on the maximum number of limit cycles of a planar differential system

In this work, we are interested in the study of the limit cycles of a perturbed differential system in r², given as follows ẋ = y, ẏ = −x − ε(1 + sinm(θ))ψ(x, y), where ε is small enough, m is a non-negative integer, tan(θ) = y/x, and ψ(x, y) is a real polynomial of degree n ≥ 1. we use the averaging theory of first order to provide an upper bound for the maximum number of limit cycles. in the end, we present some numerical examples to illustrate the theoretical results.