Integrability, degenerate centers, and limit cycles for a class of polynomial differential systems

AbstractWe consider the class of polynomial differential equations x˙ Pn(x,y)+Pn+1(x,y)+Pn+2(x,y), y˙=Qn(x,y)+Qn+1(x,y)+Qn+2(x,y), for n ≥ 1 and where Pi and Qi are homogeneous polynomials of degree i These systems have a linearly zero singular point at the origin if n > 2. Inside this class, we identify a new subclass of Darboux integrable systems, and some of them having a degenerate center, i.e., a center with linear part identically zero. Moreover, under additional conditions such Darboux integrable systems can have at most one limit cycle. We provide the explicit expression of this limit cycle

On a family of polynomial differential equations having at most three limit cycles

Agraïments/Ajudes: The first author is partially supported by by grants MTM2005-06098-C02-1. The second author is partially supported by the Spanish grant SAB-2005-0029, NSF of China (No.10571184) and SRF for ROCS, SEM. The second author also wants to express his thanks to the Departament de Matemàtiques of the Universitat Autònoma de Barcelona for the hospitality and support during the period in which this paper was started.We prove the existence of at most three limit cycles for a family of planar polynomial differential equations. Moreover we show that this upper bound in sharp. The key point in our approach is that the differential equations of this family can be transformed into Abel differential equations