Poincaré Bifurcations of Two Classes of Polynomial Systems

Abstract

Using bifurcation methods and the Abelian integral, we investigate the number of the limit cycles that bifurcate from the period annulus of the singular point when we perturb the planar ordinary differential equations of the form $\dot{x}=-yC(x,y)$, $\dot{y}=xC(x,y)$ with an arbitrary polynomial vector field, where $C(x,y)=1-x^3$ or $C(x,y)=1-x^4$.