Développement asymptotique de l’application retour d’un polycycle. (Asymptotic expansion of the return map around a polycycle).(French)Zbl 0659.58009

Dynamical systems, Proc. Symp., Valparaiso/Chile 1986, Lect. Notes Math. 1331, 140-149 (1988).
[For the entire collection see Zbl 0647.00005.]
The author describes the form of the asymptotic expansion of the return map around a separatrix cycle in two dimensions. He derives it very efficiently, using $$C^{\infty}$$ normal forms due to Roussaire. The author states that this paper is one step, “certainly the most elementary”, in the proof by himself, Ecalle, Martinet, and Ramis that a polynomial vector field on $${\mathbb{R}}^ 2$$ has only a finite number of limit cycles.
Reviewer: St.Schecter

MSC:

 37D99 Dynamical systems with hyperbolic behavior 34E05 Asymptotic expansions of solutions to ordinary differential equations

Zbl 0647.00005