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

Abstract

We consider the class of polynomial differential equations ẋ = λx + Pn(x, y), ẏ = μy + Qn(x, y) in ℝ2 where Pn(x, y) and Qn(x, y) are homogeneous polynomials of degree n > 1 and λ ≠ 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 nonexistence of limit cycles surrounding the node localized at the origin of coordinates.