Poincaré map for some polynomial systems of differential equations

One approach to the classical problem of distinguishing between a centre and a focus for a system of differential equations with polynomial right-hand sides in the plane is discussed. For a broad class of such systems necessary and sufficient conditions for a centre are expressed in terms of equations in variations of higher order. By contrast with the existing methods of investigation, attention is concentrated on the explicit calculation of the asymptotic behaviour of the Poincaré map rather than on finding sufficient centre conditions as such; this also enables one to study bifurcations of birth of arbitrarily strongly degenerate cycles.