Reversible global centres with quintic homogeneous nonlinearities

centre of a differential system in the plane R-2 is a singular point p having a neighbourhood U such that U \ {p} is filled of periodic orbits. A global centre is a centre p such that R-2 \ {p} is filled of periodic orbits. To determine if a given differential system has a centre is in general a difficult problem, but it is even harder to know if it has a global centre. In the present paper we deal with the class of polynomial differential systems of the form (x) over dot = -y + P(x, y), (y) over dot = x + Q(x,y), (1) where P and Q are homogeneous polynomials of degree n. It is known that these systems can have global centres only if n is odd and the global centres in the cases n = 1 and n = 3 are known. Here we work with the case n = 5 and we classify the global centres 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.