Semnan UniversityInternational Journal of Nonlinear Analysis and Applications2008-682213120220301On the maximum number of limit cycles of a planar differential system14621478576010.22075/ijnaa.2021.23049.2468ENSanaKarfesLaboratory of Applied Mathematics, Badji Mokhtar-Annaba University, P.O. Box 12, 23000 Annaba, AlgeriaElbahiHadidiLaboratory of Applied Mathematics, Badji Mokhtar-Annaba University, P.O. Box 12, 23000 Annaba, AlgeriaMohamed AmineKerkerLaboratory of Applied Mathematics, Badji Mokhtar-Annaba University, P.O. Box 12, 23000 Annaba, Algeria0000-0003-2215-533XJournal Article20210404In 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