Computing center conditions for vector fields with constant angular speed

Abstract

We investigate the planar analytic systems which have a center-focus equilibrium at the origin and whose angular speed is constant. The conditions for the origin to be a center (in fact, an isochronous center) are obtained. Concretely, we find conditions for the existence of a Cw-commutator of the field. We cite several subfamilies of centers and obtain the centers of the cuartic polynomial systems and of the families (-y + x(H1 + Hm), x + y(H1 + Hm)1 and (-y + x(H2 + H2n), x + y(H2 + H2n))t, with H i homogeneous polynomial in x,y of degree i. In these cases, the maximum number of limit cycles which can bifurcate from a fine focus is determined.