Fixed and moving limit cycles for Liénard equations

  • Armengol Gasull
  • Marco SabatiniEmail author


We consider a family of planar vector fields that writes as a Liénard system in suitable coordinates. It has a fixed closed invariant curve that often contains periodic orbits of the system. We prove a general result that gives the hyperbolicity of these periodic orbits, and we also study the coexistence of them with other periodic orbits. Our family contains the celebrated Wilson polynomial Liénard equation, as well as all polynomial Liénard systems having hyperelliptic limit cycles. As an illustrative example, we study in more detail a natural 1-parametric extension of Wilson example. It has at least two limit cycles, one of them fixed and algebraic and the other one moving with the parameter, presents a transcritical bifurcation of limit cycles and for a given parameter has a non-hyperbolic double algebraic limit cycle. In order to prove that for some values of the parameter the system has exactly two hyperbolic limit cycles, we use several suitable Dulac functions.


Liénard equation Limit cycle Bifurcations Invariant algebraic curve 

Mathematics Subject Classification

Primary 34C07 Secondary 37C23 34C25 37C27 



The first author is partially supported by the MINECO/FEDER MTM2016-77278-P and AGAUR 2017-SGR-1617 Grants. The second author is partially supported by GNAMPA, Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni.


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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dep. de MatemàtiquesUniv. Autònoma de BarcelonaBellaterra, BarcelonaSpain
  2. 2.Dip. di MatematicaUniv. di TrentoPovoItaly

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