Limit cycles appearing from the perturbation of a system with a multiple line of critical points
Consider the family of planar systems \[ \dot{x}=-yC(x,y)+\epsilon P(x,y);~\,\dot{y}=xC(x,y)+\epsilon Q(x,y) \] where \(P,Q\) and \(C\) are real polynomials, \(C(0,0) \neq 0\), and \(\epsilon\) is a small real parameter. The number of zeros of the abelian integral \(M(r)\) on \(x^{2}+y^{2}=r^{2}\) controls the number of limit cycles that bifurcate from the periodic orbits of the unperturbed system with \(\epsilon =0\). The authors consider the case \(C(x,y)=(1-y)^{m}\) and \(P, Q\) are of degree \(n\). They prove that \(M(r)=[\frac{m+n}{2}]-1\) when \(n<m-1\) and \(n\) when \(n \geq m-1\).
scientific article; zbMATH DE number 5967364