Dynamic Systems and Applications 17 (2008) 625-636 LIMIT CYCLES COMING FROM THE PERTURBATION OF 2-DIMENSIONAL CENTERS OF VECTOR FIELDS IN R 3 JAUME LLIBRE, JIANG YU, AND XIANG ZHANG Departament de Matem` atiques, Universitat Aut` onoma de Barcelona, 08193 Bellaterra, Barcelona, Spain jllibre@mat.uab.cat Department of Mathematics, Shanghai Jiaotong University Shanghai, 200240, P. R. China jiangyu@sjtu.edu.cn and xzhang@sjtu.edu.cn ABSTRACT. In this paper we study the limit cycles of polynomial vector fields in R 3 which bifurcates from three different kinds of two dimensional centers (non-degenerate and degenerate). The study is down using the averaging theory. AMS (MOS) Subject Classification. 37G15, 37D45 1. INTRODUCTIONS AND STATEMENT OF THE MAIN RESULTS One of the main problems in the theory of differential systems is the study of their periodic orbits, their existence, their number and their stability. In this paper the study of the existence of periodic orbits of a differential system is reduced using the averaging theory to study the zeroes of a system of functions. One of the main problems for applying the averaging theory is to transform the differential system that we want to study into the normal form for applying the averaging method. When this method cannot be applied, sometimes there are other ways to reduce the problem of studying the existence of periodic orbits to study the zeroes of a system of functions. In general these methods are called alternative methods (see for instance Section 2.4 of Chow and Hale [4]), one of these particular alternative methods is the well known Liapunov–Schmidt method. As usual a limit cycle of a differential equation is a periodic orbit isolated in the set of all periodic orbits of the differential equation. In this paper we shall study the limit cycles which bifurcate from the periodic orbits of three kinds of different 2–dimensional centers contained in a differential system of R 3 when we perturb it. These kinds of bifurcations have been studied extensively for 2–dimensional systems (see for instance the book [6] and the references quoted there), but for 3–dimensional systems there are very few results, see for instance [1, 2, 7, 8]. Received October 16, 2007 1056-2176 $15.00 c Dynamic Publishers, Inc.