Systems having liouvillian first integrals and Non-Algebraic limit cycles

Abstract

We consider the class of polynomial differential equations ẋ = Pm (x, y)+Pm+n (x, y), ẏ = Qm (x, y)+ Qm+n (x, y) for m, n ≥ 1 and where Pi and Qi are homogeneous polynomials of degree i. Inside this class, we identify a new subclass of Liouvillian integrable systems, under suitable conditions such Liouvillian integrable systems can have at most one limit cycle, and when it exists, is non-algebraic and hyperbolic. Then we study the general systems of the systems studied in [9], which allow us to find the necessary and suffi cient conditions for the existence and non-existence of limit cycles