Reversible global centres with quintic homogeneous nonlinearities

A center of a differential system in the plane R2 is an equilibrium point p having a neighbourhood U such that U \ {p} is filled of periodic orbits. A global center is a center p such that R2 \ {p} is filled of periodic orbits. To determine when a given differential system has a center is in general a difficult problem, but to determine if a given differential system has a global center is even more difficult. We deal with the class of polynomial differential systems of the form (1) ˙x = −y + P(x, y), y˙ = x + Q(x, y), with P and Q homogeneous polynomials of degree n. It is known that these systems only can have global centers if n is odd. The global centers when n is 1 or 3 have been characterized. Here for n = 5 we classify the global centers of a four parameter family of systems (1). In particular we illustrate how to study the local phase portraits of the singular points whose linear part is identically zero using only vertical blow ups.