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Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3

Published online by Cambridge University Press:  19 September 2008

F. Dumortier ,
R. Roussarie  and
J. Sotomayor
F. Dumortier
Affiliation:
Limburgs Universitair Centrum, Universitair Campus, B-3610 Diepenbeek, Belgium;
R. Roussarie
Affiliation:
Laboratoire de Topologie, U.A. 755 du CNRS, Université de Bourgogne, B.P. 138-21004-Dijon, France;
J. Sotomayor
Affiliation:
Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, CEP 22460 Jardim Botânico, Rio de Janeiro, Brazil
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Abstract

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A cusp type germ of vector fields is a C∞ germ at 0∈ℝ2, whose 2-jet is C∞ conjugate to

We define a submanifold of codimension 5 in the space of germs consisting of germs of cusp type whose 4-jet is C0 equivalent to

Our main result can be stated as follows: any local 3-parameter family in (0, 0) ∈ ℝ2 × ℝ3 cutting transversally in (0, 0) is fibre-C0 equivalent to

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 7 , Issue 3 , September 1987 , pp. 375 - 413
Copyright
Copyright © Cambridge University Press 1987

References

REFERENCES

[A1]Arnol'd, V.. Lectures on bifurcations in versal families. Russian Mathematical Surveys V, 26, 1971.Google Scholar
[A2]Arnol'd, V.. Chapitres Supplémentaires de la Théorie des Equations Différentielles Ordinaires. Ed Mir, Moscow, 1980.Google Scholar
[A3]Arnol'd, V.. Méthodes Mathématiques de la Mecanique Classique. Editions Mir, Moscow, 1976.Google Scholar
[B1]Bogdanov, R.. Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues. Seminar Petrovski, 1976. Sel. Math. Sov. 1(4), 1981.Google Scholar
[B2]Bogdanov, R.. Bifurcation of a limit cycle for a family of vector fields on the plane. Seminar Petrovski, 1976. Sel. Math. Sov. 1(4), 1981.Google Scholar
[C]Cherkas, L. A.. Structure of a successor function in the neighbourhood of a separatrix cycle of a perturbed analytic autonomous system in the plane. Translated from Differential' nye Uravneniya 17, No. 3, pp. 469–478, 03, 1971.Google Scholar
[D]Dumortier, F.. Singularities of Vector Fields. Monografias de Matematica 32, I.M.P.A., Rio de Janeiro, 1978.Google Scholar
[D, R]Dumortier, F. & Roussarie, R.. Etude Locale des champs de vecteurs à paramètres. Astérisque 59–60, Société Mathématique de France, 1978, 7–42.Google Scholar
[I]Jashenko, Yu. S. Il'. The multiplicity of limit cycles arising from perturbations of the forms w' = P 2/Q 1 of a Hamilton equation in the real and complex domain. Amer. Math. Soc. Transl. (2) 118, 1982;Google Scholar
Translation from: Trudy Sem. Petrovsk. 3, 1978, 49–60.Google Scholar
[K, H]Kopell, N. & Howard, L. N.. Bifurcations and trajectories joining critical points. Advances in Mathematics 18, 1975, 306–358.CrossRefGoogle Scholar
[M]Malgrange, B.. Ideals of Differentiate Function. Oxford Univ. Press, Oxford 1966.Google Scholar
[R]Roussarie, R.. On the number of limit cycles which appear by perturbation of a separatrix loop of planar vector fields. To be published in Bol. da Soc. Mat. Bras.Google Scholar
[Se]Seidenberg, A.. A new decision method for elementary algebra. Ann. of Math. 60 (1954), 365–374.CrossRefGoogle Scholar
[S]Sotomayor, J.. Generic one-parameter families of vector fields on two dimensional manifolds. Publ. Math. I.H.E.S. 43, 1974.CrossRefGoogle Scholar
[T1]Takens, F.. Unfolding of certain singularities of vector fields. Generalized Hopf bifurcations. J. Diff. Eq. 14, 1973, 476–493.CrossRefGoogle Scholar
[T2]Takens, F.. Forced oscillations and bifurcations. In Applications of Global Analysis I, Communications of Math. Inst. Rijksuniversiteit Utrecht 3, 1974.Google Scholar
[T3]Takens, F.. Singularities of vector fields. Publ. math. I.H.E.S. 43, 1974.CrossRefGoogle Scholar