Abstract:

We consider the class of polynomial differential equations $dot{x}=lambda x+P_n(x,y),dot{y}=lambda y+Q_n(x,y)$ in $mathbb{R}^2$ where $P_n(x,y)$ and $Q_n(x,y)$ are homogeneous polynomials of degree $n>1$ and $lambda eq 0$, i.e. the class of polynomial differential systems with homogeneous nonlinearities with a star node at the origin. We prove that these systems are Darboux integrable. Moreover, for these systems we study the existence and non-existence of limit cycles surrounding the equilibrium point located at the origin.