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

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-à Ministry of Education and Science of the Russian Federation ÍØ-5138.2014.1

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

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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

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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• http://mi.mathnet.ru/eng/izv8203
• https://doi.org/10.4213/im8203
• http://mi.mathnet.ru/eng/izv/v78/i6/p5

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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
2. A. Davydov, “Normal forms of linear second order partial differential equations on the plane”, Sci. China-Math., 61:11, SI (2018), 1947–1962
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
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
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