DĂ©veloppement 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


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


Zbl 0647.00005