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

Mardešić, Pavao; Saavedra, Mariana

doi:10.1090/S0002-9939-07-09026-0

Non-accumulation of critical points of the Poincaré time on hyperbolic polycycles
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by Pavao Mardešić and Mariana Saavedra PDF

Proc. Amer. Math. Soc. 135 (2007), 3273-3282 Request permission

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.

References

Carmen Chicone and Freddy Dumortier, Finiteness for critical periods of planar analytic vector fields, Nonlinear Anal. 20 (1993), no. 4, 315–335. MR 1206421, DOI 10.1016/0362-546X(93)90136-G
H. Dulac, Sur les cycles limites, Bull. Soc. Math. France 51 (1923), 45–188 (French). MR 1504823
Jean Écalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1992 (French, with French summary). MR 1399559
John Guckenheimer and Philip Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. MR 709768, DOI 10.1007/978-1-4612-1140-2
Yu. S. Il′yashenko, Limit cycles of polynomial vector fields with nondegenerate singular points on the real plane, Funktsional. Anal. i Prilozhen. 18 (1984), no. 3, 32–42 (Russian). MR 757247
Yu. S. Il′yashenko, Finiteness theorems for limit cycles, Algebra and analysis (Kemerovo, 1988) Amer. Math. Soc. Transl. Ser. 2, vol. 148, Amer. Math. Soc., Providence, RI, 1991, pp. 55–64. MR 1109064, DOI 10.1090/trans2/148/06
Robert Moussu, Le problème de la finitude du nombre de cycles limites, Astérisque 145-146 (1987), 3, 89–101 (French). Séminaire Bourbaki, Vol. 1985/86. MR 880027
Robert Roussarie, Bifurcation of planar vector fields and Hilbert’s sixteenth problem, Progress in Mathematics, vol. 164, Birkhäuser Verlag, Basel, 1998. MR 1628014, DOI 10.1007/978-3-0348-8798-4
Mariana Saavedra, Développement asymptotique de la fonction période, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 6, 563–566 (French, with English and French summaries). MR 1298283
Mariana Saavedra, Asymptotic expansion of the period function, J. Differential Equations 193 (2003), no. 2, 359–373. MR 1998638, DOI 10.1016/S0022-0396(03)00091-3
Mariana Saavedra, Asymptotic expansion of the period function. II, J. Differential Equations 222 (2006), no. 2, 476–486. MR 2208293, DOI 10.1016/j.jde.2005.04.011
E. Titčmarš, Teoriya funktsiĭ, 2nd ed., “Nauka”, Moscow, 1980 (Russian). Translated from the English and with a preface by V. A. Rohlin. MR 593142