Positive quadratic differential forms: linearization, finite determinacy and versal unfolding

Gutiérrez, Carlos; Guíñez, Víctor

doi:10.5802/afst.844

Cited by 8 publications

(10 citation statements)

References 1 publication

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“…We list the existing cases in the Appendix ( §6). It is worth pointing out here that Gutierrez and Guíñez [42] showed that BDEs with 1-jet (y, b 1 x + b 2 y, −y) are topologically 1-determined when the discriminant has a Morse singularity, i.e., when b 1 = 0. They also proved that any BDE with 1-jet (y, b 1 x + b 2 y, −y) is smoothly equivalent to (y,…”

Section: Bdes Of Typementioning

confidence: 85%

“…As pointed out in §3.1 there are no discrete local models under formal equivalence for BDEs of Type 2. To my knowledge, smooth equivalence has only been considered in one case in [42] (see Remark 2(2)). For topological equivalence there are several ways to proceed.…”

Section: Bdes Of Typementioning

confidence: 99%

“…The first method is computational and the calculations are sometimes long and winding. (It is used in [38,39,40,41,42,57,66,68] to obtain topological models of BDEs with singular associated surface M .) (2) There are classifications of some more degenerate singularities, motivated by geometric problems.…”

Section: Bdes Of Typementioning

confidence: 99%

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Pairs of foliations on surfaces

Tari¹

2010

Real and Complex Singularities

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We survey in this paper results on a particular set of Implicit Differential Equations (IDEs) on smooth surfaces, called Binary/Quadratic Differential Equations (BDEs). These equations define at most two solution curves at each point on the surface, resulting in a pair of foliations in some region of the surface. BDEs appear naturally in differential geometry and in control theory. The examples we give here are all from differential geometry. They include natural families of BDEs on surfaces. We review the techniques used to obtain local models of BDEs (formal, analytic, smooth and topological). We also discuss some invariants of BDEs and present a framework for studying their bifurcations in generic families.

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Section: Bdes Of Typementioning

confidence: 85%

Section: Bdes Of Typementioning

confidence: 99%

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Pairs of foliations on surfaces

Tari¹

2010

Real and Complex Singularities

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“…Determining models of configurations associated with BDEs has been also addressed in many works; for example, in [2,4,5,6,7,14,21,22] the classification of BDEs is performed up to topological, formal, analytic or smooth equivalences. We refer to [20] for a survey on these topics.…”

Section: (Communicated By Rafael De La Llave)mentioning

confidence: 99%

Binary differential equations with symmetries

Manoel¹,

Tempesta²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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This paper introduces the study of occurrence of symmetries in binary differential equations (BDEs). These are implicit differential equations given by the zeros of a quadratic 1-form, a(x, y)dy 2 +b(x, y)dxdy +c(x, y)dx 2 = 0, for a, b, c smooth real functions defined on an open set of R 2 . Generically, solutions of a BDE are given as leaves of a pair of foliations, and the action of a symmetry must depend not only whether it preserves or inverts the plane orientation, but also whether it preserves or interchanges the foliations. The first main result reveals this dependence, which is given algebraically by a formula relating three group homomorphisms defined on the symmetry group of the BDE. The second main result adapts methods from invariant theory of compact Lie groups to obtain an algorithm to compute general expressions of equivariant quadratic 1-forms under each compact subgroup of the orthogonal group O(2).

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“…Since the publication of [22], several authors have produced a considerable amount of research with generalizations and extensions of the results to surfaces in other ambient spaces, such as Euclidean and semi-Euclidean spaces. See, for instance [2,11,13,16,17,18,19,20,21] just to mention a few. In particular, P. Bayard and F. Sánchez-Bringas study in [2] the principal configurations for spacelike surfaces in Minkowski 4-space and they find the 1-jet of the differential equation of the principal curvature lines with respect to a null normal vector field.…”

Section: Introductionmentioning

confidence: 99%