On the limit cycles of a polynomial differential system with a homogeneous non-linearity of the third order

Background. The presence or absence of limit cycles of polynomial systems constitutes the second part of Hilbert's well-known 16$^{th}$ problem, which has not yet been completely solved. The purpose of this work is to study the existence of limit cycles surrounding the origin of a polynomial differential system with a linear node and a homogeneous third-order nonlinearity containing two parameters. Materials and methods. The latest research on limit cycles of polynomial systems is applied. Results. Areas on the parameter plane are found that correspond to the presence of limit cycles in the neighborhood of the origin. Conclusions. The application of new methods made it possible to reveal the conditions for the existence of limit cycles for the system under consideration.