NEW FAMILY OF CUBIC HAMILTONIAN CENTERS MART ´ IN-EDUARDO FR ´ IAS-ARMENTA 1 AND JAUME LLIBRE 2 Abstract. We characterize the 11 non topological equivalent classes of phase portraits in the Poincar´ e disc of the new family of cubic poly- nomial Hamiltonian differential systems with a center at the origin and Hamiltonian H = 1 2 ( (x + ax 2 + bxy + cy 2 ) 2 + y 2 ) , with a 2 + b 2 + c 2 = 0. 1. Introduction For a given family of real planar polynomial differential systems depending on parameters one of the main problems is the characterization of their centers and their phase portraits. The notion of center goes back to Poincar´ e in [16]. He defined it for differential systems on the real plane; i.e. a singular point surrounded by a neighborhood fulfilled of closed orbits with the unique exception of the singular point. The classification of the centers of the real polynomial differential sys- tems started with the quadratic ones with the works of Kapteyn [10, 11], Bautin [3], Vulpe [21], Schlomiuk [18, 19], ˙ Zo l¸ adek in [23], ... Schlomiuk, Guckenheimer and Rand in [20] described a brief history of the problem of the center in general, and it includes a list of 30 papers covering the history of the center for the quadratic polynomial differential systems (see pages 3, 4 and 13). While the centers and all their phase portraits have been characterized for all the quadratic polynomial differential systems, this is not the case for the polynomial differential systems of degree larger than 2, but for such systems there are many partial results. Thus the centers for cubic polynomial differ- ential systems of the form linear with homogeneous nonlinearities of degree 3 were classified by Malkin [12], and Vulpe and Sibirskii [22], and their phase portraits when they are Hamiltonian have been classified by Colak, Llibre and Valls in [6, 7]. Moreover for polynomial differential systems which are linear with homogeneous nonlinearities of degree k> 3 the centers are not classified, but there are partial results for k =4, 5 see Chavarriga and Gin´ e [4, 5], respectively. 2010 Mathematics Subject Classification. 34C05. Key words and phrases. isochronous center, Hamiltonian system. 1 This is a preprint of: “New family of cubic Hamiltonian centers”, Mart´ ın-Eduardo Fr´ ıas-Armenta, Jaume Llibre, Bol. Soc. Mat. Mexicana, 2016. DOI: [10.1007-s40590-016-0126-6]