Limit cycles of Lienard systems.
by
Makhlouf Amar
Department of mathematics;Faculty of sciences;University of Annaba;Annaba;ALGERIA.
Coauthors: Badi Sabrina (Department of Mathematics, Faculty of Sciences, University Badji Mokhtar, Elhadjar, BP 12, ANNABA, ALGERIA)

We study the existence of limit cycles of planar systems and especially the Lienar systems.We give an other proof of the theorem proved by L.Perko concerning the number of limit cycles of the system:

x¢=y-F(x)

y¢=-x

with F(x)=e(a₁x+...+a_2m+1x^2m+1), 0<e<<1

by using averaging method.We give the asymptotic solutions by using the Lindsted method.We can construct polynomial systems with as many cycles as we like.We review the main and recents results of limit cycles of planar

Polynomial systems (sixtenth problem of Hilbert).Wegive a table on the number of limit cycles for different values of the degree of the polynoms.We determin the number of small amplitude limit cycles that can be bifurcated from the origin for the systems:

x¢=y-F(x)

y¢=-g(x)

with F(x)=a₁x+a₂x²+...+a_ux^u

g(x)=x+b₂x²+...+b_vx^v

The Lyapunov quantities are significant in this study.

An algorithm is presented for bifurcating small-amplitude limit cycles out of a critical point of this system.

Classification AMS: 34C05-34C25-34D20.

Key words :limit cycle-Lienard system.

Date received: March 5, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caky-23.