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 Tr. Mat. Inst. Steklova, 2002, Volume 236, Pages 134–141 (Mi tm283)

Singularities of Limiting Directions of Generic Higher Order Implicit ODEs

A. A. Davydov

Abstract: An implicit differential equation of order $n$ is defined as a zero level of a smooth function on the $(n+2)$-dimensional space with a two-dimensional distribution which is the result of natural Goursat prolongation procedure from a standard contact structure in the space of directions on the plane. The solution of this equation is an immersed curve which lies in this level and is tangent to this distribution. Generic metamorphoses of cones of possible directions on the plane of all solutions are classified. This classification is closely related to the classification of generic singularities of first-order implicit differential equations on the plane and to the classification of generic singularities of limiting direction fields of dynamic inequalities on surfaces.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2002, 236, 124–131

Bibliographic databases:

UDC: 517.9
Received in December 2000

Citation: A. A. Davydov, “Singularities of Limiting Directions of Generic Higher Order Implicit ODEs”, Differential equations and dynamical systems, Collected papers. Dedicated to the 80th anniversary of academician Evgenii Frolovich Mishchenko, Tr. Mat. Inst. Steklova, 236, Nauka, MAIK «Nauka/Inteperiodika», M., 2002, 134–141; Proc. Steklov Inst. Math., 236 (2002), 124–131

Citation in format AMSBIB
\Bibitem{Dav02} \by A.~A.~Davydov \paper Singularities of Limiting Directions of Generic Higher Order Implicit ODEs \inbook Differential equations and dynamical systems \bookinfo Collected papers. Dedicated to the 80th anniversary of academician Evgenii Frolovich Mishchenko \serial Tr. Mat. Inst. Steklova \yr 2002 \vol 236 \pages 134--141 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm283} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1931013} \zmath{https://zbmath.org/?q=an:1020.34003} \transl \jour Proc. Steklov Inst. Math. \yr 2002 \vol 236 \pages 124--131 

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This publication is cited in the following articles:
1. Takahashi M., “On implicit second–order ordinary differential equations: Completely integrable and Clairaut type”, Journal of Dynamical and Control Systems, 13:2 (2007), 273–288
2. Takahashi M., “On completely integrable first order ordinary differential equations”, Real and Complex Singularities, 2007, 388–418
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