Bifurcation of limit cycles by perturbing a periodic annulus with multiple critical points

In this paper, we consider the planar system ẋ = -yF(x, y) + eP(x, y), ẏ = xF(x, y) + eQ(x, y), where the set {F(x, y) = 0} consists of m nonzero points (ai, bi)(i = 1, …, m) with multiple multiplicities, P(x, y) and Q(x, y) are arbitrary real polynomials. We study the number of limit cycles bifurcating from the periodic annulus surrounding the origin by using Abelian integrals and residue integration.