Qual. Theory Dyn. Syst. 7 (2009), 369–403 c  2008 Birkh¨auser Verlag Basel/Switzerland 1575-5460/020369-35, published online 28 December 2008 DOI 10.1007/s12346-008-0021-2 Qualitative Theory of Dynamical Systems Phase Portraits of Reversible Linear Differential Systems with Cubic Homogeneous Polynomial Nonlinearities Having a Non-degenerate Center at the Origin Claudio A. Buzzi, Jaume Llibre and Jo˜ ao C. R. Medrado Abstract. In this paper we classify the global phase portraits of all reversible linear differential systems with cubic homogeneous polynomial nonlinearities defined in the plane and having a non degenerate center at the origin. The reversibility is given by a linear involution having a fixed set of dimension 1. Mathematics Subject Classification (2000). 34C05, 58F14. Keywords. Reversible vector fields, cubic vector fields, phase portrait. 1. Introduction and setting the problem We consider polynomial vector fields X : R 2 → R 2 given by X(x)= L(x)+ g(x), where L is its linear part and g is a homogeneous polynomial vector field of degree three satisfying the following two hypotheses: (H1) The vector field X is reversible with respect to a linear involution ϕ : R 2 → R 2 ; (i.e., ϕ ◦ X = −X ◦ ϕ) such that the dimension of the fixed point set of ϕ is one. (H2) The eigenvalues of L are ±αi with α = 0. In this paper we classify the global phase portraits of all these vector fields X. Cubic polynomial vector fields having a center have been investigated in- tensively, and many papers have been published about these systems, see for in- stance [3, 6, 12] and [13]. The first author is partially supported by a FAPESP–BRAZIL grant 2007/04307–2. The second author is partially supported by the grants MEC/FEPER MTM 2005–06098–C02-01 and CIRIT 2005SGR 00550. All authors are partially supported by the joint project CAPES–MECD grants 071/04 and HBP2003–0017.