Open AccessCenter conditions for a class of planar rigid polynomial differential systems
Author(s) -
Jaume Llibre,
Roland Rabanal
Publication year - 2015
Publication title -
discrete and continuous dynamical systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.289
H-Index - 70
eISSN - 1553-5231
pISSN - 1078-0947
DOI - 10.3934/dcds.2015.35.1075
Subject(s) - center (category theory) , polynomial , eigenvalues and eigenvectors , focus (optics) , singularity , planar , order (exchange) , class (philosophy) , differential (mechanical device) , mathematics , constant (computer programming) , pure mathematics , mathematical analysis , combinatorics , discrete mathematics , physics , computer science , quantum mechanics , chemistry , computer graphics (images) , artificial intelligence , thermodynamics , crystallography , programming language , finance , optics , economics
El títol de la versió pre-print de l'article és: Explicit focal basis for some planar rigid polynomial differential systemsAgraïments: The first author was partially partially supported by CAPES/DGU grant number BEX 12566/12-8.In general the center–focus problem cannot be solved, but in the case that the singularity has purely imaginary eigenvalues there are algorithms to solving it. The present paper implements one of these algorithms for the polynomial dierential systems of the form x = −y + xf (x)g(y), y = x + yf (x)g(y), where f (x) and g(y) are arbitrary polynomials. These dierential systems have constant angular speed and are also called rigid systems. More precisely, this paper gives focal bases of these systems, and then necessary and sucient conditions in order to have an uniform isochronous center. In particular, the existence of a focus with the highest order is also studied