A Complement to the Uniqueness of the Limit Cycle of a Family of Systems with Homogeneous Components

Abstract

Consider the number of limit cycles of a family of systems with homogeneous components: \( {\dot{x}}=y, {\dot{y}}=-x^3+\alpha x^2y+y^3. \) We show that there is an \(\alpha ^*<0\) such that the system has exactly one limit cycle for \(\alpha \in (\alpha ^*,0),\) while no limit cycle for the else region. This completes a previous result and also gives a positive answer to the second part of Gasull’s 3rd problem listed in the paper (SeMA J 78(3):233–269, 2021). To obtain this result, we mainly analyse the behavior of the heteroclinic separatrices at infinity.