Dra. García-Saldaña, Johanna

One of the classical and difficult problems in the qualitative theory of differential systems in the plane is the characterization of their centers. In this paper we characterize the linear and nilpotent global centers of polynomial differential systems with quintic homogeneous terms, with the symmetry (x,y,t) → (−x,y,−t) and without infinite singular points.

The notion of center goes back to Poincar´e and Dulac, see [10, 6]. They defined a center for a vector field on the real plane as a singular point having a neighborhood filled of periodic orbits with the exception of the singular point. The problem of distinguishing when a monodromic singular point is a focus or a center, known as the focus-center problem started precisely with Poincar´e and Dulac and is still active nowadays with many questions still unsolved. These last years also the centers are perturbed for studying the limit cycles bifurcating from their periodic solutions, see for instance. We recall that a global center for a vector field on the plane is a singular point p having R 2 filled of periodic orbits with the exception of the singular point. The easiest global center is the linear center ˙x = −y, ˙y = x. It is known (see [11, 2]) that quadratic polynomial differential systems have no global centers. The global degenerated centers of homogeneous or quasihomogeneous polynomial differential systems were characterized in [4] and [8], respectively. However the characterization of the global centers in the cases that the center is nilpotent or of linear-type has not been done. This is the first paper in which such classification is done for the linear-type centers for the systems having a linear part at the origin with purely imaginary eigenvalues and cubic homogeneous nonlinearities.

In this paper, we characterize the global nilpotent centers of polynomial differential systems of the linear form plus cubic homogeneous terms.