Explicit focal basis for some 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, 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.