On the approximation of the limit cycles function

L. Cherkas, Belorussian State University of Informatics and Radioelectronics, Minsk, Belarus
A. Grin, Grodno State University, Grodno, Belarus
K. R. Schneider, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

E. J. Qualitative Theory of Diff. Equ., No. 28. (2007), pp. 1-11.

Abstract: 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 Poincar\'e function and represents a continuation method. Finally, we demonstrate the effectiveness of the proposed procedure by means of a Li\'enard system.

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