The 16th Hilbert problem restricted to circular algebraic limit cycles
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Ramírez, Rafael Orlando (Universitat Rovira i Virgili. Departament d'Enginyeria Informàtica i Matemàtiques)
Ramírez, Valentín (Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II)
Sadovskaia, Natalia (Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II)

Date:	2016
Abstract:	We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S admits at most S - 1 invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S - 1 algebraic limit cycles given by circles, and this number is reached.
Grants:	Ministerio de Economía y Competitividad MTM2008-03437 Ministerio de Economía y Competitividad TSI2007-65406-C03-01 Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-568 European Commission 318999 European Commission 316338
Note:	Agraïments: FEDER-UNAB10-4E-378 and Consolider CSD2007-00004 "ES"
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Darboux integrability ; Invariant algebraic circles ; Planar polynomial differential system ; Polynomial vector fields
Published in:	Journal of differential equations, Vol. 260 (2016) , p. 5726-5760, ISSN 1090-2732

DOI: 10.1016/j.jde.2015.12.019

Postprint 32 p, 789.9 KB

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