Global Phase portraits of some Reversible cubic Centers with Collinear or Infinitely Many Singularities

The center problem for polynomial differential equations in the plane is one of the celebrated and longstanding problems in the qualitative theory of planar differential equations (see e.g. the works [Dulac, 1908], [Kapteyn, 1911], [Kapteyn, 1912], [Bautin, 1954], [Vulpe & Sibirskĭı, 1988], [Schlomiuk, 1993], [Żo la̧dek, 1994a] and [Żo la̧dek, 1994b]). It asks to distinguish a linear center between a focus and a center of a polynomial vector field of a given degree. It is solved completely only for quadratic vector fields; for cubic vector fields it is an open problem. This problem is closely related to another celebrated problem from the beginning of the twentieth century that is still one of the challenges for the twenty first century known as Hilbert’s sixteenth problem, which asks essentially for the maximum number H(n) of limit cycles of a polynomial vector field only depending on its degree n (see e.g. [Hilbert, 1902], [Smale, 1998]). Until now even H(2) is not known. Limit cycles, i.e. isolated periodic orbits, can be found for instance by perturbing a center. The mechanism of simultaneous perturbation of different centers is used to construct concrete polynomial vector fields having a certain configuration of limit cycles. As such lower bounds for the socalled Hilbert number H(n) are deduced. For instance in [Shi, 1980] Shi Songling used this mechanism to find that H(2) ≥ 4; in particular to find the fourth limit cycle for a concrete quadratic vector field an unbounded period annulus (‘center at infinity’) was involved in the simultaneous bifurcation of quadratic centers. In [Blows & Rousseau, 1993] and [Caubergh et al., 2011a] the problem of characterizing a center at the origin and simultaneously at infinity is solved for the following subfamily of the cubic differential equations