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.
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