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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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


Authors: Freddy Dumortier and Peter Fiddelaers
Journal: Trans. Amer. Math. Soc. 326 (1991), 101-126
MSC: Primary 58F14; Secondary 58F36
DOI: https://doi.org/10.1090/S0002-9947-1991-1049864-0
MathSciNet review: 1049864
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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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DOI: https://doi.org/10.1090/S0002-9947-1991-1049864-0
Keywords: Quadratic planar vector fields, singularities, codimension, versal unfoldings, bifurcations
Article copyright: © Copyright 1991 American Mathematical Society