130 YEARS OF THE EFFORT IN THE SOLVING OF THE POINCARÉ’S CENTER-FOCUS PROBLEM

It is well known that many mathematical models use differential equation systems and apply the qualitative theory of differential equations, introduced by Poincaré and Liapunov. One of the problems that persists in order to control the behavior of systems of this type, is to distinguish between a focus or a center (the center-focus problem). The solving of this problem goes through the computation of the Poincaré-Liapunov constants. In the case of polynomial right-hand sides it follows from Hilbert’s theorem on the fi niteness of bases of polynomial ideals that in this sequence only fi nitely many are essential and that the remaining ones are consequences of them. Hence, this problem is divided in two parts: in the fi rst, to estimate the number of essential constants; in the second, to determine the minimal upper border of the indexes of a complete system of essential constants. The fi rst part is called the weak center-focus problem. The problem of exitimation the maximal number of algebraically independent essential constants is called the generalized center-focus problem. Recently M. N.Popa and V. V. Pricop have solved the generalized center-focus problem. The present article contains: some moments related to the history of the center-focus problem; the contribution of the Sibirschi’s school in the solving of the center-focus problem; methodological aspects of the Popa – Pricop solution of the generalized center-focus problem. The problem of the estimation of the minimal upper border of the indexes of a complete system of algebraically independent essential constants is open. Another open problem consists on determining what differential systems are integrable.