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

Abstract

We consider the class of polynomial differential equations x˙ P n ( x,y)+ P n+1 ( x,y)+ P n+2 ( x,y), y˙= Q n ( x,y)+ Q n+1 ( x,y)+ Q n+2 ( x,y), for n ≥ 1 and where P i and Q i 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.