Algebraic limit cycles of degree 4 for quadratic systems

Javier Chavarriga, Jaume Llibre, Jordi Sorolla

Research output: Contribution to journalArticleResearchpeer-review

20 Citations (Scopus)

Abstract

Yablonskii (Differential Equations 2 (1996) 335) and Filipstov (Differential Equations 9 (1973) 983) proved the existence of two different families of algebraic limit cycles of degree 4 in the class of quadratic systems. It was an open problem to know if these two algebraic limit cycles where all the algebraic limit cycles of degree 4 for quadratic systems. Chavarriga (A new example of a quartic algebraic limit cycle for quadratic sytems, Universitat de Lleida, Preprint 1999) found a third family of this kind of algebraic limit cycles. Here, we prove that quadratic systems have exactly four different families of algebraic limit cycles. The proof provides new tools based on the index theory for algebraic solutions of polynomial vector fields. © 2004 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)206-244
JournalJournal of Differential Equations
Volume200
DOIs
Publication statusPublished - 10 Jun 2004

Keywords

  • Algebraic limit cycles
  • Quadratic systems
  • Quadratic vector fields

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