Limit cycles of a second-order differential equation

We provide an upper bound for the maximum number of limit cycles bifurcating from the periodic solutions of ẍ+x=0, when we perturb this system as follows ẍ+ε(1+cosmθ)Q(x,y)+x=0,where ε>0 is a small parameter, m is an arbitrary non-negative integer, Q(x,y) is a polynomial of degree n and θ=arctan(y∕x). The main tool used for proving our results is the averaging theory.