A complement to the uniqueness of the limit cycle of a family of systems with homogeneous components (Q6554400)

The purpose of this work is to derive the number of limit cycles of the following system \N\[\N\frac{dx}{dt} = y, \quad \frac{dy}{dt} = - x^3+\alpha x^2y+y^3\N\]\Nfor all real values of the parameter \(\alpha\). The authors prove the existence of a value \(\alpha^*<0\) such that the system has a unique limit cycle for \(\alpha \in (\alpha^*,0)\) while it has no limit cycles for all other real values of \(\alpha\). It completes previous results devoted to the mentioned problem. The limit cycle appears from a Hopf bifurcation and finally disappears in a heteroclinic loop at infinity. To obtain this result, the behavior of the heteroclinic separatrices at infinity is analysed.