Center conditions for a class of planar rigid polynomial differential systems

Abstract

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 \[ \dot x= -y + x f(x) g(y),\quad \dot y= x+y f(x) g(y), \] where $f(x)$ and $g(y)$ are arbitrary polynomials. These differential systems have constant angular speed and are also called rigid systems. More precisely, in this paper we give the center conditions for these systems, i.e. the necessary and sufficient conditions in order that they have an uniform isochronous center. In particular, the existence of a focus with the highest order is also studied.