Dynamics of the polynomial differential systems with homogeneous nonlinearities and a star node

Abstract

We consider the class of polynomial differential equations x˙=λx+Pn(x,y), y˙=λy+Qn(x,y), in R2 where Pn(x,y) and Qn(x,y) are homogeneous polynomials of degree n>1 and λ≠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.