@article{10272/25439,
year = {2009},
month = {9},
url = {https://hdl.handle.net/10272/25439},
abstract = {We study the analytic system of differential equations in the plane
which can be written, in a suitable coordinates system, as
$$(\dot{x},\dot{y})^T=\sum_{i=0}^{\infty} \F_{q-p+2is},$$
where $p,q\in \mathbb{N}, p\le q,\ \ s=(n+1)p-q>0, \ n\in
\mathbb{N}$ and $\F_{i}=(P_i,Q_i)^T$ are quasi-homogeneous vector
fields of type ${\t}=(p,q)$ and degree $i$, with
$\F_{q-p}=(y,0)^T$ and $Q_{q-p+2s}(1,0)<0.$ The origin of this
system is a nilpotent and monodromic isolated singular point. We
show the Taylor expansion of the return map near the origin for
this system, which allow us to generate small amplitude limit
cycles bifurcating from the critical point.
 Also, as an application of the theoretical
procedure, we characterize the centers and we generate limit cycles
of small amplitude from the origin of several families. Finally, we
give a new family integrable analytically  which includes the
centers of the systems studied.},
organization = {This work has been partially supported by  Ministerio de
Ciencia y Tecnología, Plan Nacional I+D+I co-financed with
FEDER funds, in the frame of the project MTM2007-64193 and by 
Consejería de Educación y Ciencia de la Junta de
Andalucía (FQM-276 and EXC/2008)},
publisher = {Elsevier},
keywords = {Ecuaciones diferenciales},
title = {Local bifurcation of limit cycles and integrability of a class of nilpotent systems of differential equations},
doi = {10.1016/j.amc.2009.04.077},
author = {Algaba Durán, Antonio and García García, Cristóbal and Reyes Columé, Manuel},
}