The number of limit cycles in perturbations of polynomial systems with multiple circles of critical points

This paper investigates the problem for limit cycle bifurcations of system x˙=yF(x,y)+εp(x,y), y˙=−xF(x,y)+εq(x,y), where F(x,y)F(x,y) consists of multiple circles and p(x,y),q(x,y)p(x,y),q(x,y) are polynomials of degree n. The upper bound for the maximal number of limit cycles emerging from the period annulus surrounding the origin is provided in terms of n and the involved multiplicities of circles by using the first order Melnikov function. Furthermore, Hopf bifurcation for a cubic system of this type is discussed.