On the Limit Cycles of the Polynomial Differential Systems with a Linear Node and Homogeneous Nonlinearities

We consider the class of polynomial differential equations ẋ = λx + Pn(x, y), ẏ = μy + Qn(x, y) in R where Pn(x, y) and Qn(x, y) are homogeneous polynomials of degree n > 1 and λ 6= μ, i.e. the class of polynomial differential systems with a linear node with different eigenvalues and homogeneous nonlinearities. For this class of polynomial differential equations we study the existence and non–existence of limit cycles surrounding the node localized at the origin of coordinates.