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Cyclicit√© finie des polycycles hyperboliques de champs de vecteurs du plan mise sous forme normale. (Finite cyclicity under normal forms of hyperbolic polycycles of vector fields on the plane). (French) Zbl 0719.58031
Bifurcations of planar vector fields, Proc. Meet., Luminy/Fr. 1989, Lect. Notes Math. 1455, 272-314 (1990).
[For the entire collection see Zbl 0707.00011.]
This paper is devoted to a study of families of vector fields \(X_{\lambda}\) \((\lambda \in {\mathbb{R}}^ N)\) on the plane such that \(X_ 0\) has a hyperbolic polycycle (i.e. a graph whose singular points are all hyperbolic saddle points). The main purpose of the work is to present a normal form for the “first return mapping” associated to the polycycle” of \(X_{\lambda}\), for \(\lambda\) close to 0.

MSC:
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
37G99 Local and nonlocal bifurcation theory for dynamical systems
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms