Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T00:56:33.398Z Has data issue: false hasContentIssue false
  • English
  • Français

Un théorème de préparation pour fonctions à développement Tchébychévien

Published online by Cambridge University Press:  19 September 2008

Pierre Joyal
Pierre Joyal
Affiliation:
Departement d'informatique et de mathématique, Université du Québec à Chicoutimi, 555, boulevard d'Université, Montréal, Québec, CanadaG7H 2B1
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article, we prove a preparation theorem for functions which admit a certain type of expansion called Chebychev expansion. Taylor expansions are particular cases of Chebychev expansions. The result is based on an approach essentially different from those used for the classical preparation theorems. It has applications in bifurcation theory of vector fields.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 14 , Issue 2 , June 1994 , pp. 305 - 329
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Barbançon, G.. Théorème de Newton pour les fonctions de classe Cr. Ann. Sci. Ec. Norm. Sup. 4 5 (1972), 435–158.CrossRefGoogle Scholar
[2]Dangelmayr, G. & Guckenheimer, J.. On a four parameter family of planar vector fields. Arch. Rat. Mech. Anal. 97 (1987), 321–352.CrossRefGoogle Scholar
[3]Dumortier, F., Roussarie, R. & Sotomayor, J.. Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp of codimension 3. Ergod. Th. & Dynam. Sys. 7 (1987), 375–413.CrossRefGoogle Scholar
[4]Joyal, P.. Generalized Hopf bifurcation and its dual: generalized homoclinic bifurcation. SIAM J. Appl. Math. 48 (1988), 481–96.Google Scholar
[5]Joyal, P.. The Cusp of order n. SIAM J. Appl. Math. 88 (1990), 1–14.Google Scholar
[6]Joyal, P.. Generalized homoclinic bifurcation, à paraître dans SIAM J. Appl. Math. 107(1) (1994), 1–45.Google Scholar
[7]Karlin, S. & Studden, W. J.. Tchebycheff Systems: With Applications in Analysis and Statistics Interscience: New York, 1966.Google Scholar
[8]Lassalle, M. G.. Une démonstration du théorème de division pour les fonctions différentiables Topology 12 (1973), 41–62.Google Scholar
[9]Petrov, G. S.. Elliptic integrals and their nonoscillation. Funktsional'nyi Analiz i Ego Prilozheniya 20 (1986), 46–49.Google Scholar
[10]Roussarie, R.. On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields. Bol. Soc. Bras. Mat. 17 (1986), 67–101.CrossRefGoogle Scholar
[11]Rousseau, C. & Schlomiuk, D.. Generalized Hopf bifurcations and applications to planar quadratic systems. Ann. Polici Mathematici XLIX (1986), 1–16.Google Scholar
[12]Takens, F.. Unfolding of certain singularities of vector fields: generalised Hopf bifurcations. J. Diff. Eq. 14 (1973), 476–493.CrossRefGoogle Scholar