Limit cycles for cubic systems with a symmetry of order 4 and without infinite critical points

In this paper we study those cubic systems which are invariant under a rotation of $2\pi/4$ radians. They are written as $\dot{z}=\varepsilon z+p\,z^2\bar{z}-\bar{z}^3,$ where $z$ is complex, the time is real, and $\varepsilon=\varepsilon_1+i\varepsilon_2$, $p=p_1+ip_2$ are complex parameters. When they have some critical points at infinity, i.e. $\vert p_2\vert\le 1$, it is well-known that they can have at most one (hyperbolic) limit cycle which surrounds the origin. On the other hand when they have no critical points at infinity, i.e. $\vert p_2\vert>1,$ there are examples exhibiting at least two limit cycles surrounding nine critical points. In this paper we give two criteria for proving in some cases uniqueness and hyperbolicity of the limit cycle that surrounds the origin. Our results apply to systems having a limit cycle that surrounds either 1, 5 or 9 critical points, the origin being one of these points. The key point of our approach is the use of Abel equations.