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 left{ begin{array}{l} dot{x}=y, dot{y}=-x-varepsilon (1+sin ^{m}(theta )) psi (x,y),% end{array}% right. where ε is small enough, m is a non-negative integer, tan(θ)=yx, 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.