Smooth normal forms of folded elementary singular points

Davydov, A. A.; Ortiz-Bobadilla, Laura

doi:10.1007/bf02255893

Cited by 21 publications

(30 citation statements)

References 5 publications

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“…Clearly, the integral curve γ b from Lemma 1 satisfies the required conditions. The representations (7) and (12) show that the curves π(K ) and π(γ 1−b ) are semicubic parabolas on the (x, y)-plane having the common cusp at O with the same tangential direction ∂ x . To determine the mutual arrangement of π(K ) and π(γ 1−b ) in a neighborhood of O, it is convenient to represent the semicubic parabolas π(K ) and π(γ 1−b ) in the form of a sole algebraic equation:…”

Section: Resultsmentioning

confidence: 99%

On Pleated Singular Points of First-Order Implicit Differential Equations

Chertovskih

Remizov

2013

J Dyn Control Syst

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We study phase portraits of a first order implicit differential equation in a neighborhood of its pleated singular point that is a non-degenerate singular point of the lifted field. Although there is no a visible local classification of implicit differential equations at pleated singular points (even in the topological category), we show that there exist only six essentially different phase portraits, which are presented. 1 This approach goes back to H. Poincaré (Mémoire sur les courbes définies par leséquations différentielles. -J. Math. Pures Appl., Sér 4, 1885) and A. Clebsch (Ueber eine Fundamentalaufgabe der Invariantentheorie. -Göttingen Abh. XVII, 1872). The latter paper contains geometric interpretation of differential equations (both ordinary and partial) and related theory of connexes, which is quite similar to the lifting; an account of these ideas is contained also in the famous book "Vorlesungenüber höhere Geometrie" by F. Klein.

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Section: Resultsmentioning

confidence: 99%

On Pleated Singular Points of First-Order Implicit Differential Equations

Chertovskih

Remizov

2013

J Dyn Control Syst

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“…The problem is that the resonances for saddles are everywhere dense and generically a resonance saddle is not C ∞ -linearizable. Normal forms for a generic folded resonance and folded elementary singular points were obtained in [17,18] and [15,16], respectively. The formulation of the respective results is analogous to the classification theorems above.…”

Section: Theorem 35 (Seementioning

confidence: 99%

Normal forms of linear second order partial differential equations on the plane

Davydov

2018

Sci. China Math.

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The paper is devoted to the theory of normal forms of main symbols for linear second order partial differential equations on the plane. We discuss the results obtained in the last decades and some problems, which are important both for the development of this theory and the applications. The reduction theorem, which was used to obtain many of recent results in the theory, is included in the paper in the parametric form together with proof. There is a feeling that the theorem still has potential to get progress in the solution of open problems in the theory.

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“…Smooth normal forms for BDEs in a neighbourhood of a folded singularity were established in [8,10] (see also [7,9]). For our purposes, a topological classification is sufficient.…”

Section: Binary Differential Equationsmentioning

confidence: 99%

“…The set V E is given by A 2 − K 5 B 2 = 0. A calculation shows that A 2 − K 5 B 2 = −49κ 10 2 a 4 30 /16 when κ 1 = 0. Hence, in general, S G ⊂ V E .…”

Section: The Torsion Of the Characteristic Curvesmentioning

confidence: 99%

On the characteristic curves on a smooth surface

Oliver

2011

Journal of the London Mathematical Society

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We study some symmetries between two classical pairs of foliations defined on smooth surfaces in ℝ3: the asymptotic curves and the characteristic curves. The asymptotic curves exist in hyperbolic regions of surfaces and have been well studied. The characteristic curves are the analogy of the asymptotic curves in elliptic regions. We produce results on the characteristic curves mirroring those in Uribe‐Vargas (Mosc. Math. J. 6 (2006) 731–768) on the asymptotic curves. By considering cross‐ratios of Legendrian lines in the manifold of contact elements to the surface, we show that certain properties of the characteristic curves are invariant under projective transformations. We establish an analogy of the Beltrami–Enepper theorem. We exhibit the existence of a curve of points of zero torsion of the characteristic curves and determine its behaviour near cusps of Gauss and umbilic points.

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Smooth normal forms of folded elementary singular points

Cited by 21 publications

References 5 publications

On Pleated Singular Points of First-Order Implicit Differential Equations

On Pleated Singular Points of First-Order Implicit Differential Equations

Normal forms of linear second order partial differential equations on the plane

On the characteristic curves on a smooth surface

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