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Non-accumulation of critical points of the Poincaré time on hyperbolic polycycles

Authors: Pavao Mardesic and Mariana Saavedra
Journal: Proc. Amer. Math. Soc. 135 (2007), 3273-3282
MSC (2000): Primary 34C07; Secondary 34C25, 34M35
Published electronically: June 22, 2007
MathSciNet review: 2322759
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Abstract: We call Poincaré time the time associated to the Poincaré (or first return) map of a vector field. In this paper we prove the non-accumulation of isolated critical points of the Poincaré time $ T$ on hyperbolic polycycles of polynomial vector fields. The result is obtained by proving that the Poincaré time of a hyperbolic polycycle either has an unbounded principal part or is an almost regular function. The result relies heavily on the proof of Il'yashenko's theorem on non-accumulation of limit cycles on hyperbolic polycycles.

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  • 1. C. Chicone; Dumortier, Freddy, Finiteness for critical periods of planar analytic vector fields. Nonlinear Anal. 20 (1993), no. 4, 315-335. MR 1206421 (94b:58082)
  • 2. H. Dulac, Sur les cycles limites, Bull. Soc. Math. France 51 (1923), 45-188. MR 1504823
  • 3. J. Écalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac. (French) [Introduction to analyzable functions and constructive proof of the Dulac conjecture], Actualités Mathématiques. Hermann, Paris, (1992). ii+340 pp. MR 1399559 (97f:58104)
  • 4. J. Guckenheimer; Ph. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983. xvi+453 pp. MR 709768 (85f:58002)
  • 5. Yu. S. Il'yashenko, Limit cycles of polynomial vector fields with nondegenerate singular points on the real plane, Funct. Anal. and Appl., Vol. 18, No. 3, 1984, 199-209. MR 757247 (86a:34054)
  • 6. Yu. S. Il'yashenko, Finiteness Theorems for Limit Cycles, Translations of Mathematical Monographs, Volume 94 (1991). MR 1109064 (92c:58110)
  • 7. R. Moussu, Le problème de la finitude du nombre de cycles limites. Séminaire Bourbaki, Vol. 1985/86. Astérisque, No. 145-146 (1987), 3, 89-101. MR 880027 (88g:58159)
  • 8. R. Roussarie, Bifurcation of planar vector fields and Hilbert's sixteenth problem. Progress in Mathematics, 164. Birkhäuser Verlag, Basel, 1998. xviii+204 pp. MR 1628014 (99k:58129)
  • 9. M. Saavedra, Développement asymptotique de la fonction période thèse, Dijon (1995). MR 1298283 (95g:58158)
  • 10. M. Saavedra, Asymptotic expansion of the period function, J. Diff. Eq. 193 (2003), 359-373. MR 1998638 (2004i:34074)
  • 11. M. Saavedra, Asymptotic expansion of the period function II, J. Diff. Eq. 222 (2006), 476-486. MR 2208293
  • 12. E. Titchmarsh. The theory of functions, Second edition. Translated from the English and with a preface by V. A. Rohlin. ``Nauka'', Moscow, 1980. 464 pp. MR 593142 (82b:30001)

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Additional Information

Pavao Mardesic
Affiliation: Institut de Mathématique de Bourgogne, U.M.R. 5584 du C.N.R.S. Université de Bourgogne, B.P. 47 870 21078 Dijon Cedex, France

Mariana Saavedra
Affiliation: Departamento de Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Chile

Keywords: Critical period, finiteness, non-accumulation, quasi-analyticity, Dulac problem
Received by editor(s): July 5, 2006
Published electronically: June 22, 2007
Additional Notes: This work was partially supported by Fondecyt Projects 1061006 and 7060107, Escuela de Graduados de la Universidad de Concepción and Proyecto Fundación Andes C13955/12
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2007 American Mathematical Society

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