RT Journal Article
T1 Local bifurcation of limit cycles and integrability of a class of nilpotent systems of differential equations
A1 Algaba Durán, Antonio
A1 García García, Cristóbal
A1 Reyes Columé, Manuel
K1 Ecuaciones diferenciales
AB We study the analytic system of differential equations in the planewhich 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 vectorfields 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 thissystem is a nilpotent and monodromic isolated singular point. Weshow the Taylor expansion of the return map near the origin forthis system, which allow us to generate small amplitude limitcycles bifurcating from the critical point. Also, as an application of the theoreticalprocedure, we characterize the centers and we generate limit cyclesof small amplitude from the origin of several families. Finally, wegive a new family integrable analytically  which includes thecenters of the systems studied.
PB Elsevier
SN 1873-5649 (electrónico)
SN 0096-3003
YR 2009
FD 2009-09
LK https://hdl.handle.net/10272/25439
UL https://hdl.handle.net/10272/25439
LA eng
NO Algaba, A., García, C., & Reyes, M. (2009). Local bifurcation of limit cycles and integrability of a class of nilpotent systems of differential equations. In Applied Mathematics and Computation (Vol. 215, Issue 1, pp. 314–323). Elsevier BV. https://doi.org/10.1016/j.amc.2009.04.077
NO This work has been partially supported by  Ministerio deCiencia y Tecnología, Plan Nacional I+D+I co-financed withFEDER funds, in the frame of the project MTM2007-64193 and by Consejería de Educación y Ciencia de la Junta deAndalucía (FQM-276 and EXC/2008)
DS Repositorio Institucional de la Universidad de Huelva
RD 13 jun 2026