Geodesic
Implicit ordinary differential equations: bifurcations and sharpening of equivalence
Izvestiya. Mathematics , Tome 78 (2014) no. 6, pp. 1063-1078.

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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, normal form, linear equation of mixed type, characteristic, contact equivalence, generating function of a contact vector field.
Mots-clés : formal change of variables, bifurcation 
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I. A. Bogaevsky. Implicit ordinary differential equations: bifurcations and sharpening of equivalence. Izvestiya. Mathematics , Tome 78 (2014) no. 6, pp. 1063-1078. http://geodesic.mathdoc.fr/item/IM2_2014_78_6_a1/

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