Averaging methods of arbitrary order, periodic solutions and integrability
Giné, Jaume (Universitat de Lleida. Departament de Matemàtica)
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Wu, Kesheng (Shanghai Jiao Tong University. Department of Mathematics)
Zhang, Xiang (Shanghai Jiao Tong University. Department of Mathematics)

Date:	2016
Abstract:	In this paper we provide an arbitrary order averaging theory for higher dimensional periodic analytic differential systems. This result extends and improves results on averaging theory of periodic analytic differential systems, and it unifies many different kinds of averaging methods. Applying our theory to autonomous analytic differential systems, we obtain some conditions on the existence of limit cycles and integrability. For polynomial differential systems with a singularity at the origin having a pair of pure imaginary eigenvalues, we prove that there always exists a positive number N such that if its first N averaging functions vanish, then all averaging functions vanish, and consequently there exists a neighborhood of the origin filled with periodic orbits. Consequently if all averaging functions vanish, the origin is a center for n = 2. Furthermore, in a punctured neighborhood of the origin, the system is C^ completely integrable for n > 2 provided that each periodic orbit has a trivial holonomy. Finally we develop an averaging theory for studying limit cycle bifurcations and the integrability of planar polynomial differential systems near a nilpotent monodromic singularity and some degenerate monodromic singularities.
Grants:	Ministerio de Ciencia e Innovación MTM2011-22877 Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-1204 Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-568 Ministerio de Economía y Competitividad MTM2013-40998-P European Commission 316338 European Commission 316339
Note:	Agraïments: The third and fourth authors are partially supported by NNSF of China grant number 11271252, by RFDP of Higher Education of China grant number 20110073110054. The fourth author also is supported by the innovation program of Shanghai Municipal Education Commission grant 15ZZ012.
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Averaging method ; Differential systems ; Integrability ; Limit cycle ; Polynomial differential systems
Published in:	Journal of differential equations, Vol. 260 (2016) , p. 4130-4156, ISSN 1090-2732

DOI: 10.1016/j.jde.2015.11.005

Postprint 30 p, 812.6 KB

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