Center conditions for a class of planar rigid polynomial differential systems

Abstract

El títol de la versió pre-print de l'article és: Explicit focal basis for some planar rigid polynomial differential systems

Agraïments: The first author was partially partially supported by CAPES/DGU grant number BEX 12566/12-8.

In general the center-focus problem cannot be solved, but in the case that the singularity has purely imaginary eigenvalues there are algorithms to solving it. The present paper implements one of these algorithms for the polynomial differential systems of the form x = -y + xf (x)g(y), y = x + yf (x)g(y), where f (x) and g(y) are arbitrary polynomials. These differential systems have constant angular speed and are also called rigid systems. More precisely, this paper gives focal bases of these systems, and then necessary and sufficient conditions in order to have an uniform isochronous center. In particular, the existence of a focus with the highest order is also studied.