The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems

Héctor Giacomini (1) , Jaume Giné , Jaume Llibre (2)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 UAB - Departament de Matemàtiques [Barcelona]

In this work we study the centers of planar analytic vector fields which are limit of linear type centers. It is proved that all the nilpotent centers are limit of linear type centers and consequently the Poincaré--Liapunov method to find linear type centers can be also used to find the nilpotent centers. Moreover, we show that the degenerate centers which are limit of linear type centers are also detectable with the Poincaré--Liapunov method.

Domaines: Systèmes dynamiques [math.DS]; Analyse classique [math.CA]

Héctor Giacomini, Jaume Giné, Jaume Llibre. The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems. Journal of Differential Equations, 2006, 227 (2), pp.406-426. ⟨10.1016/j.jde.2006.03.012⟩. ⟨hal-00805747⟩