Mathematics – Dynamical Systems
Scientific paper
2008-11-27
J. Math. Anal. Appl. 357 (2009) 69-76
Mathematics
Dynamical Systems
Scientific paper
10.1016/j.jmaa.2009.04.004
We study the stratum in the set of all quadratic differential systems $\dot{x}=P_2(x,y), \dot{y}=Q_2(x,y)$ with a center, known as the codimension-four case $Q_4$. It has a center and a node and a rational first integral. The limit cycles under small quadratic perturbations in the system are determined by the zeros of the first Poincar\'e-Pontryagin-Melnikov integral $I$. We show that the orbits of the unperturbed system are elliptic curves, and $I$ is a complete elliptic integral. Then using Picard-Fuchs equations and the Petrov's method (based on the argument principle), we set an upper bound of eight for the number of limit cycles produced from the period annulus around the center.
Gavrilov Lubomir
Iliev Iliya D.
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