Semnan University International Journal of Nonlinear Analysis and Applications 2008-6822 13 1 2022 03 01 On the maximum number of limit cycles of a planar differential system 1462 1478 5760 10.22075/ijnaa.2021.23049.2468 EN Sana Karfes Laboratory of Applied Mathematics, Badji Mokhtar-Annaba University, P.O. Box 12, 23000 Annaba, Algeria Elbahi Hadidi Laboratory of Applied Mathematics, Badji Mokhtar-Annaba University, P.O. Box 12, 23000 Annaba, Algeria Mohamed Amine Kerker Laboratory of Applied Mathematics, Badji Mokhtar-Annaba University, P.O. Box 12, 23000 Annaba, Algeria Journal Article 2021 04 04 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<br />\[\left\{<br />\begin{array}{l}<br />\dot{x}=y, \\<br />\dot{y}=-x-\varepsilon (1+\sin ^{m}(\theta ))\psi (x,y),%<br />\end{array}%<br />\right.\]<br />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 \(n\geq1\). 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. https://ijnaa.semnan.ac.ir/article_5760_8a9b3bf64826dfec536fc0f11d6a15d0.pdf