Averaging theory at any order for computing periodic orbits

Jaume Giné, Maite Grau, Jaume Llibre

Research output: Contribution to journalArticleResearchpeer-review

39 Citations (Scopus)


We provide a recurrence formula for the coefficients of the powers of ε in the series expansion of the solutions around ε=0 of the perturbed first-order differential equations. Using it, we give an averaging theory at any order in ε for the following two kinds of analytic differential equation: dxdθ=∑k≥1εkFk(θ,x), dxdθ=∑k≥0εkFk(θ,x). A planar polynomial differential system with a singular point at the origin can be transformed, using polar coordinates, to an equation of the previous form. Thus, we apply our results for studying the limit cycles of a planar polynomial differential systems. © 2013 Elsevier B.V. All rights reserved.
Original languageEnglish
Pages (from-to)58-65
JournalPhysica D: Nonlinear Phenomena
Publication statusPublished - 1 Jan 2013


  • Averaging theory
  • First-order analytic differential equations
  • Limit cycles
  • Periodic orbits
  • Polynomial differential equations


Dive into the research topics of 'Averaging theory at any order for computing periodic orbits'. Together they form a unique fingerprint.

Cite this