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  (mathbb{R}^2), given as follows[left{begin{array}{l}dot{x}=y, \dot{y}=-x-varepsilon (1+sin ^{m}(theta ))psi (x,y),%end{array}%right.]where (varepsilon) is small enough, (m) is a non-negative integer, (tan (theta )=y/x), and (psi (x,y)) is a real polynomial of degree (ngeq1). 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.