Criteria on the existence of limit cycles in planar polynomial differential systems

Abstract

We summarize known criteria for the non-existence, existence and on the number of limit cycles of autonomous real planar polynomial differential systems, and also provide new results. We give examples of systems which realize the maximum number of limit cycles provided by each criterion. In particular we consider the class of differential systems of the form $\dot{x}=P_n(x,y)+P_m(x,y),\dot{y}=Q_n(x,y)+Q_m(x,y),$ where $n,m$ are natural numbers with $m>n\ge 1$ and $(P_i,Q_i)$ for $i=n,m$, are quasi-homogeneous vector fields.