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Izvestiya: Mathematics, 2014, Volume 78, Issue 6, Pages 1063–1078
DOI: https://doi.org/10.1070/IM2014v078n06ABEH002720
(Mi im8203)
 

This article is cited in 7 scientific papers (total in 7 papers)

Implicit ordinary differential equations: bifurcations and sharpening of equivalence

I. A. Bogaevsky

M. V. Lomonosov Moscow State University
References:
Abstract: We obtain a formal classification of generic local bifurcations of an implicit ordinary differential equation at its singular points as a single external parameter varies. This classification consists of four normal forms, each containing a functional invariant. We prove that every deformation in the contact equivalence class of an equation germ which remains quadratic in the derivative can be obtained by a deformation of the independent and dependent variables. Our classification is based on a generalization of this result for families of equations. As an application, we obtain a formal classification of generic local bifurcations on the plane for a linear second-order partial differential equation of mixed type at the points where the domains of ellipticity and hyperbolicity undergo Morse bifurcations.
Keywords: implicit ordinary differential equation, formal change of variables, normal form, linear equation of mixed type, characteristic, bifurcation, contact equivalence, generating function of a contact vector field.
Received: 23.12.2013
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2014, Volume 78, Issue 6, Pages 5–20
DOI: https://doi.org/10.4213/im8203
Bibliographic databases:
Document Type: Article
UDC: 517.922+517.956.6
MSC: Primary 34A09; Secondary 34A26, 34C23
Language: English
Original paper language: Russian
Citation: I. A. Bogaevsky, “Implicit ordinary differential equations: bifurcations and sharpening of equivalence”, Izv. RAN. Ser. Mat., 78:6 (2014), 5–20; Izv. Math., 78:6 (2014), 1063–1078
Citation in format AMSBIB
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  • https://www.mathnet.ru/eng/im8203
  • https://doi.org/10.1070/IM2014v078n06ABEH002720
  • https://www.mathnet.ru/eng/im/v78/i6/p5
  • This publication is cited in the following 7 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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    Abstract page:761
    Russian version PDF:686
    English version PDF:33
    References:93
    First page:33
     
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