Darboux integrability and algebraic limit cycles for a class of polynomial differential systems

Abstract

This paper deals with the existence of Darboux first integrals for the planar polynomial differential systems $$\mathop x\limits^. $$ = λ x−y +P n+1(x, y)+xF 2n (x, y), $$\mathop y\limits^. $$ = x+λ y +Q n+1(x, y)+yF 2n (x, y), where P i (x, y), Q i (x, y) and F i (x, y) are homogeneous polynomials of degree i. Within this class, we identify some new Darboux integrable systems having either a focus or a center at the origin. For such Darboux integrable systems having degrees 5 and 9 we give the explicit expressions of their algebraic limit cycles. For the systems having degrees 3, 5, 7 and 9 and restricted to a certain subclass we present necessary and sufficient conditions for being Darboux integrable.