Darboux integrability and Algebraic limit cycles for a class of polynomial differential Systems
Cao, Jinlong (Shanghai Jiao Tong University. Department of Mathematics)
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Zhang, Xiang (Shanghai Jiao Tong University. Department of Mathematics)

Date:	2014
Abstract:	This paper deals with the existence of Darboux first integrals for the planar polynomial differential systems ˙x = λx - y + Pn+1(x, y) + xF2n(x, y), ˙y = x + λy + Qn+1(x, y)+yF2n(x, y), where Pi(x, y), Qi(x, y) and Fi(x, y) are homogeneous polynomials of degree i. Inside this class we identify some new Darboux integrable systems having either a focus or a center at the origin. For such Darboux integrable systems having degrees 5 and 9 we give the explicit expressions of their algebraic limit cycles. For the systems having degrees 3, 5, 7 and 9 we present necessary and sufficient conditions for being Darboux integrable.
Grants:	Ministerio de Economía y Competitividad MTM2005-06098-C02-01 Ministerio de Economía y Competitividad SAB2006-0098
Note:	Agraïments: The first and third authors are partially supported by NNSF of China grant No. 10671123 and by a CICYT grant No. 2005SGR 00550. The third author is also partially supported by NCET of China grant No. 050391.
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Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Algebraic limit cycles ; Darboux first integral
Published in:	Science China Mathematics, Vol. 57 Núm. 4 (2014), p. 775-794, ISSN 1869-1862

DOI: 10.1007/s11425-014-4772-8

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