November 19, 2004 17:45 WSPC/Trim Size: 9in x 6in for Proceedings ws-gaiko03 GLOBAL ANALYSIS OF A CANONICAL CUBIC SYSTEM * V. A. GAIKO † Department of Mathematics, Belarusian State University of Informatics and Radioelectronics, L. Beda Street 6, Apt. 4, 220040 Minsk, Belarus E-mail: vlrgk@cit.org.by W. T. VAN HORSSEN Department of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands E-mail: W.T.vanHorssen@EWI.tudelft.nl We costruct a canonical Kukles system and carry out the global qualitative analysis of its special case corresponding to a generalized Li´ enard equation. 1. Introduction We consider a so-called Kukles system ˙ x = y, ˙ y = -x+δy +a 1 x 2 +a 2 xy +a 3 y 2 +a 4 x 3 +a 5 x 2 y +a 6 xy 2 +a 7 y 3 . (1) I. S. Kukles was the first who began to study (1) and tried to solve the center-focus problem for this system almost 60 years ago: he gave the necessary and sufficient conditions for O(0, 0) to be a center for (1) with a 7 = 0 [1]. Later, system (1) was studied by many mathematicians. For example, in [2] necessary and sufficient center conditions for an arbitrary system (1) when a 7 6= 0 were conjectured. In [3] this conjecture was proved, and in [4] the center conditions were simplified. There are also some global results for system (1). For example, in [5] global qualitative pictures and bifurcation diagrams of a reduced Kukles system (a 7 = 0) were given. In [6] the global analysis of a system (1) with two weak foci was carried out. In [7] the number of singular points under the * This work is supported by the Netherlands Organization for Scientific Research, NWO. † Present Address: Department of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands. E-mail: vlrgk@yahoo.com. 1071