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Finitude des cycles-limites et accelero-sommation de l’application de retour. (Finitude of limit cycles and accelerated summation of the return mapping). (French) Zbl 0729.34016

Bifurcations of planar vector fields, Proc. Meet., Luminy/Fr. 1989, Lect. Notes Math. 1455, 74-159 (1990).
[For the entire collection see Zbl 0707.00011.]
The article presents a proof of the Dulac proposition that the limit cycles of an analytic vector field X in the plane \({\mathbb{R}}^ 2\) cannot accumulate. Since the accumulation can take place only on a polycycle \((S_ 1,S_ 2,...,S_ r)\) one considers the transition functions between the vertices \(S_ i\), \(g_ i: x_ i\to x_{i+1}\) \((i=1,...,r- 1)\) and their composition \(f=g_ n\circ...g_ 2\circ g_ 1\), which is called the function of return. It is to show that there exists an interval on which the function f is either identity or fixed point free. By the substitution \(z_ i=1/x_ i\) \((i=1,...,r-1)\) the function f transforms to the function F with which the author associates a certain geometric object \(\tilde F\) which he calls a “transseries” and which is a generalization of an asymptotic series. Using certain operations on transseries he finally proves the non-oscillation property of the function f. Several examples are also given.
Reviewer: L.Janos (Praha)

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37-XX Dynamical systems and ergodic theory

Citations:

Zbl 0707.00011