The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems
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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.
Is part ofJournal of Differential Equations, 2006, vol. 227, núm. 2, p. 406-426
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García, I. A. (Isaac A.); Giacomini, Héctor; Giné, Jaume; Llibre, Jaume (Elsevier, 2016-04-25)We prove that all the nilpotent centers of planar analytic differential systems are limit of centers with purely imaginary eigenvalues, and consequently the Poincaré--Liapunov method to detect centers with purely imaginary ...
Grau Montaña, Maite; Llibre, Jaume (Elsevier, 2015)We consider a planar autonomous real analytic differential system with a monodromic singular point p. We deal with the center problem for the singular point p. Our aim is to highlight some relations between the divergence ...
Giné, Jaume; Llibre, Jaume; Valls, Claudia (Shanghai Normal University & Wilmington Scientific Publisher, 2017)For the polynomial differential system x˙=−y, y˙=x+Qn(x,y), where Qn(x,y) is a homogeneous polynomial of degree n there are the following two conjectures done in 1999. (1) Is it true that the previous system for n≥2 has a ...