Abstract:

We consider the class of polynomial differential equations
[x' = a x + P_n(x,y), y'= b  y + Q_n(x,y)] in $mathbb{R}^2$ where $P_n$ and $Q_n$ are homogeneous polynomials of degree $n>1$ and $a$ is different from $b,$ 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.