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Izv. RAN. Ser. Mat., 2014, Volume 78, Issue 6, Pages 5–20 (Mi izv8203)  

This article is cited in 4 scientific papers (total in 4 papers)

Implicit ordinary differential equations: bifurcations and sharpening of equivalence

I. A. Bogaevsky

M. V. Lomonosov Moscow State University

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.

Funding Agency Grant Number
Russian Foundation for Basic Research 11-01-00960-а
Ministry of Education and Science of the Russian Federation НШ-5138.2014.1


DOI: https://doi.org/10.4213/im8203

Full text: PDF file (701 kB)
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English version:
Izvestiya: Mathematics, 2014, 78:6, 1063–1078

Bibliographic databases:

UDC: 517.922+517.956.6
MSC: Primary 34A09; Secondary 34A26, 34C23
Received: 23.12.2013

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

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. C. Nabarro, A. Saloom, “On the Singularities of Families of Curve Congruences on Lorentzian Surfaces”, J. Dyn. Control Syst., 22:3 (2016), 507–530  crossref  mathscinet  zmath  isi  elib  scopus
    2. A. Davydov, “Normal forms of linear second order partial differential equations on the plane”, Sci. China-Math., 61:11, SI (2018), 1947–1962  crossref  mathscinet  zmath  isi  scopus
    3. A. A. Davydov, Yu. A. Kasten, “On nonlocal normal forms of linear second order mixed type PDEs on the plane”, Control Systems and Mathematical Methods in Economics: Essays in Honor of Vladimir M. Veliov, Lecture Notes in Economics and Mathematical Systems, 687, eds. G. Feichtinger, R. Kovacevic, G. Tragler, Springer-Verlag Berlin, 2018, 15–25  crossref  mathscinet  zmath  isi
    4. A. A. Davydov, Yu. A. Kasten, “On Structural Stability of Characteristic Nets and the Cauchy Problem for a Tricomi–Cibrario Type Equation”, Proc. Steklov Inst. Math., 304 (2019), 146–152  mathnet  crossref  crossref  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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