Quartic rigid systems in the plane and in the Poincaré sphere (Q6572425)

The article considers the planar family of rigid systems of the form \N\[\N\dot{x}=-y+xP(x,y), \quad \dot{y}=x+yP(x,y),\N\] \Nwhere \(P\) is any polynomial with monomials of the first and third degree. The authors use the Poincaré-Lyapunov compactification [\textit{F. Dumortier} et al., Qualitative theory of planar differential systems. Berlin: Springer (2006; Zbl 1110.34002)]. It transforms a planar system into two copies of a vector field defined on a sphere called the Poincaré-Lyapunov sphere. The main difference between the Poincaré-Lyapunov sphere and the Poincaré disk is that the trajectories of the system on the Poincaré-Lyapunov sphere can intersect the equator. That is, the equator contains information about the dynamics of the system at infinity. Therefore, the Poincaré-Lyapunov sphere allows us to detect properties of the system that are not visible either in the finite part of the plane or on the Poincaré disk. For example, homoclinic or heteroclinic connections or limit cycles in the neighborhood of infinity. It is proved that the system can have a limit cycle on the sphere that intersects the equator at two symmetric points. An example of a system with one limit cycle in the finite part of the plane and three limit cycles on the Poincaré-Lyapunov sphere is given. The authors pose seven open questions concerning the limit cycles of the system, both in the finite part of the plane and on the Poincaré-Lyapunov sphere. One more question seems interesting. Those parts of the limit cycle crossing the equator that are on one of the Poincaré disks should also be called isolated. Can this be detected if limited to just the disk?