Averaging methods of arbitrary order, periodic solutions and integrability

Jaume Giné, Jaume Llibre, Kesheng Wu, Xiang Zhang

Research output: Contribution to journalArticleResearchpeer-review

19 Citations (Scopus)

Abstract

© 2015 Elsevier Inc. All rights reserved. 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 dif-ferential 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 Nsuch that if its first Naveraging 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.
Original languageEnglish
Pages (from-to)4130-4156
JournalJournal of Differential Equations
Volume260
Issue number5
DOIs
Publication statusPublished - 1 Mar 2016

Keywords

  • Averaging method
  • Differential systems
  • Integrability
  • Limit cycle
  • Polynomial differential systems

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