Maximum number of limit cycles of a second order differential system

In this paper, we study the limit cycles of a perturbed differential in R ², given by {equation*} \{ {array}{ccl} {.}{x} y ,\\ {.}{y} -x-ε (1+sin^n (θ) cos^m (θ))H(x,y), {array} . {equation*} where ε is a small parameter, m and n are non-negative integers, tan(θ)=y/x, and $P(x,y)$ is a real polynomial of degree n ≥ 1. Using Averaging theory of first order we provide an upper bound for the maximum number of limit cycles. Also, we provide some examples to confirm and illustrate our results.