Multiplicity of limit cycles and analytic m-solutions for planar differential systems

Armengol Gasull, Jaume Giné, Maite Grau

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5 Citations (Scopus)


This work deals with limit cycles of real planar analytic vector fields. It is well known that given any limit cycle Γ of an analytic vector field it always exists a real analytic function f0 (x, y), defined in a neighborhood of Γ, and such that Γ is contained in its zero level set. In this work we introduce the notion of f0 (x, y) being an m-solution, which is a merely analytic concept. Our main result is that a limit cycle Γ is of multiplicity m if and only if f0 (x, y) is an m-solution of the vector field. We apply it to study in some examples the stability and the bifurcation of periodic orbits from some non-hyperbolic limit cycles. © 2007 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)375-398
JournalJournal of Differential Equations
Publication statusPublished - 15 Sep 2007


  • Bifurcation
  • Invariant algebraic curve
  • Multiple limit cycle
  • Planar vector field
  • Stability
  • m-Solution


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