\mathbb{R}})\), and essentially means that there are no saddle-connections for \(\bar X| S^ 2\times \{0\}.\)
The key tool in the proof will be a result of local normal linearization along a codimension 1 submanifold M providing a \(C^ 0\) conjugacy having a normal derivative along M equal to 1.
Reviewer: P.Bonckaert

MSC:
58A20 Jets in global analysis
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Moise, Geometric Topology in Dimensions 2 and 3 (1977) · Zbl 0349.57001 · doi:10.1007/978-1-4612-9906-6
[2] de Melo, Ergod. Th. & Dynam. Sys. 7 pp 415– (1987)
[3] DOI: 10.1016/0022-0396(85)90075-0 · Zbl 0619.58007 · doi:10.1016/0022-0396(85)90075-0
[4] DOI: 10.2307/1998338 · Zbl 0484.58020 · doi:10.2307/1998338
[5] Urbina, Finite determinacy in ?
[6] Palis, Geometric Theory of Dynamical Systems (1982) · doi:10.1007/978-1-4612-5703-5
[7] DOI: 10.1016/0022-0396(87)90103-3 · Zbl 0633.58031 · doi:10.1016/0022-0396(87)90103-3
[8] DOI: 10.2307/2372774 · Zbl 0083.31406 · doi:10.2307/2372774
[9] DOI: 10.1016/0022-0396(83)90039-6 · Zbl 0468.58017 · doi:10.1016/0022-0396(83)90039-6
[10] DOI: 10.2307/2373513 · Zbl 0197.20701 · doi:10.2307/2373513
[11] DOI: 10.1007/BF01389422 · Zbl 0619.58033 · doi:10.1007/BF01389422
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.