Quadratic models for generic local 3-parameter bifurcations on the plane

Dumortier, Freddy; Fiddelaers, Peter

doi:10.1090/S0002-9947-1991-1049864-0

Quadratic models for generic local $3$-parameter bifurcations on the plane
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by Freddy Dumortier and Peter Fiddelaers

Trans. Amer. Math. Soc. 326 (1991), 101-126

DOI: https://doi.org/10.1090/S0002-9947-1991-1049864-0

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Abstract:

The first chapter deals with singularities occurring in quadratic planar vector fields. We make distinction between singularities which as a general system are of finite codimension and singularities which are of infinite codimension in the sense that they are nonisolated, or Hamiltonian, or integrable, or that they have an axis of symmetry after a linear coordinate change or that they can be approximated by centers. In the second chapter we provide quadratic models for all the known versal $k$-parameter unfoldings with $k = 1,2,3$, except for the nilpotent focus which cannot occur as a quadratic system. We finally show that a certain type of elliptic points of codimension $4$ does not have a quadratic versal unfolding.

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Quadratic models for generic local $3$-parameter bifurcations on the plane
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Abstract: