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Symétrie et forme normale des centres et foyers dégénérés

Published online by Cambridge University Press:  19 September 2008

Robert Moussu
Robert Moussu
Affiliation:
Departement de Mathematiques, Laboratoire de Topologie, Universite de Dijon, 21004 Dijon Cedex
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Abstract

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Consider an analytic differential equation ο = adx + bdy with an algebraically isolated singularity and without a separatrix. The germ at 0∈ℝ2 of the 1-jet y dy is either a focus or a centre. The equation has C∞ normal form of the type = with F(x) = F(−x) if and only if the germ is a centre.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 2 , Issue 2 , June 1982 , pp. 241 - 251
Copyright
Copyright © Cambridge University Press 1982

References

REFERENCES

[1]Bendixson, I.. Sur les points singuliers des équations différentielles. Ofv. Kangl. Vetenskaps. Akade. Eörhandlinger. Stokholm. 9 (1898), 635–658.Google Scholar
[2]Dumortier, F., Rodriguez, P. & Roussarie, R.. Germs of diffeomorphisms in the plane. Lecture Notes in Math. No. 902. Springer: Berlin, 1975.Google Scholar
[3]Dumortier, F.. Singularities of vector fields on a plane. J. Diff. Equns. 23 (1977), 53–106.CrossRefGoogle Scholar
[4]Lyapunov, A. M.. Stability of Motion, pp. 123–194. Academic Press: New York, 1966.Google Scholar
[5]Moussu, R.: Sur un théorème de Poincaré-Lyapunov. Astérisque. (To appear.)Google Scholar
[6]Moussu, R.. Sur l'existence d'int´grales premières pour un germe de forme de Pfaff. Ann. Inst. Fourier 26 (1976) 229–237.CrossRefGoogle Scholar
[7]Mattel, J. F. & Moussu, R.. Holonomie et intégrates premières. Ann. Scient. Ec. Norm. Sup. Ser. 4.13 (1980), 469–523.Google Scholar
[8]Poincaré, H.. Mémoire sur les courbes définies par une équation differentielle. J. Math. Pures et Appl. 3–7 (1881), 375–422.Google Scholar
[9]Roussarie, R.. Modéles locaux de champs et formes. Astérisque 30 (1975).Google Scholar
[10]Seidenberg, A.. Reduction of singularities of the differential equation A dy = B dx. Amer. J. Math. 79 (1968), 248–269.CrossRefGoogle Scholar
[11]Takens, F.. Normal forms for certain singularities of vector fields. Ann. Inst. Fourier-Grenoble. 23 (1973) 163–195.CrossRefGoogle Scholar
[12]Takens, F.. Singularities of vector fields. Publ. Math. I.H.E.S. 43 (1974), 47–100.CrossRefGoogle Scholar
[13]Takens, F.. Forced oscillations and bifurcations. Comm. Math. Inst. Rijkcuniversiteit Utrecht 3 (1974), 1–59.Google Scholar