WIAS Preprint No. 1196, (2007)

On the approximation of the limit cycles function


  • Cherkas, Leonid
  • Grin, Alexander
  • Schneider, Klaus R.

2010 Mathematics Subject Classification

  • 34C05 34C07 65L99


  • Family of limit cycles, multiple limit cycle, Liénard system




We consider planar vector fields depending on a real parameter. It is assumed that this vector field has a family of limit cycles which can be described by means of the limit cycles function $l$. We prove a relationship between the multiplicity of a limit cycle of this family and the order of a zero of the limit cycles function. Moreover, we present a procedure to approximate $l(x)$, which is based on the Newton scheme applied to the Poincaré function and represents a continuation method. Finally, we demonstrate the effectiveness of the proposed procedure by means of a Liénard system. The obtained result supports a conjecture by Lins, de Melo and Pugh.

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