131 research outputs found
Perturbations of quadratic centers of genus one
We propose a program for finding the cyclicity of period annuli of quadratic
systems with centers of genus one. As a first step, we classify all such
systems and determine the essential one-parameter quadratic perturbations which
produce the maximal number of limit cycles. We compute the associated
Poincare-Pontryagin-Melnikov functions whose zeros control the number of limit
cycles. To illustrate our approach, we determine the cyclicity of the annuli of
two particular reversible systems.Comment: 28 page
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
The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems
In this work we study the centers of planar analytic vector fields which are
limit of linear type centers. It is proved that all the nilpotent centers are
limit of linear type centers and consequently the Poincar\'e--Liapunov method
to find linear type centers can be also used to find the nilpotent centers.
Moreover, we show that the degenerate centers which are limit of linear type
centers are also detectable with the Poincar\'e--Liapunov method.Comment: 24 pages, no figure
On the cyclicity of weight-homogeneous centers
Let W be a weight-homogeneous planar polynomial differential system with a
center. We find an upper bound of the number of limit cycles which bifurcate
from the period annulus of W under a generic polynomial perturbation. We apply
this result to a particular family of planar polynomial systems having a
nilpotent center without meromorphic first integral.Comment: 13 pages, no figure
On integrability and cyclicity of cubic systems
In this paper we study the integrability of a few families of the complex cubic system. We have obtained necessary and sufficient conditions for existence of a local analytic first integral. Sufficiency of the obtained conditions was proven using different methods: time-reversibility, Darboux integrability and others. Using the obtained results on integrability of complex cubic system, we have obtained results for corresponding real cubic systems. Then the study of bifurcation of limit cycles from each component of the center variety of real system was performed
Limit cycles from a monodromic infinity in planar piecewise linear systems
Planar piecewise linear systems with two linearity zones separated by a
straight line and with a periodic orbit at infinity are considered. By using
some changes of variables and parameters, a reduced canonical form with five
parameters is obtained. Instead of the usual Bendixson transformation to work
near infinity, a more direct approach is introduced by taking suitable
coordinates for the crossing points of the possible periodic orbits with the
separation straight line. The required computations to characterize the
stability and bifurcations of the periodic orbit at infinity are much easier.
It is shown that the Hopf bifurcation at infinity can have degeneracies of
co-dimension three and, in particular, up to three limit cycles can bifurcate
from the periodic orbit at infinity. This provides a new mechanism to explain
the claimed maximum number of limit cycles in this family of systems. The
centers at infinity classification together with the limit cycles bifurcating
from them are also analyzed.Comment: 24 pages, 5 figure
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