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

Consider the number of limit cycles of a family of systems with homogeneous components: \( {\dot{x}}=y, {\dot{y}}=-x^3+\alpha x^2y+y^3. \) $\dot{x}=y,\dot{y}=-x^3+\alpha x^{2}y+y^3.$ We show that there is an \(\alpha ^*<0\) ${\alpha }^{\ast }<0$ such that the system has exactly one limit cycle for \(\alpha \in (\alpha ^*,0),\) $\alpha \in ({\alpha }^{\ast },0),$ 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.