105 research outputs found
On the number of limit cycles which appear by perturbation of two-saddle cycles of planar vector fields
We prove that every heteroclinic saddle loop (a two-saddle cycle) occurring
in an analytic finite-parameter family of plane analytic vector fields, may
generate no more than a finite number of limit cycles within the family.Comment: 21 pages, 10 figures, a new section explaining the so called "Petrov
trick" in the context of the paper is added. The paper will appear in
"Functional Analysis and Its Applications" (2013
Quadratic perturbations of quadratic codimension-four centers
We study the stratum in the set of all quadratic differential systems
with a center, known as the
codimension-four case . 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 . We show that the orbits of the unperturbed system are elliptic
curves, and 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
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