Limit cycles bifurcating from a perturbed quartic center

We consider the quartic center x˙=-yf(x,y),y˙=xf(x,y), with f(x, y) = (x + a) (y + b) (x + c) and abc ≠ 0. Here we study the maximum number σ of limit cycles which can bifurcate from the periodic orbits of this quartic center when we perturb it inside the class of polynomial vector fields of degree n, using the averaging theory of first order. We prove that 4[(n − 1)/2] + 4 ⩽ σ ⩽ 5[(n − 1)/2] + 14, where [η] denotes the integer part function of η.

► We study polynomial perturbations of a quartic center.
► We get simultaneous upper and lower bounds for the bifurcating limit cycles.
► A higher lower bound for the maximum number of limit cycles is obtained.
► We obtain more limit cycles than the number obtained in the cubic case.