First integrals of local analytic differential systems

Jaume Llibre, Chara Pantazi, Sebastian Walcher

Research output: Contribution to journalArticleResearchpeer-review

26 Citations (Scopus)

Abstract

We investigate formal and analytic first integrals of local analytic ordinary differential equations near a stationary point. A natural approach is via the Poincaré-Dulac normal forms: If there exists a formal first integral for a system in normal form then it is also a first integral for the semisimple part of the linearization, which may be seen as "conserved" by the normal form. We discuss the maximal setting in which all such first integrals are conserved, and show that all first integrals are conserved for certain classes of reversible systems. Moreover we investigate the case of linearization with zero eigenvalues, and we consider a three-dimensional generalization of the quadratic Dulac-Frommer center problem. © 2011 Elsevier Masson SAS.
Original languageEnglish
Pages (from-to)342-359
JournalBulletin des Sciences Mathematiques
Volume136
Issue number3
DOIs
Publication statusPublished - 1 Apr 2012

Keywords

  • Center
  • Formal first integral
  • Normal form
  • Primary
  • Reversible system

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