Quadratic models for generic local parameter bifurcations on the plane
Authors:
Freddy Dumortier and Peter Fiddelaers
Journal:
Trans. Amer. Math. Soc. 326 (1991), 101126
MSC:
Primary 58F14; Secondary 58F36
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 parameter unfoldings with , except for the nilpotent focus which cannot occur as a quadratic system. We finally show that a certain type of elliptic points of codimension does not have a quadratic versal unfolding.
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 [C]
 W. A. Coppel, A survey of quadratic systems, J. Differential Equations 2 (1966) 293304. MR 0196182 (33:4374)
 [D1]
 F. Dumortier, Singularities of vector fields on the plane, J. Differential Equations 23 (1977), 53106. MR 0650816 (58:31276)
 [D2]
 , Singularities of vector fields, Monogr. Mat. 32, IMPA, Rio de Janeiro, 1978.
 [DRS1]
 F. Dumortier, R. Roussarie, and J. Sotomayor, Generic parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case, Ergodic Theory Dynamical Systems 7 (1987), 375413. MR 912375 (89g:58149)
 [DRS2]
 F. Dumortier, R. Roussarie, and J. Sotomayor, Generic parameter families of planar vector fields, unfoldings of saddle, focus and elliptic singularities with nilpotent linear parts, Lecture Notes in Math., SpringerVerlag (to appear).
 [G.H.]
 J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Appl. Math. Sci. 42, SpringerVerlag, 1983. MR 709768 (85f:58002)
 [R]
 C. Rousseau, Example of a quadratic system with two cycles appearing in a homoclinic loop bifurcation, J. Differential Equations 66 (1987), 140150. MR 871575 (88b:34041)
 [R.A.]
 R. Rand and D. Armbruster, Perturbation methods, bifurcation theory and computer algebra, Appl. Math. Sci. 65, SpringerVerlag, 1987. MR 911274 (89a:58083)
 [S]
 D. Schlomiuk, Personal Communication.
 [T1]
 F. Takens, Singularities of vector fields, Publ. Math. IHES 43 (1974), 47100. MR 0339292 (49:4052)
 [T2]
 , Unfoldings of certain singularities of vector fields. Generalized Hopf bifurcations, J. Differential Equations 14 (1973), 476493. MR 0339264 (49:4024)
 [Y]
 Ye Yanqian et al., Theory of limit cycles, Transl. Math. Monos., vol. 66, Amer. Math. Soc., Providence, R.I., 1986. MR 854278 (88e:58080)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199110498640
PII:
S 00029947(1991)10498640
Keywords:
Quadratic planar vector fields,
singularities,
codimension,
versal unfoldings,
bifurcations
Article copyright:
© Copyright 1991
American Mathematical Society
