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.